Physics of Emergence and Organization This Page Intentionally Left Blank Physics of Emergence and Organization

Physics of Emergence and Organization This page intentionally left blank Physics of Emergence and Organization

editors Ignazio Licata Institute for Scientific Methodology, Palermo, Italy Ammar Sakaji Ajman University, Abu Dhabi

World Scientific

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ISBN-13 978-981-277-994-6 ISBN-10 981-277-994-9

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Lakshmi - Physics of Emergence.pmd 1 7/22/2008, 10:01 AM v

Foreword

Gregory J. Chaitin IBM Research, P O Box 218, Yorktown Heights, NY 10598, USA [email protected] http://www.umcs.maine.edu

When I was a child the most exciting field of science was fundamental physics: relativity theory, quantum mechanics and cosmology. As we all know, fundamental physics is now in the doldrums. The most exciting field of science is currently molecular biology. We are being flooded with fasci- nating data (for example, in comparative genomics), and the opportunities for applications seem limitless. A mathematician or physicist looking with admiration at the exciting developments in molecular biology might be forgiven for wondering how theory is faring in this flood of data and medical applications. In a way, molecular biology feels more like software engineering than a fundamental science. Nevertheless there are fundamental questions in biology. Here are some examples: a) Is life pervasive in the universe, or are we unique? b) What are consciousness and thought, and how widespread are they? c) Can human intelligence be greatly amplified through genetic engineering?

These are extremely diﬃcult questions, but as the articles in this special issue attest, the human being is a theory-building as well as a tool-using animal. The desire for fundamental understanding cannot be suppressed. Someday we will have a theoretical understanding of fundamental biological concepts. In order to do that we will probably have to drastically change mathematics and physics. This is already happening, as the notions of complexity, computation and information develop and spread. vi Foreword

What the mathematics and physics of complex systems may ultimately be like, nobody can say, but as the articles in this issue show, we are starting to get some interesting glimpses. The only thing I can safely predict is that the future is unpredictable. There will no doubt be a lot of surprises.

Gregory Chaitin vii

Preface

It was Philip W. Anderson in his famous 1972 paper More is Different who first threw doubts upon the reductionist approach as the fundamental methodology in Physics. Since then, the problem to build a Theory of Emer- gence has grown more and more as a matter of great importance both in Physics and in a broad interdisciplinary area. If, initially, emergence could be framed within a na¨ıve, “objective” world scheme and thus easily seized by mere formal and computational models, today widespread, epistemological awareness has rooted the idea that the most genuine and radical features of emergence cannot be separated from the observer’s choices. In this way, the fundamental lesson of Quantum Physics extends to the whole theoretical field transversally crossing old and new subjects. A general theory of the observer/observed relationships thus represents the ideal framework within which the connections among different areas can be discussed. Such kind of theory can actually be regarded as an authentic “Theory of Everything” in systemic sense and can go side by side with the more traditional unified theories of particles and forces. From the Physics viewpoint, the matter implies many fundamental questions connected to the different ways in which classical and quantum systems exhibit emergence. The formation and evolution of structures find their natural, conceptual context in the theories of critical phenomena and collective behaviors. It is getting clear that information takes on different connotations depending on whether we consider a phase transition from the classical or quantum viewpoint, as well as that the connection between Physics and computation finds its deepest significance in regarding physical systems as systems performing effective computation. This is the reason why the recent researches on Quantum Computing are just one eighth of an iceberg which will lead not only to new technological perspectives, but — above all — to a different comprehension of the traditional questions about the foundations of Physics. Another relevant problem concerns the relationships among the different description levels of a system in that complex and viii Preface not completely colonized middle land represented by the mesoscopic realm, where Physics meets the other research fields. Among such questions, one emerges as the most drastic: whether to extend the syntax of both Quantum Theory and Quantum Field Theory in order to build a general theory of the interaction between the observer and the external world, or whether such action will rather lead to include Quantum Theory itself as a particular case within a more general epistemological perspective. This project has been conceived within the yearly tradition of the Special Issues of Electronic Journal of Theoretical Physics. We intended to give birth to an open space hosting quite different conceptions so as to offer the researchers a significant state of the art scenario in such exciting topics. Such a project would have never turned into a book without the fruitful discussions with my friend Ammar J. Sakaji, who shared both the conceptual choices and hard work. I would also like to thank the authors who perfectly grasped the spirit of our project and have presented precious contributions — our debate will not stop! I extend my thanks to the EJTP Editorial Board, and Teresa Iaria whose artwork on science and art relationships has provided the cover image. Finally, a warm acknowledgement to all my friends and colleagues who — during these years — have developed these topics with me . . . I am glad I can say that all of them are — directly or indirectly — present in this volume.

Ignazio Licata November 2007 ix

Contents

Foreword v Gregory J. Chaitin

Preface vii Ignazio Licata

Emergence and Computation at the Edge of Classical and Quantum Systems 1 Ignazio Licata

Gauge Generalized Principle for Complex Systems 27 Germano Resconi

Undoing Quantum Measurement: Novel Twists to the Physical Account of Time 61 Avshalom C. Elitzur and Shahar Dolev

Process Physics: Quantum Theories as Models of Complexity 77 Kirsty Kitto

A Cross-disciplinary Framework for the Description of Contextually Mediated Change 109 Liane Gabora and Diederik Aerts

Quantum-like Probabilistic Models Outside Physics 135 Andrei Khrennikov

Phase Transitions in Biological Matter 165 Eliano Pessa

Microcosm to Macrocosm via the Notion of a Sheaf (Observers in Terms of t-topos) 229 Goro Kato x Contents

The Dissipative Quantum Model of Brain and Laboratory Observations 233 Walter J. Freeman and Giuseppe Vitiello

Supersymmetric Methods in the Traveling Variable: Inside Neurons and at the Brain Scale 253 H.C. Rosu, O. Cornejo-P´erez, and J.E. P´erez-Terrazas

Turing Systems: A General Model for Complex Patterns in Nature 267 R.A. Barrio

Primordial Evolution in the Finitary Process Soup 297 Olof G¨ornerup and James P. Crutchﬁeld

Emergence of Universe from a Quantum Network 313 Paola A. Zizzi

Occam’s Razor Revisited: Simplicity vs. Complexity in Biology 327 Joseph P. Zbilut

Order in the Nothing: Autopoiesis and the Organizational Characterization of the Living 339 Leonardo Bich and Luisa Damiano

Anticipation in Biological and Cognitive Systems: The Need for a Physical Theory of Biological Organization 371 Graziano Terenzi

How Uncertain is Uncertainty? 389 Tibor Vamos

Archetypes, Causal Description and Creativity in Natural World 405 Leonardo Chiatti 1

Emergence and Computation at the Edge of Classical and Quantum Systems

Ignazio Licata ISEM, Institute for Scientiﬁc Methodology, Palermo, Italy [email protected]

The problem of emergence in physical theories makes necessary to build a general theory of the relationships between the observed system and the observing system. It can be shown that there exists a correspondence between classical systems and computational dynamics according to the Shannon–Turing model. A classical system is an informational closed system with respect to the observer; this characterizes the emergent processes in classical physics as phenomenological emergence. In quantum systems, the analysis based on the computation theory fails. It is here shown that a quantum system is an informational open system with respect to the observer and able to exhibit processes of observational, radical emergence. Finally, we take into consideration the role of computation in describing the physical world.

Keywords: Intrinsic Computation; Phenomenological and Radical Emergence; Informational Closeness and Openness; Shannon–Turing Computation; Bohm– Hiley Active Information PACS(2006): 03.50.z; 03.67.a; 05.45.a; 05.65.+b; 45.70.n; 45.05.+x; 45.10.b; 03.65.w; 03.67.a; 03.70.+k; 89.75.k; 89.75.Fb

1. Introduction The study of the complex behaviors in systems is one of the central problems in Theoretical Physics. Being related to the peculiarities of the system under examination, the notion of complexity is not univocal and largely interdisciplinary, and this accounts for the great deal of possible approaches. But there is a deeper epistemological reason which justifies such intricate “archipelago of complexity”: the importance of the observer’s role in detecting complexity, that is to say those situations where the system’s collective behaviors give birth to structural modifications and hierarchical arrangements. This consideration directly leads to the core of the emergence question in Physics. We generally speak of emergence when we observe a “gap” between the formal model of a system and its behaviors. In other words, 2 Ignazio Licata the detecting of emergence expresses the necessity or, at least, the utility for the creation of a new model able to seize the new observational ranges. So the problem of the relationship among different description levels is put and two possible situations arise: 1) phenomenological emergence, where the observer operates a “semantic” intervention according to the system’s new behaviors, and aiming at creating a new model—choosing the state variables and dynamical description—which makes the description of the observed processes more convenient. In this case the two descriptive levels can be always—in principle—connected by opportune “bridge laws”, which carry out such task by means of a finite quantity of syntactic information; 2) radical emergence, where the new description cannot be connected to the initial model. Here we usually observe a breaking of the causal chain (commonly describable through opportune symmetries), and irreducible forms of unpredictability. Hence, the link between the theoretical corpus and the new model could require a different kind of semantics of the theory, such as a new interpretation and a new arrangement of the basic propositions and their relationships. Such two distinctions have to be considered as a mere exemplification, actually more varied and subtler intermediate cases can occur. The relationships between Newtonian Dynamics and the concept of entropy can be taken into consideration as an example of phenomenological emergence. The laws of Classical Dynamics are time-reversal, whereas entropy defines a “time arrow”. In order to connect the two levels, we need a new model based on Maxwell–Boltzmann statistics as well as on a refined probabilistic hypothesis centered, in turn, on space-time symmetries—because of the space-time isotropy and homogeneity there do not exist points, directions or privileged instants in a de-correlation process between energetic levels. So, a “conceptual bridge” can be built between the particle description and entropy, and consequently between the microscopic and macroscopic analysis of the system. But this connection does not cover all the facets of the problem, and thus we cannot regard it as a “reduction” at all. In fact in some cases, even within the closed formulation of classical physics, entropy can decrease locally, and after all the idea to describe a perfect gas in molecular terms would never cross anybody’s mind! Another example regards the EPR-Bell correlations and the non-locality role in Quantum Mechanics. Within the Copenhagen Interpretation the non-local correlations are experimentally observed but they are not considered as facts of the theory. In the Bohm Interpretation the introduction of the quantum potential makes possible to bring non-locality within the Emergence and Computation at the Edge of Classical and Quantum Systems 3 theory. It should not be forgotten that historically the EPR question comes out as a gedanken experiment between Einstein and Bohr about the “elements of physical reality” in Quantum Mechanics. Only later, thanks to Bohm analysis and Bell’s Inequality on the limits of local hidden variable theories, such question developed into an experimental matter. Nor Ein- stein neither Bohr would expect to observe really the “ghost-like-action-at- a-distance”. It is useful to remember that in Bohm theory the introduction of non-locality does not require any additional formal hypotheses but the standard apparatus provided by the Schrödingerequation. Besides, if on one hand the new interpretative perspective provides a different comprehension of the theory, on the other hand it puts some problems about the so-called “pacific coexistence” between Restricted Relativity and Quantum Mechanics. In both of the briefly above-examined cases we can see how the phenomenological and radical features of emergence are strongly intertwined with the development dynamics of physical theories and how the general problem of emergence points up questions of fundamental importance for the physical world description, such as the updating mechanism of the theories and the crucial role of the observer in choosing models and their interpretations. In particular, it is worth noticing that the relationship between the observer and the observed is never a merely “one-way” relationship and it is unfit to be solved in a single direction, which would lead to epistemological impoverishment. This relationship has rather to be considered as an adaptive process in which the system’s internal logic meets our modalities to acquire information about it in order to build theories and interpretations able to shape up a system’s description. The problems related to the emergence theory, conceived as a general theory of the relationships between observing system and observed system, will be here taken into consideration, and will be tested on some evolution models of both classical and quantum systems. Finally, we will develop some considerations about the logic limits of the theories and the computability role in describing the physical world. 4 Ignazio Licata

2. The Observers in Classical Physics: Continuous Systems For our aims, an informational or logical closed formal system will be intended as a model of physical system such that: 1) the state variables and the evolution laws are individuated; 2) it is always possible to obtain the values of the variable states at each instant; 3) thanks to the information obtained by the above mentioned two points, it is always possible to connect univocally the input and the output of the system and to forecast its asymptotic state. So, a logical closed formal system is a deterministic system with respect to a given choice of the state variables. Let us consider a classical system and see how we can regard it as logical closed with respect to the observation procedures and its ability to show emergent processes. To be more precise, the values of the state variables express the intrinsic characteristic properties of the classical object and they are not affected by the measurement. Such fact, as it is known, can be expressed by saying that in a classical system the measurement made on all state variables are commutative and contextually compatible, i.e. all the measurement apparatuses connected to different variables can always be used without interfering one with the other and without any loss of reciprocal information. This assumption, supported by the macroscopic observations, leads to the idea of a biunivocal correspondence between the system, its states and the outcomes of the measurements. Hence, the logic of Classical Physics is Boolean and orthocomplemented, and it formalizes the possibility to acquire complete information about any system’s state for any time interval. The description of any variation in the values of the state variables, at each space-time interval, defines the local evolutionary feature of a classical system, either when it is “embedded” within the structure of a system of differential equations or within discrete transition rules. The peculiar independence of a classical system’s properties from the observer has deep consequences for the formal structure of classical physics. It is such independence which characterizes the system’s local, causal determinism as well as the principle of distinguishability of states in the phase space according to Maxwell–Boltzmann distribution function. This all puts very strong constraints to the informational features of classical physics and its possibility to show emergence. The correspondence between the volume in a classical system’s phase space and Shannon information via Shaw’s Theorem (Shaw, 1981) allows to combine the classification of the thermodynamic schemes (isolated, closed, open) in a broader and more elaborate vision. Three cases are possible: Emergence and Computation at the Edge of Classical and Quantum Systems 5 a) Information-conserving systems (Liouville’s Theorem); b) Information-compressing systems, ruled by the second principle of thermodynamics and consequently by the microscopic principle of correlation weakening among the constituents of the system individuated by the Maxwell-Boltzmann statistics (see Rumer & Ryvkin, 1980). These systems, corresponding to the closed ones, have a finite number of possible equilibrium states. By admitting a more general conservation principle of information and suitably redefining the system’s boundaries for the (b)- type systems, we can connect the two kind of systems (a) and (b), and so coming to the conclusion that in the latter a passage from macroscopic information to microscopic information takes place; c) Information-amplifying systems which show definitely more complex behaviors. They are non-linear systems where the variation of a given order parameter can cause macroscopic structural modifications. In these systems, the time dependence between the V volume of the phase space and the I information is given by: dI/dt = (1/V ) dV /dt. The velocity of information production is strictly linked to the kind of non-linearity into play and can thus be considered as a measure of complexity of the systems. Two principal classes can be individuated: c-1) Information-amplifying systems in polynomial time, to which the dissipative systems able to show self organization processes belong (Prigogine, 1994; Haken, 2004); c-2) Information-amplifying systems in exponential time; they are structurally unstable systems (Smale, 1966), such as the deterministic chaotic systems. Both the information-amplifying types belong to the open system classes, where an infinite number of possible equilibrium states are possible. Despite their behavioral diversity, the three classical dynamic systems formally belong to the class of logical closed models, i.e. because of the deep relationships between local determinism, predictability and computation, they allow to describe the system by means of recursive functions. If we consider the evolution equations as a local and intrinsic computation,we will see that in all of the three examined cases it is possible to characterize the incoming and outgoing information so as to define univocally the output/input relationships at each time (Cruchtfield, 1994). In dissipative systems, for example, information about the self-organized stationary state is already contained in the structure of the system’s equations. The actual setting up of the new state is due to the variation of the order parameter in addition to the boundary conditions. Contrary to what is often stated, even the behavior of structural unstable systems — and highly sensible to initial 6 Ignazio Licata conditions — is asymptotically predictable as strange attractor. What is lacking is the connection between global and local predictability, but we can always follow step-by-step computationally the increase of information within a predefined configuration. Our only limit is the power of resolution of the observation tool and/or the number of computational steps. In both cases we cannot speak of intrinsic emergence, but of emergence as detection of patterns.

3. The Observers in Classical Physics: Discrete Systems The discrete systems such as Cellular Automata (CA) (Wolfram, 2002) represent interesting cases. They can be considered as classical systems as well, because the information on the evolution of the system’s states is always available for the observer in the same way we saw for the continuous systems. On the other hand, their features are quite different than those of such systems in relation to emergent behavior. The Wolfram–Langton classification (Langton, 1990) identifies four fundamental classes of cellular automata. At the λ parameter’s varying — a sort of generalized energy — they show up the following sequence: Class I (evolves to a homogeneous state)→ Class II (evolves to simple periodic or quasi-periodic patterns)→Class IV (yields complex patterns of localized structures with a very long transient, for ex. Conway Life Game)→Class III (yields chaotic aperiodic patterns). It is known that cellular automata can realize a Universal Turing Ma- chine (UTM). To this general consideration, the Wolfram–Langton classification adds the analysis of the evolutionary behaviors of discrete systems, so building an extreme interesting bridge between the theory of dynamical systems and its computational facets. The I, II, III classes can be directly related to the information- compressing systems, the dissipative-like polynomial amplifiers and the structural unstable amplifiers, respectively.This makes the CA a powerful tool in simulating physical systems and the privileged one among the discrete models. The correspondence between a continuous system and the class IV appears to be more problematic. This class looks rather like an intermediate dynamic typology between unstable systems and dissipative systems, able to show a strong peculiar order/chaos mixture. It suggests they are systems which exhibit emergence on the edge of chaos in special way with respect to the case of the continuous ones. (Bak et al., 1988). Although this problem is still questionable from the conceptual and formal viewpoint, it is possible to individuate at least a big and significant Emergence and Computation at the Edge of Classical and Quantum Systems 7 difference between CA and continuous systems. We have seen that information is not erased in information-compressing systems, but a passage from macroscopic to microscopic information takes place. Therefore, the not time reversal aspects of the system belong more to the phenomenological emergence of entropy at descriptive level than the local loss of information about its states. In other words, for the conservation principle and the distinguishability of states, the information about any classical particles is always available for observation. It is not true for CA, there the irreversible erasing of local states can take place, such as in some interactions among gliders in Life. The situation is analogous to the middle-game in chess, where some pieces have been eliminated from the game. In this case, it is impossible to univocally reconstruct the opening initial conditions. Never- theless, it is always possible to individuate at least one computational path able to connect the initial state to the final one, and it is possible to show, thanks to the finite number of possible paths, that one of such paths must be the one that the system has actually followed. Consequently, if on the one hand the erasing of information in CA suggests more interesting possibilities of the discrete emergence in relation to the continuous one, on the other its characteristics are not so marked to question about the essential classical features of the system. In fact, in more strictly physical terms, it is also possible for the observer to locally detect the state erasing without losing the global describability of the dynamic process in its causal features. Some interesting formal analogies between the unpredictability of structural unstable systems and the halting problem in computation theory can be drawn. In both cases, there is no correlation between local and global predictability, and yet the causal determinism linked to the observer’s possibility to follow step-by-step the system’s evolution is never lost. Far from simply being the base for a mere simulation of classical systems, such point illuminates the deep connection between computation and classical systems. Our analysis has provided broad motives for justifying the following definition of classical system: a classical system is a system whose evolution can be described as an intrinsic computation in Shannon–Turing sense. It means that any aspect of the system’s “unpredictability” is not connected to the causal structure failing and any loss of information can be individuated locally by the observer. All this is directly linked to what we called the classical object’s principle of indifference to the measurement process and can be expressed by saying that classical systems are informational closed with respect to the observer. Such analysis, see in the following, is not valid for the quantum systems. 8 Ignazio Licata

This is one of the reasons which makes the introduction of the indeterminism in discrete systems quite problematic (Friedkin, 1992; Licata, 2001).

4. Quantum Events and Measurement Processes 4.1. The Centrality of Indeterminism The Heisenberg Uncertainty Principle is the essential conceptual nucleus of Quantum Physics and characterizes both its entire formal development and the debate on its interpretations. For a quantum system, the set of values of the state variables is not accessible to the observer at each moment. It is so necessary to substitute the “state variable” concept with the system state notion, characterized by the wave-function or state vector |Ψi. The relation between the system state and the variables is given by the Born rule. For example, given a 0 state variable q we can calculate its probability value through the state vector’s general form. It leads to explicitly introduce in the formalism the measurement operations as operators which are — differently from the classical ones — non-commutative and contextually incompatible. So, Quantum Logic differs from the Classical one because it is relatively orthocomplemented and not Boolean, thus orthomodular (D’Espagnat, 1999). A direct formal consequence is that for the quantum objects there exists a Principle of indistinguishability of states which characterizes the Fermi–Dirac as well as the Bose–Einstein quantum statistics. In this way, the observer takes on a radical new role, which can be clarified by means of the relational notion of quantum event (Healy, 1989; Bene, 1992; Rovelli, 1996; for the notion of quantum contextuality as a frame see also Griffiths, 1995; for the collapse problem from the logic viewpoint see: Dalla Chiara, 1977). Let us consider a quantum system S and an observer O. The interaction between the two systems, here indicated as O+S, will give a certain value q of the corresponding state variable q. So, we can say — by para- phrasing Einstein — that the quantum event q is the element of physical reality of the system S with respect to O. If we describe S and O as two quantum systems, for ex. two-state systems — another system O’ will find the value q0relative to the system S+O. Therefore, a quantum event always expresses an interaction between two systems and, according to the relational approach, this is all we can know about the physical world, because a hypothetical comparison between O and O’ will be a quantum event, too. Despite the fact that it cannot be fully considered as an Emergence and Computation at the Edge of Classical and Quantum Systems 9

“interpretation” yet, the relational view clearly shows the irreducibility of the role of the observer in QM as well as the new symmetry with the observed system. In spite of such radically not classic feature, the so-called “objective reality” described by the quantum world has not to be questioned at all; it is precisely the peculiar nature of the quantum objects which imposes a new relationship with measurements on the theory and leads to a far diﬀerent notion of event than the classical one.

4.2. The Collapse Postulate When we consider the “collapse postulate” of the wave function, a radical asymmetry will be unavoidable. In this case, we have to consider on the one hand the temporal evolution of the wave function U, provided by the rigorously causal, deterministic and time-reversal Schrödingerequation, and on the other the reduction processes of the state vector R, where the measurement’s outcome is macroscopically fixed, such as the position of an electron on a photographic plate. The processes R are not-causal and time asymmetrical. Different standpoints are possible about the role of the processes R in QM, it recalls the 1800s mathematical debate about the structure of Euclidean geometry and in particular the fifth postulate position. Without any claim to exhaustiveness, we can individuate here three main standpoints about R: A) The wave function contains the available information on the physical world in probabilistic form; the wave function is not referred to an “objective reality”, but — due to the intrinsically relational features of the theory — only to what we can say about reality. Consequently, the “collapse postulate” is simply an expression of our peculiar knowledge of the world of quantum objects; B) The wave function describes what actually happens in the physical world and its probabilistic nature derives from our perspective of observers; R, as well as the entire QM, is the consequence of the fact that the most part of the needed knowledge is structurally unavailable; C) The wave function partially describes what happens in the physical processes; in order to comprehend its probabilistic nature and the postulate R in particular, we need a theory connecting U and R. The above three standpoints are subtly different and yet connected. To clarify these interrelations we will give some examples. (A) reflects the traditional Copenhagen interpretation. We are not interested here in the 10 Ignazio Licata old ontological debate about reality, but rather we are going to focus on the formal-logical structure of each theory. According to the supporters of (A) — or at least one of its manifold forms — QM is coherent and complete, and the collapse postulate assures the connection between U and the observations, even if this connection cannot be deduced by U. It means that in order to provide a quantum event, the “Von Neumann chain” breaks at the observer level. The price to pay for such readings of the theory is that the non-local correlations come out as a compromise between causality (conservation laws and Relativity) and quantum casuality (R irreducibility). In the class (B) quite different interpretations are taken into consideration, such as the Everett Theory on the relative states (DeWitt & Graham, 1973; Deutsch, 1985) and the Bohm–Hiley theory of Implicate Order (Bohm & Hiley, 1995). The Everett Many Worlds Theory regards U as the description of the splitting of the branches in the multiverse (Deutsch, 1998), and R simply as the observer’s act of classically detecting its own belonging to one of the branches of the quantum multiverse by the measurement. The Everett Theory can be considered to lie at the bottom of the current relational approaches as well as, indirectly, the so-called decoherent histories and, in general, many of the quantum cosmology conceptions (Gell–Mann & Hartle, 1993; Omnès,1994). Unfortunately, the idea of the multiverse does not solve all the R-related problems. The real problem is to clarify why the universes interfere one with the other and show entanglement with respect to any observer at a given scale. So the idea to reduce the quantum superposition to a classical collection of observers fails. That is why, in the decoherent histories, the problem shifts to the emergence — a coarse-grained one, at least — of classicality from quantum processes. For our aims it is particularly relevant the recent observation that a coherent theory of multiverse implies the existence of objects carrying not classical information, such as the ghost-spinors (Palesheva, 2001; 2002; Guts, 2002). Deutsch and Hayden have shown that non-locality can be regarded as locally inaccessible information and such conception can clarify the decoherence processes (Deutsch and Hayden, 2000; Hewitt–Horsman & Vedral, 2007). In Bohm–Hiley Theory — which has to be kept separate from the minimalist and mechanic reading often made (Berndl et al., 1995) the radical not-classical aspects are likewise considered as a kind of information which is unknown in the classical conception and linked to the fundamentally not-local structure of the quantum world. The irreducibility of R is thus the consequence of the introduction of quantities depending Emergence and Computation at the Edge of Classical and Quantum Systems 11 on the quantum potential and representing not casual events, but rather “potentialities” in a formally very strict sense. The class (C) instead includes all those theories which tend to reconcile U with R by introducing new physical process, the so-called OR (Objec- tive Reduction) (see for ex. Longtin & Mattuck, 1984; Kàrolyhàzy1974; 1986; Pearle, 1989; Ghirardi et al., 1986; Diòsi,1989; Percival, 1995; Pen- rose, 1996a). The OR theories focus on the relationship between quantum objects and classical, macroscopic measurement apparatuses (i.e. made up of a number of particles equal to at least 1023). The structure of U is modified by introducing new terms and parameters able to obtain a spontaneous localization. The current state of these theories is very complex. In these approaches some conflicts with Relativity and suggestions for the quantum gravity stay side by side, so making difficult to say what is ad hoc and what will turn out to be fecund. Nevertheless, an acknowledgement is due to the theories of class (C) because they identify the asymmetry between U and R as a radical emergence within the structure of physics and it is a demand for new theoretical proposals able to comprehend the border between quantum and classical systems. However, we have to say that the theories of spontaneous localization do not introduce any new element useful to understand the non-local correlations, but they make the question much more intricate. In fact, if we consider an EPR-Bell experiment with quantum erasing, it is difficult to reconcile it with the thermodynamic irreversibility implicit in the localization concept (see Yoon-Ho et al., 2000; and also Callaghan & Hiley, 2006a ; 2006b). Obviously, the standpoints on the singular role of R in the axioms of QM are much more elaborate, and there are also theories with intermediate features with respect to the above mentioned three classes. In the Cini–Serva theory of correspondence between QM and classical statistical mechanics, for instance, the measurement effect is just to reduce the statistical ensemble and consequently our uncertainty on the system’s conditions as well. It is thus eliminated a physically meaningless superposition so as to find the minimum value of the wave packet compatible with quantum indeterminacy (Cini & Serva, 1990; 1992). This theory comprises some typical features of both the class (B) “realism” — yet rejecting its analysis about the nature of the “hidden” information — and the class (C), since it is centered on the localization problem at the edge between micro and macro physics, but it does not introduce any new reduction mechanism. Finally – like in the standard interpretation of the class (A) – , in the Cini–Serva theory, the not-separability aspects of quantum correlations are regarded not as a 12 Ignazio Licata background dynamics, but as being due to the intrinsical casual character of quantum behaviors. The na¨ıve (i.e. pre-Bell) theory of hidden variables can be considered as the precursor of the class (B) philosophy, even if the idea of restoring the classical mechanical features has been replaced today by a not-mechanical conception in Bohm sense (Bohm, 1951).

4.3. Quantum Observers and Non-local Information Let’s try now to focus on the formal facets of QM with respect to the observer. In the quantum context it is impossible for the observer a biunivocal correspondence between the notion of system state, described by the state vector, and the value assignment to the state variables at any instant; it imposes to give up the indifference principle of the system state with respect to the observer, typical of the classical systems, and it leads to the breakdown of the causal, local determinism. In QM formalism, such aspect is expressed by the fact that a closure relation is only valid for the eigenvalues (see for ex. Heylighen, 1990). So we can naturally characterize the quantum systems as informational open systems with respect to the observer. Now the problem is what meaning we have to give to such logical openness. To be more precise, if we want both no modification in the formalism and a broader comprehension of the QM non-local features, we have to focus our attention on the (B)-type theories and to define a suitable non-local and classically irreducible information able at the same to extend the concept of intrinsic computation to quantum systems, too. Once again, computation theory shows to be very useful in characterizing the physical systems. In fact, considering the irreducibility of an outcome R to the evolutionary structure of U, we can state that it is impossible to apply an algorithmic causal structure to two quantum events relative to an observer O at different times. We can so conclude that a quantum system is a system where the quantum events cannot be correlated one with the other by means of a Shannon-Turing-type computational model. This proves the intrinsically “classic” roots of the computation concept. We will see in the following that the Bohm-Hiley active information is a really useful approach. But first we have to briefly review the emergent processes in Quantum Field Theory (QFT).

5. Emergence in Quantum Field Theory (QFT) The idea according to which the QFT distinctive processes are those exhibiting intrinsic emergence and not mere detection of patterns is widely Emergence and Computation at the Edge of Classical and Quantum Systems 13 accepted by the community of physicists by now (Anderson and Stein, 1985; Umezawa, 1993; Pessa, 2006; Vitiello, 2001; 2002). The central idea is that in quantum systems with infinite states, as it is known, different not unitarily equivalent representations of the same system, and thus phase transitions structurally modifying the system, are possible. This takes place by means of spontaneous symmetry breaking (SSB), i.e. the process which changes all the fundamental states compatible with a given energy value. Generally, when a variation of a suitable parameter occurs, the system will switch to one of the possible fundamental states, so breaking the symmetry. This causes a balance manifesting itself as long-range correlations associated with the Goldstone–Higgs bosons which stabilize the new configuration. The states of bosonic condensation are, in every respect, forms of the system’s macroscopic coherence, and they are peculiar of the quantum statistics, formally depending on the indistinguishability of states with respect to the observer. The new phase of the system requires a new descriptive level to give account for its behaviors, and we can so speak of intrinsic emergence. Many behaviors of great physical interest such as the phonons in crystal, the Cooper pairs, the Higgs mechanism and the multiple vacuum states, the inflation and the “cosmic landscape” formation in quantum cosmology can be included within the SSB processes. It is so reasonable to suppose that the fundamental processes for the formation of structures essentially and crucially depend on SSB. On the other hand, the formal model of SSB appears problematic in many respects when we try to apply it to systems with a finite number of freedom degrees. Considering the neural networks as an approximate model of QFT is an interesting approach (Pessa & Vitiello, 1999; 2004).The generalized use of the QFT formalism as a theory of emergent processes requires new hypotheses about the system/environment interface. The most ambitious challenge for this formalism is surely the Quantum Brain Theory (Ricciardi & Umezawa, 1967; Vitiello, 2001). A formally interesting aspect of QFT is that it allows, within limits, to frame the question of reductionism in a clear and not banal way, i.e. to give a clear significance to the relations among different descriptive levels. In fact, if we define the phenomenology linked to a given range of energy and masses as the descriptive level, it will be possible to use the renormalization group (RG) as a tool of resolution to pass from a level to another by means of a variation of the group’s parameters. We thus obtain a succession of descriptive levels — a tower of Effective Field Theory (EFT) — having a fixed cut-off each, able to grasp the peculiar aspects of the investigated 14 Ignazio Licata level (Lesne, 1998; Cao, 1997; Cao & Schweber, 1993). In this way each level is connected to the other ones by a rescaling of the kind Λ0 → Λ(σ) = σΛ0, where Λ0 is the cut-off parameter relative to the fixed scale of the energies/masses into play. The universality of the SSB mechanism is deeply linked to such aspect, and the possibility to use the QFT formalism as a general theory of emergence puts the problem of extending the EFT “matryoshka-like” structures allowed by the RG to different systems. In what sense can the intrinsic emergence of SSB be compared to the phenomenological detection of patterns and which are the radically quantum features in the same sense we pointed out for ordinary QM? Also in the case of the SSB processes, the phase transition is led by an order parameter towards a globally predictable state, i.e. we know that there exists a value beyond which the system will reach a new state and will exhibit macroscopic correlations. Once again a prominent role is played by the boundary conditions (all in all, a phonon is the dynamic emergence occurring within a crystal lattice, but it is meaningless out of the lattice). Moreover, in the SSB there exists a transient phase whose description is widely classic. Where the analogy falls down and we can really speak of an irreducibly not-classic feature is in bosonic condensation which is a non-local phenomenon. While in a classical dissipative system is possible, in principle, to obtain information about the “fine details” of a bifurcation and to know where “the ball will fall”, in a process of SSB it is impossible for reasons connected to the nature of the quantum roulette! In this sense the radical features of emergence in QFT and in QM are of the same nature and they demand a new information theory able to take into account the non-local aspects.

6. Quantum Information from the Structure of Quantum Phase Spaces The concept of active information has been developed by Bohm and Hiley and the Birbeck College group within a wide research programme aimed at comprehending the QM non-local features (see for ex. Bohm & Hiley, 1995; Hiley, 1991; Hiley et al., 2002; Monk & Hiley, 1993; 1998; Brown & Hiley, 2004). This theory is formally equivalent to the standard one and does not introduce any additional hypothesis. The features often described as “Bohmian mechanics” and aimed at a “classic” visualization of the trajectories of quantum objects are not essential. The theory rather tends to grasp the essentially not-mechanic nature of quantum processes (Bohm, 1951), in the sense of a topology of not-separability quite close, in formalism and spirit, to non-commutative geometries (Demeret et al., 1997; Emergence and Computation at the Edge of Classical and Quantum Systems 15

Bigatti, 1998). It is known that the theory first started from splitting the Schrödingerordinary equation in real and¡ ± imaginary¢ ¡ part in± order¢ to obtain the quantum potential Q (r, t) = − ~2 2m ∇2R (r, t) R (r, t) . This potential QP, unknown in classical physics, contains in nuce the QM nonlocal features and individuates an infinite set of phase paths; in particular, the QP gets a contextual nature, that is to say it brings global information about the quantum system and its environment. We underline that such notion is absolutely general and can be naturally connected to the Feynman path integrals. Our aim is to briefly delineate the geometrical aspects of the quantum phase spaces from which the QP draws its physical significance.

Let us consider an operator O and the wave function ψ (O). By using the polar form of the wave function we can write the usual equations of probability and energy conservation in operatorial form: dρ dS 1 i + [ρ, H] = 0; ρ + [ρ, H] = 0, (1) dt − dt 2 + where ρ = Ψ∗ (O)i hΨ(O) is the density operator which makes available the entropy S of the system as S = trρ ln ρ, i.e. as the maximum obtainable information from the system by means of a complete set of observations (Aharonov & Anandan, 1998). Now let us choose a x-representation for the (1) and we will get: ∂ρ ∂S (∇ S)2 + ∇ j = 0; + r + Q (r, t) + V (r, t) = 0. (2) ∂t r ∂t 2m Instead, in a p-representation the (1) take the form: ∂ρ ∂S p2 + ∇ j = 0; + + Q (p, t) + V (∇ S, t) = 0. (3) ∂t p ∂t 2m p From the probability conservation in (2) and (3), two sets of trajectories in the phase space can be ﬁnally derived, with both R and Re for real:

∗ ∗ ∇rS = Re [Ψ (r, t) P Ψ(r, t)] = pR; ∇pS = Re [Φ (p, t) XΦ(p, t)] = xR. (4) The conceptual fundamental point is that we obtain the quantum potential only when we have chosen a representation for the (1). Such a construction is made necessary by the non-commutative structure of the phase spaces of the conjugate variables, and it implies that the observer has to follow a precise process of information extracting via the procedure state preparation→state selection→measurement. So the quantum potential can be regarded as the measure of the active information extracted from what 16 Ignazio Licata

Bohm and Hiley have called implicate order to the information prepared for the observation in the explicate order. The process algebra (Symplectic Clifford algebra) between implicate and explicate order is thus a dynamics of quantum information. It is worth noticing that in the (4), given a representation, the conjugate variable is a “beable” variable (Bell, 1987), i.e. a construction depending on the choice made by the observer. Such choice is in no way “subjective”, but it is deeply connected with the phase space structure in QM. The Bohm and Hiley’s reading restores the natural symplectic symmetry between x and p, but, unlike the classical case, it does so by geometrically justifying the complementarity notion. To better understand the dynamics of quantum information and its relation with Shannon–Turing classical information, let us consider the notions of both Implicate and Explicate Order and enfolded and unfolded information. (Licata & Morikawa, 2007). For our aims, it will suffice here to define the Implicate Order as the non-commutative structure of the conjugate variables of the quantum phase space. Therefore, it is impossible to express such structure in terms of space- time without making a choice within the phase space beforehand, so fixing an explicate order. Let us designate the implicate order with E and the explicate order with E’; we will say that E is a source of enfolded information, whereas E’ contains the unfolded information extracted from E. The relation between the two orders is given by: Z E0 (t) = G (t, τ)E (τ) dτ, (5)

implicate where G (t, τ) is the Green’s Function and τ is the unfolding parameter. The inverse operation, from the explicit space-time structure to the implicate order, is so given by: Z E (τ) = G (τ, t)E0(t)dt. (6)

exp licate The passage from (5) to (6) — and vice versa — has a simple physical significance. The unfolding corresponds to the state selection, and the enfolding to the state preparation. Let us consider a typical two-state system, for ex. a spin 1/2 particle: |Ψi = a |↑i + b |↓i . (7) Here a and b are a measure of the active information extracted from the implicate order by choosing the state variable. It has to be noticed that the Emergence and Computation at the Edge of Classical and Quantum Systems 17 state preparation itself , as well as the measurement, modifies the system contextual information, so defining a new relationship between the background order E (τ) and the foreground one E0 (t). In other words, being chosen a variable, the information of the previous explicate order vanishes into the implicate order. During the measurement, information becomes enactive, that is to say that the information contained in the superposition state (7) is destroyed, thus selecting between a and b via an artificial unfolding. Here the term “artificial” means that the measurement — or the “collapse” — has not a privileged position within the theory, but it is only one of the ways which the configuration of the system active information can change into. For example, a creation and annihilation process of particles can be considered as a “spontaneous” process of unfolding/enfolding. What is really important is the relationship between the contextuality of active information and the non-commutativity of the phase space. We can sum up all this by saying that in Bohm–Hiley theory a quantum event is the expression of a deeper quantum process connecting the description in terms of space and time with the intrinsic non-local one of QM. We can say that the implicate order is a realization of the Wheeler “pre-space” (Wheeler, 1980). The quantum potential, as well as the appearing of “trajectories” in the (4), in no way restores the classic view. In fact, the algebraic structure of QM clearly tells us that both the sets of trajectories are necessary to comprehend the quantum processes, and describing them in the explicate order implies a complementarity, and consequently a structural loss of information. Any pretence about the centrality of x-representation is arbitrary and it is only based on a classical prejudice, i.e. the “position” regarded as the “existence”. In no way a quantum system has to be considered as less “real” and “objective” than a classical one. It is the observer role which changes just in relation to the peculiar nature of quantum processes. So the trajectories have not to be regarded as “mysterious” or “surreal” lines of force violating Relativity, but as informational configurations showing the intrinsic not-separability of the quantum world in the space-time foreground. From what we have said above, it clearly appears that the problem of the emergence of the classical world from the quantum one cannot be dealt with by using a classical limit for the quantum potential, despite the statistical interest of such pragmatic approach (see for ex. Allori & Zangh`ı,2001). The limit of classicality has rather to be faced as a problem of relationships between algebra and metric in the sense of the principle of reconstruction of Gel’fand (Demaret et al., 1997). Another significant point, which we can 18 Ignazio Licata only mention here, is the importance of the unfolding parameter in quantum cosmology, for example in passing from a timeless quantum De Sitter-like universe to a post inflationary time evolution (Callander & Weingard, 1996; Nelson Pinto-Neto, 2000; Lemos & Monerat, 2002; Licata, 2006; Chiatti & Licata, 2007). The Shannon–Turing information comes into play when a syntactic, local analysis of the system is possible on a well-defined channel. In QM, it is possible only when a system has been prepared in a set of orthogonal wave functions, thus making a selection of the enfolded information which can be analyzed in terms of usual quantum bits. Elsewhere we have pointed out how such crucial difference between contextual active information and q-bit information can be used to explain forms of quantum computation much more powerful than the one based on quantum gates (Licata, 2007). A different and more immediate way to understand the limits of the classical computation theory in QM is to take into consideration again the trajectories in the (4). As the two sets are complementary, the thought experiment of “rewinding” the trajectories of an unmeasured quantum system is structurally unable to provide information about the details of evolution. In fact, let us suppose we have chosen, for example, a family of trajectories in x-representation and we — yet ideally — have made a computational analysis on it, all the same we can say nothing on the computation of the trajectories in the p-representation: just like the variable on which it is centred on, the computation on p-representation appears to be a “beable computation”! It brings out a deep connection between non-commutativity, non-locality and the new kind of quantum time-asymmetry related to the fact that the logic of the process of unfolding/enfolding described by (5) and (6) — that is to say τ → t and vice versa — implies an uncomputable modification of the system information. We can conclude that the features of informational openness of QM are a direct consequence of the non-commutative algebra which imposes upon the observer to make a choice on the available information and it prevents from describing the intrinsic computation of a quantum systems via a Shannon–Turing model because of the contextual nature of the active information.

7. On the Relationships Between Physics and Computation In the last years an important debate on the relationships between physical systems, formal models and computation has been developed. In our analysis, we have used the classical computation theory not only as Emergence and Computation at the Edge of Classical and Quantum Systems 19 simulation means, but rather as a conceptual tool able to let us understand the formal relationships between the system’s behaviours and the information that the observer can get about it. We have seen that it is possible to give a computational description of the classical systems centred on the possibility to be always able to identify the information in local way. So, emergence in classical systems is fundamentally of computational kind and there exists a strict correspondence between classical systems and computational dynamics (Baas & Emmeche, 1997). This is not in contradiction with the appearing of non-Turing features in some classical systems (see for ex. Siegelmann, 1998; Calude, 2004) because it has been proved that sets of interactive Turing Machines (TMs) with oracles can show computational abilities superior than those of a TM in the strict sense (Kieu & Ord, 2005; Collins, 2001). Actually, if we look at the “oracles” as metastable configurations fixed during the evolution of a system, this is the minimum required baggage to understand the biological evolution and mind. QM does not seem to be necessary to comprehend the essential features of life and cognition, in contrast to what Penrose claims (Penrose, 1996). In QM, we deal instead with radical processes of observational emergence which cannot be overcome by the construction of a new model. The Bohm–Hiley theory helps us to understand such radical features of QM by the Implicate/Explicate Order process algebra and the non-commutative structure of the quantum phase spaces. The failure of a TM-based observer in describing quantum systems and the consequent recourse to a probabilistic structure are related to the fact that any observer belongs to the explicate order and it has to use a space-time structure to build up the concept of physical event. From this view, the role of computation in physics appears to be even more profound than what the Turing Principle suggests (Deutsch, 1998). In fact, the Shannon–Turing theory here appears as a computability general constraint on the causal structures which can be defined in space-time and as the extreme limit of any description in terms of space-time (Markopoulou–Kalamera, 2000a; 2000b; Tegmark, 2007). Processing information is what all physical systems do. The way to process information depends on the system’s nature, and it is not surprising that the Shannon–Turing information naturally fits to not-quantum systems or, at the utmost, to q-bit systems because it has been conceived within a classical context. In quantum systems instead, the breakdown of the classical computation model calls into play a new concept of information, in the same way as more than once in the history of physics, a conceptual crisis has required new strategies to comprehend the unity of 20 Ignazio Licata the physical world. The Bohm–Hiley concept of active information allows to extend the concept of intrinsic computation to quantum systems, too. In order to make such notion efficacious, a great deal of new ideas will be necessary. Already in 1935, Von Neumann wrote in a letter to Birkhoff: “I would like to make a confession which may seem immoral: I do not believe in Hilbert space anymore.” (Von Neumann, 1935). In fact the separable Hilbert spaces are not fit for representing infinite systems far from equilibrium. It is thus necessary a QFT able, at least, to represent the semigroups of irreversible transformations — i.e. operators for the absorption and dissipation of quanta — by not-separable Hilbert spaces. Here, a new scenario connecting non-commutativity, irreversibility and information emerges. The “virtuous circle” of systems-models-computation finds its ultimate significance not only in the banal fact that we build models of the physical world so as to be “manageable”, but also because we are constrained to built such models based on family of observers in the explicate order. It suggests that an authentic “pacific co-existence” between Relativity and QM will require a general mechanism to obtain space-time transformation groups via an unfolding parameter as a boost from a more complex underlying background structure. In this context, the hyperspherical group approach appears to be interesting (see Licata, 2006; Chiatti & Licata, 2007). The physical world has no need for observers for its structure and evolution. But the nature of the quantum processes makes the relationship between the observer and the observed irreducibly participatory, such that the description of any physical system is necessarily influenced by unbounded and contextual information. Such kind of information is not cooped up in the “Hilbert cage” and has less to do with what is usually meant by quantum computing. Far from being a “shadow” information, it fixes the fundamental non-local character of the quantum world and the limit of Shannon–Turing information in describing the intrinsic computation of quantum systems. In contrast, it is just this radical uncomputable feature which individuates the Boolean characteristics of the observers in the space-time and allows, thanks to the spontaneous symmetry breaking processes, the increasing complexity of the physical universe.

References 1. Aharonov, Y., Anandan, J. (1998) Meaning of the Density Matrix, e.print in ArXiv:quant-ph/9803018v1. Emergence and Computation at the Edge of Classical and Quantum Systems 21

2. Allori, V. , Zangh`ı,N. (2001) On the Classical Limit of Quantum Mechanics, e.print in ArXiv:quant-ph/0112009v1. 3. Anderson, P.W., Stein, D.L. (1985) Broken Symmetry, Emergent Properties, Dissipative Structure, Life. Are They Related? In Yates, F. E.(ed), Self- Organizing Systems: The Emergence of Order, Plenum Press, pp 445-457. 4. Baas, N. A., Emmeche, C. (1997) On Emergence and Explanation, in Intel- lectica,2, 25, pp 67–83. 5. Bak , P., Tang, C., Wiesenfeld, K. (1988) Self-Organized Criticality in Phys- ical Review A38, pp 364–374. 6. Bell, J.S. (1987) Speakable and Unspeakable in Quantum Mechanics, Cam- bridge Univ. Press. 7. Bene, G. (1992) Quantum Reference Systems: A New Framework for Quan- tum Mechanics in PhysicaA, 242, pp 529–560. 8. Berndl, K., Daumer, M., Dürr,D., Goldstein, S., Zanghi, N. (1995) A Survey on Bohmian Mechanics, in Nuovo Cim. B 110, pp 737-750. 9. Bigatti, D. (1998) Non-Commutative Spaces in Physics and Mathematics, e.print in ArXiv:hep-th/0006012. 10. Bohm, D. (1989) Quantum Theory, Prentice Hall, 1951; reprint Dover. 11. Bohm, D. , Hiley, B. (1995) The Undivided Universe, Routledge. 12. Brown, M.R. , Hiley, B.J. (2004) SchrödingerRevisited: an Algebraic Ap- proach, e.print in ArXiv:quant-ph/0005026v5. 13. Callaghan, R.E., Hiley, B.J. (2002) Quantum Trajectories, Real, Surreal or an Approximation to a Deeper Process, e.print in ArXiv:quant-ph/0010020. 14. Callaghan, R.E., Hiley, B.J. (2006a) What is Erased in the Quantum Era- sure?, in Found. of Phys., 36, 12, 1869-1883. 15. Callaghan, R.E., Hiley, B.J. (2006b) Delayed Choice Experiments and the Bohm Approach, in Phys.Scr., 74, 336-348. 16. Callender, C., Weingard, R. (1996) Time, Bohm’s Theory and Quantum Cosmology, in Phil.of Science, 63, 3, 470-474. 17. Calude, C.S. , Paun, G. (2004) Bio-steps beyond Turing, in BioSystems, 77, 175–194. 18. Cao, T.Y. , Schweber, S.S. (1993) The Conceptual Foundations and the Philosophical Aspects of Renormalization Theory, in Synthese, 97, 33–108. 19. Cao, T.Y. (1997) Conceptual Developments of 20 th Century Field Theories, Cambridge Univ. Press. 20. Chiatti, L., Licata, I. (2007) Big-Bang Structure, Cosmological Term and Time Arrow in Fantappié-ArcidiaconoProjective Cosmology, to be publ. in Elect. Jour. of Theor. Phys. 21. Cini, M., Serva, M. (1990) State Vector Collapse as a Classical Statistical Effects of Measurement, in Cini, M., J.M. Lévy-Leblond(eds),Quantum Mechanics without Reduction, Hilgher, 1990, 103. 22. Cini, M., Serva, M. (1992) Measurement in Quantum Mechanics and Clas- sical Statistical Mechanics, in Physics Letters A, 167, 319–325. 23. Collins, J.C. (2001) On the Compatibility between Physics and Intelligent Organism, e.print in ArXiv:physics/0102024v1. 22 Ignazio Licata

24. Cruchtfield, J.P. (1994) The Calculi of Emergence: Computation, Dynamics and Induction, in Physica D, 75, pp 11–54. 25. D’Espagnat, B. (1999) Conceptual Foundations of Quantum Mechanics, Westview Press; 2 ed. 26. Dalla Chiara, M.L. (1977) Logical Self-Reference, Set Theoretical Paradoxes and the Measurement Problem in Quantum Mechanics, in Jour. of. Phil. Logic, 6, pp 331–347. 27. Demaret, J., Heller, M., Lambert, D. (1997) Local and Global Properties of the World, in Found. of Science, 2 , pp 137–176. 28. Deutsch, D. (1985) Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer, in Proceedings of the Royal Society of Lon- don,A 400, pp 97–117. 29. Deutsch, D. (1998) The Fabric of Reality: The Science of Parallel Universes and Its Implications, Penguin. 30. Deutsch, D., Hayden, P. (2000) Information Flow in Entangled Quantum Systems, in Proceedings of the Royal Society A, 456, pp 1759-1774. 31. DeWitt, B., Graham, N. (Eds) (1973) The Many-Worlds Interpretation of Quantum Mechanics, Princeton Univ. Press. 32. Diòsi,L. (1989) Models for Universal Reduction of Macroscopic Quantum Fluctuations, in Phys. Rev. A40, pp 1165-1174. 33. Fredkin, E. (2003) An Introduction to Digital Philosophy, in International Journal of Theoretical Physics, 42, (2), pp 189–247. 34. Freeman, W.J., Vitiello, G. (2005) Nonlinear Brain Dynamics and Many- Body Field Dynamics, in Electromagnetic Biology and Medicine, 24, pp 1–9. 35. Gell–Mann, M., Hartle, J. (1993) Classical Equations for Quantum Systems, in Phys. Rev.D, 47, 3345-3382. 36. Ghirardi , G.C., Rimini, A., Weber, T. (1986) Unified Dynamics for Micro- scopic and Macroscopic Systems, in Phys. Rev. D, 34, pp 470–491. 37. Griffiths, R.B. (1996) Consistent Histories and Quantum Reasoning, in Phys. Rev. A, 54, pp 2759–2774. 38. Guts, A.K. (2001) Topos-theoretic Model of the Deutsch Multiverse, in Mathematical Structures and Modeling, 8, 76-90; also available in arXiv:physics/0203071. 39. Haken, H. (2004) Synergetics : Introduction and Advanced Topics, Springer. 40. Hiley, B.J. (1991) Vacuum or Holomovement, in Saunders, S., Brown, H.R.(eds), The Philosophy of Vacuum, Clarendon Press, pp 217–249. 41. Hiley, B.J, Maroney, O.J.E. (2000) Consistent Histories and the Bohm Ap- proach, e.print in arXiv:quant-ph/0009056. 42. Healey, R. (1989) The Philosophy of Quantum Mechanics; An Interactive Interpretation, Cambridge Univ. Press. 43. Hewitt-Horsman, C., Vedral, V. (2007) Developing the Deutsch–Hayden approach to Quantum Mechanics, in New J. Phys. 9,135. 44. Heylighen, F. (1990) Classical and Non-Classical Representation in Physics II:Quantum Mechanics, in Cybernetics and Systems, 21, pp 477–502. Emergence and Computation at the Edge of Classical and Quantum Systems 23

45. Károlyházy, F., Frenkel, A., Lukács,B. (1986) On the Possible Role of Gravity on the Reduction of the Wave Function, in Penrose, R. & Isham, C.I.(eds.) Quantum Concepts in Space and Time, Oxford Univ. Press, pp 109–128. 46. Kieu, T.D., Ord, T., (2005), The Diagonal Method and Hypercomputation, in The Brit. Jour. for the Phil. of Sci. 56, (1), pp 147–156. 47. Langton, C. (1991) Computation at the Edge of Chaos: Phase Transitions and Emergent Computation, in Stephanie Forest (ed.) Emergent Computa- tion, MIT Press, pp 12–37. 48. Lesne, A. (1998) Renormalization Methods. Critical Phenomena, Chaos, Fractal Structures, Wiley. 49. Lemos, N.A., Monerat, G. A. (2002) Quantum Cosmology in the Bohm- De Broglie Interpretation, in Anais do XXIII Encontro de Fisica de Particulas e Campos, Aguas de Lindoia. 50. Licata, I. (2003) The Big Computer. Complexity and Computability in Physical Universe, in Determinism, Holism and Complexity (Benci,V., Cer- rai, P., Freguglia, P., Israel, G., Pellegrini, C., (eds.), Springer. 51. Licata, I. (2006) Universe Without Singularities. A Group Approach to De Sitter Cosmology, in Licata, I. (ed), Majorana Legacy in Contemporary Physics, EJTP Special Issue, 3, (10), pp 211–224, Di Renzo, Roma. 52. Licata, I. (2007) Effective Physical Processes and Active Information in Quantum Computing, in QuantumBioSystems, 1, pp 51-65. 53. Licata, I., Morikawa, R. (2007) Unfolding and Enfolding Information Dy- namics in Quantum Teleportation, to be publ. in QuantumBioSystems, 2. 54. Longtin, L., Mattuck, R.D. (1984) Relativistically Covariant Bohm-Bub Hidden-Variable Theory for Spin Measurement of a Single Particle, in Found. of Phys., 14, (8), pp 685–703. 55. Markopoulou-Kalamera,F. (2000a) The Internal Description of a Causal Set: What the Universe Looks like from the Inside, in Comm. in Math. Physics, 211, (3), pp 559–583. 56. Markopoulou-Kalamera,F. (2000b) An Insider’s Guide to Quantum Causal Histories,in Nuclear Physics B, Proc. Suppl. 88, 1–3, 308-313. 57. Monk, N.A.M., Hiley, B.J. (1998) A Unified Algebraic Approach to Quan- tum Theory, in Found. Phys. Lett., 11, pp 371–377. 58. Monk, N.A.M., Hiley, B.J (1993) Quantum Phase Space and the Discrete Weyl Algebra, in Mod.Phys. Lett., A8, pp 3225–3233. 59. Nicolis, G., Prigogine, I.. (1989) Exploring Complexity, W.H. Freeman & Company. 60. Omnès,R. (1994) The Interpretation of Quantum Mechanics, Princeton Univ. Press. 61. Palesheva, E. (2001) Ghost Spinors, Shadow Electrons and the Deutsch Multiverse, e.print in arXiv:gr-qc/0108017v2. 62. Palesheva, E. (2002) Ghost Spinors in Quantum Particles Interference, in Mathematical Structures and Modeling, 9, 142-157, also available in arXiv:quant-ph/0207083v1. 24 Ignazio Licata

63. Pearle, P. (1989) Combining Stochastic Dynamical State-Vector Reduction with Spontaneous Localisation, in Phys. Rev. A, 39, pp 2277–2289. 64. Penrose, R. (1996a) On Gravity’s Role in Quantum State Reduction, in Gen. Rel. and Grav., 28, 581–600. 65. Penrose, R. (1996) Shadows of the Mind, Oxford Univ. Press. 66. Percival, I.C. (1995) Quantum Spacetime Fluctuations and Primary State Diffusion, in Proc. Royal Soc. A, 451, pp 503–513. 67. Pessa, E., Vitiello, G. (1999) Quantum Dissipation and Neural Net Dynam- ics, in Bioelectrochemistry and Bioenergetics, 48, (2) , pp 339–342. 68. Pessa, E., Vitiello, G. (2004) Quantum Noise Induced Entanglement and Chaos in the Dissipative Quantum Model of Brain, in Int. J. Mod. Phys B., 18, pp 841–858. 69. Pessa, E. (2006) Physical and Biological Emergence:Are They Different? In Minati,G., Pessa, E.Abram, A.(eds), Systemics of Emergence, Springer, pp 355–374. 70. Pinto-Neto, N. (2000) Quantum Cosmology: How to Interpret and Obtain Results, in Braz. J. of Phys., 30, 2, 330-345. 71. Prigogine, I. (1994) Order Out of Chaos, Bantam. 72. Ricciardi, L.M., Umezawa, H. (1967) Brain and Physics of Many-Body Prob- lems, in Kybernetik, 4, 44-48; reprint in Globus, G. G., Pribram, K. H., Vitiello, G.,( eds.), Brain and Being, John Benjamins, 2005, pp 255–266. 73. Rovelli, C. (1996) Relational Quantum Mechanics, in Int. Jour. Theor. Phys., 35, 1637-1678. 74. Rumer, Y. B., Ryvkin, M. S. (1980) Thermodynamics, Statistical Mechanics and Kinetics, MIR, Moscow. 75. Shaw, R. (1981) Strange attractors, chaotic behavior, and information flow, in Natursforsch, 36a, pp 80–112. 76. Siegelmann,H.T. (1998) Neural Network and Analog Computation: Beyond the Turing Limit, Birkhauser. 77. Smale, S. (1966) Structurally Stable Systems are Not Dense, in American Journal of Mathematics, 88, pp 491–496. 78. Stuart, C. I. J., Takahashi,Y., Umezawa, H. (1979) Mixed System Brain Dynamics: Neural Memory as a Macroscopic Ordered State, in Found. Phys., 9, 301-327. 79. Tegmark, M. (2007) The Mathematical Universe, e.print in ArXiv:gr- qc/0704.0646v. 80. Umezawa, H. (1993) Advanced Field Theory. Micro, Macro, and Thermal Physics, AIP. 81. Vitiello, G. (2002) Quantum Field Theory and System Theory, in Minati, G., Pessa, E. (eds), Emergence in Complex, Cognitive, Social, and Biological Systems, Kluwer/Plenum, pp 383-389. 82. Vitiello, G. (2001) My Double Unveiled. The dissipative Quantum Model of Brain, John Benjamins. 83. Von Neumann, J. (1935) letter to G. Birkhoff, Nov.13, in Proc. of Symp. of Pure Math., 2, AMS, 1961, 158. Emergence and Computation at the Edge of Classical and Quantum Systems 25

84. Wheeler, J.A. (1980) Pregeometry: Motivations and Prospects, in Marlow, A.R. (ed), Quantum Theory and Gravitation, Academic Press, pp 1–11. 85. Wolfram, S. (2002) A New Kind of Science, Wolfram Media. 86. Yoon-Ho K., Yu, R., Kulik, S.P., Shih, Y.H, Scully, M.O. (2000) A Delayed Choice Quantum Eraser in Phys.Rev.Lett., 84, pp 1–5. This page intentionally left blank 27

Gauge Generalized Principle for Complex Systems

Germano Resconi Catholic University, Via Trieste,17, Brescia, Italy [email protected]

In this paper we want to extend the traditional idea of the gauge to different parts of the physics. And also to complex system of agents and also to the biological systems. After the introduction of the fundamental definitions of the Gauge Generalized Principle , we move to the different domain as physics, agents , symmetry and so on to show the applicability of the gauge general principle.

Keywords: Complex Systems; Biological Systems; Gauge Generalized Principle PACS(2006): 89.75.k; 89.75.Fb; 87.10.+e; 11.10.z; 11.15.q

1. Compensation as a Prototypical Cognitive Operation The ﬁrst step toward an understanding of the general properties of the cognitive process “compensation” requires taking into consideration the mental path which leads from an explicit description of events taking place in the surrounding environment to an implicit description of events themselves, realised through the introduction of some invariance principle. To this regard , the conceptual structure, and the history itself, of classical mechanics is very instructive. As it is well known, the explicit description of motion of, let us say, a point particle, is given by Newton’s dynamic equations: F d2x = (1) m dt2 1 v x x0 That is the invariant form for the transformation − = 0 . 0 1 ct ct 0 1 1 β x x v For the Lorenz transformation − = 0 , β = . √1−β2 β 1 ct ct c − The (1) is false and d2x F is diﬀerent from . dt2 m 28 Germano Resconi

Now we want to restore the original invariant form (1) also when the velocity is near to the speed of light ( Lorenz transformation ) . To obtain this result we “compensate” the mechanical system by the change of the definition of the derivative operator in this way d d 1 1 β2 d 1 d D = = + [ − − ] = . dτ dt 1 β2 dt 1 β2 dt p− − So we have D2 = d ( d ) = 1 p d ( 1 d ) andpthe equation dτ dτ √1−β2 dt √1−β2 dt F 1 d 1 d = D2x = ( )x (2) m 1 β2 dt 1 β2 dt − − is the relativistic form of the Newtonp equation.p We remark that formally the (2) is equal to the (1). So the (2) is a “compensate” expression of (1). The (2) formally is the copy of (1). With the compensate principle we can use the Newton equation as a prototype theory for the relativistic phenomena. In conclusion from ordinary mechanics we can generate the relativistic mechanics without loss the previous knowledge but with use of the suitable compensation for which rules in classical mechanics is again true in the relativistic mechanics. Now we know that Newton’s equation (1) can be easily cast in the form of a conservative principle, such as m dx 2 ( k ) + U = const (3) 2 dt Xk provided that we suppose the existence of a function U such that ∂U Fi = . (4) −∂ xi Even if , from a formal point of view, (4) is perfectly equivalent to (1), its meaning and role in mechanics is deeper. Namely, by assuming as starting point a conservative principle like (4), we don’t obtain, as consequence, a single particular system of dynamic equation (1), but a whole class of such systems. Besides, it is easy to think of a generalisation of (4), by introducing a Hamiltonian function, which lets us deal with a class of mechanical phenomena much wider the one which could be studied by reasoning only to original Newton’s equations (1). In conclusion the (3) is true for a set of trajectories obtained by a set of homomorphism transformations C. The coherence condition in (3) does not take into consideration possible incoherence in the Newton low given Gauge Generalized Principle for Complex Systems 29 by (1), but more general incoherence related to the connections among all possible trajectories. We remember that for the relativistic case the Newton law is not true, but as we know we can again write the (3), the conservative law exists also for the relativity dynamics. In turn, this generalisation let us, via the methods of Calculus of Variations, to discover more powerful Extremum Principle lying behind the whole apparatus of analytical mechanics, such as the celebrated least action Hamilton Principle. In the Conservative Me- chanical Systems the canonical transformations change one trajectory to another trajectory that is again solution of the differential equation that represents the conservative condition, Hamiltonian differential equation. So if we consider natural all the trajectories that are solution of the Hamil- tonian equations, conservative system, the canonical transformation moves from one natural transformation to another natural transformation. The mechanical system is always close because we cannot consider any type of trajectories but only the natural trajectories. In the Extremum Principle we open the mechanical system to no natural mechanical trajectories. In this way all possible geometric transformation in the space-time can be considered. The Extremum Principle introduces another new transformation that changes the general transformation in the canonical transformation . In the previous cases It has to be underlined that here we deal with two different types of invariance: one is connected to the numerical value of a particular function such as, in the case of mechanics, the Hamiltonian one, and another is connected to the so called “form invariance” of the function appearing in the extremum principle ( in the case of mechanics, it is the Lagrangian action function). To each type of invariance is associated a particular type of transformations: these latter are the canonical transformations when we deal with the numerical value of the Hamiltonian, and generic global space time or dependent variable transformations, when we deal with form invariance of the action. Besides, to each type of invariance is associated a particular equivalence relation between different mechanical systems. In conclusion the fusion of the concept of Invariance and the Calculus of Variations generate a lot of new types of incoherence and also many different types of compensation. From the Newton law, to the Invariance and the Calculus of Variation there is a continuous movement of increase of generality and at the same time the logic structure of coherence and compensation become more and more complex. To describe completely all the logic structure from New- ton to the Calculus of Variation it is necessary a logic structure of five 30 Germano Resconi dimensions. The Calculus of Variation is at so high level that we can create the general relativity and also the quantum mechanics by this mathematical instrument. In this way we enlarged the framework originally adopted by Newton. Namely, in this latter case we had only one possible type of transformations, those given by the application of the time evolution operator to the initial data. In analytical mechanics, instead, we have two more transformations available on a same particular mechanical system: the one corresponding to a representational change as regards canonical variables, and the one corresponding to a reference frame change, or to a change of dependent variables. We must ask ourselves why the mental path which goes from dynamic equation (1) to Hamilton’s Principle was so advantageous for the progress of science knowledge that it is considered the prototype itself of generalization processes introduced in every domain of theoretical physics. The answer is that the existence of an Extremum Principle, besides the realization of an economy of thinking ( the implicit description it gives is much more compact than the explicit one offered by dynamical equations), gives us the possibility of reasoning about phenomena at higher level, where the details of the particular functional expressions appearing in (1) become irrelevant. In other words, we can discuss about general features of mechanical systems, prove universal theorems regarding them, being at the same time able to derive, through Euler–Lagrange equations, the consequences of these theorems within particular mechanical systems.

2. Sources as Compensate Term for Change of Fields At first sight, the logical structure of mechanics we sketched in the previous paragraph appears as the best one to capture both general and particular feature of dynamical behaviours. For this reason it could be seen as the prototype of every theoretical construction designed to link together both abstract and concrete aspects of phenomena taking place in a given domain. The only problems to be solved, in order to grant for an efficient operation of such a structure, seem to be of technical mathematical nature. However, behind this apparent perfection a serious drawback is hidden: the need for the introduction of external sources, as an indispensable tool for forecasting the details of dynamical evolution of mechanical systems. As regards traditional classical mechanics, where the only macroscopic field is substantially the gravitational field, without the introduction of sources we would obtain trivial models, unable to account for experimental Gauge Generalized Principle for Complex Systems 31 data, such as the one e.g. relative to planetary motion. The introduction by hand of suitable external sources is nothing but a process of adaptation of models to the state of fact observed in the external world. In other words, it is a measure of our ignorance in forecasting all possible phenomena from a global point of view when we lack for an explanation of an observed dynamical trajectory, we place an external source in a suitable space-time region to give us the possibility of reconciling the structure of our model with what we observed. For this reason the introduction of external sources into a theory, such a classical mechanics, cannot be considered as a true process of “compensation” for our ignorance, or lack of information, about the system we are studying, because it is not connected to some well-defined transformation between our old model and a new model. On the contrary, the introduction of external sources is nothing but an “adaptation”. Which leads to a model logically different from the one without external sources. Besides, this introduction is done in an essentially local way, without taking into account the true nature of the model and its transformations. To explain the “compensate” meaning of the external sources, we refer to M. Jessel before (1963) and after G. Resconi (1986) who have discovered the possibility to use the sources as adaptation at the real worlds. Given the equation of the mechanical waves ∂2ϕ 1 ∂2ϕ = S (5) ∂x2 − c2 ∂t2 where c is the velocity of the mechanical waves, ϕ(x,t) is the field of the waves and S are the sources that generate the field. The wave equation (5) can be decomposed in two differential equations of the first order in this way

c−2 ∂ϕ + ρ ∂η = Q ∂ t ∂ x (6) ρ ∂η + ∂ϕ = F ∂ t ∂ x where η is an auxiliary function, Q is a monopole source and F a dipole source (Jessel 1973) It is easy to prove that

S = ∂ F ∂ Q. (7) x − t Now we introduce the transformation T in the invariant form (4) or (5). So we obtain the new form

− ∂ (Tϕ) ∂ (Tη) c 2 + ρ = T Q ∂ t ∂ x (8) ∂ (Tη) ∂ (Tϕ)  ρ + = T F. ∂ t ∂ x  32 Germano Resconi

Now because the (8) can be written in this way c−2 ϕ ∂T + ρ η ∂T + T (c−2 ∂ϕ + ρ ∂η ) = T Q ∂ t ∂ x ∂ t ∂ x ρ η ∂T + ϕ ∂ T + T (ρ ∂η + ∂ ϕ ) = T F ∂ t ∂ x ∂ t ∂ x the (8) is true for all the transformations T where (9) is true c−2ϕ ∂ T + ρ η ∂ T = 0 ∂ t ∂ x (9) ρ η ∂ T + ϕ ∂ T = 0 ∂ t ∂ x when we multiply the first equation by ∂ T and the second by ∂ T we ∂t ∂ x obtain from the (9) the condition ∂ T 1 ∂ T ( )2 ( )2 = 0. (10) ∂ x − c2 ∂ t The differential equation (10), as we know gives the characteristic function of the wave equation, The field (ϕ ,η) does not change its structure after the transformation T. The transformation T for which (9) is true changes one solution of (5) in another solution of (5). So for T the fields are inside the same set of solutions where (5) is invariant. When (9) is not true to restore the invariance form (5) we change the sources in a way that ∂2T ϕ 1 ∂2T ϕ 2T ϕ = = T 2ϕ + S = S + S . (11) ∇ ∂x2 − c2 ∂t2 ∇ C C We will introduce the word “compensation” to denote every cognitive operation able to use an old model in a new domain, by maintaining the previous formal structure, except for some compensating terms. So we use the compensation mechanism to set up the fracture between two domains of research.

3. Quantum Mechanics and Compensation by Gauge Transformation Let us consider the simplest case in which each structure is associated to only one operator, denoted by T in the first structure and by T 0 in the second structure. Besides, let us denote by x a generic element belonging to the first structure and by ϕ the law of biunivocal correspondence between the objects of the two structures. Then, the two structures are isomorphous if Tx is the object corresponding, through ϕ , just to T 0(ϕ x). Therefore, an isomorphism between the two structures exists if and only if the following relation is satisfied (see Figure 1): Gauge Generalized Principle for Complex Systems 33

ϕ (T x) = T 0(ϕ x) , in turn implying: T 0 = ϕ T ϕ −1 .

Fig. 1. Graphical representation of an isomorphism.

As regards gauge transformations, the simplest (abelian) case assumes the form: ψ’ (x, t) = ψ (x, t) exp ( i ϕ ) where the symbols ψ (x, t ) and ψ’ (x, t ) denote, respectively, a (scalar) field and its gauge-transformed version, whereas ϕ is a space-time dependent phase and i denotes the imaginary unit. The importance of the gauge transformations case stems from the fact that, if the phase is space-time dependent, the compensation mechanism imposing invariance (isomorphism) with respect to gauge transformations gives rise to the introduction of the concept of force and, most generally, of interaction. In other words, the presence of an interaction compensates for the changes induced by a gauge transformation. A simple example is related to the behavior of a system in Quantum Mechanics, which is described through a wave function ψ, which is a solution for a dynamical equation prescribing its space-time evolution: the celebrated Schrödinger equation for a single particle (h = 1), ∂ψ 1 i = m ( i )2 ∂2 ψ + Uψ, ∂t 2 − j j X The wave function carries no physical meaning per se: only its square modulus is associated to a measurable quantity, interpreted as a probability. This implies that the wave function can be represented by a complex scalar, function of space-time coordinates, which has the form ψ = ρ eiθ, where ρ and θ are, respectively, its modulus and its phase. As only ρ has a physical meaning, we expect that the quantum-mechanical description of phenomena is invariant with respect to phase transformations, leaving ρ unaffected, and 34 Germano Resconi defined by ψ’ = ψ eiϕ. What happens if we let ϕ depend on space-time coordinates? It is easy to see that we will obtain a new equation no longer invariant with respect to phase transformations, that is:

∂ψ ∂ϕ i ∂t = e ∂t ψ = = 1 m ( i )2 ∂2ψ 2ie (∂ ϕ)∂ ψ + 2 − j j − j j j 2 2 2 ie ∂ nϕ P+ e (∂j ϕ)P ψ + Uψ. − j j j However, the differenceh Pbetween thePtwo equationsi o will disappear when we deal with a particle interacting with an electromagnetic field. In this case the Schrödinger equation takes the form 0 ∂ψ 1 0 0 0 i = m D D ψ + (eφ + U) ψ , ∂t 2 j j j X where 0 0 0 0 D ψ = i∂ ψ eA ψ j − j − j 0 in which Aj is the usual 3-dimensional vector potential and φ is the scalar 0 electric potential. If we define the transformed values of Aj and φ by: 0 Aj = Aj + (∂j ϕ ), φ = φ’ - (∂ϕ/ ∂ t), (sometimes called gauge transformations of second kind ) we will find that the Schrödinger equation for the transformed wave function ψ will assume the form: ∂ψ0 1 i = m D D ψ + (eφ + U) ψ, ∂t 2 j j i X where D ψ = i ∂ ψ eA ψ. j − j − j The generalized derivative Dj coincides with the old derivative of the classical physics plus a compensating term that is proportional to the electromagnetic potential. Associated to the local change of the phase, we have a new interpretation of the four-potential Aj . It appears as a compensating term whose role is that of keeping unchanged the previous formal Schrödinger structure. Moreover, the generalized derivative Dj is the analogous of the covariant derivative introduced in General Relativity, and is known as ∇λ ”gauge-covariant derivative”. Summarizing the previous arguments, we can conclude that the dynamical evolution equation (that is the Schrödinger equation) of a suitable basic field (substratum), described by a wave function ψ, is invariant in form with respect to local gauge transformations of the first kind if and only if it is Gauge Generalized Principle for Complex Systems 35 subjected to an electromagnetic interaction whose field potential transforms according to a gauge transformation of the second kind. In other words, this gauge transformation ”compensates” for the effects of the gauge transformation of the first kind on the substratum itself. As this ”compensating” transformation is possible only in presence of an electromagnetic field, we can say that this latter acts as a ”compensating” field. We thus have two equivalent representations of the dynamical evolutions of ψ. Alternatively, we can say that there is no electromagnetic field, but that the Schrödinger’s equation is not invariant with respect to gauge transformations of the first kind, or we may say that Schrödinger’s equation is invariant in form with respect to these latter, but that such an invariance is due to the presence of an electromagnetic field. Starting from the latter representation, it is possible to build a general theoretical apparatus for describing all physical interactions and for formulating unified theories of these interactions. As it is well known, such an approach was very successful in the last forty years and opened the way to a number of new exciting discoveries within the domains of particle physics and condensed matter physics (see, e.g. Frampton, 1987).

4. Compensation of Fields in the Physical Medium (Coupling Constant) Despite the successes obtained by such an approach, it needs a suitable generalization. Namely it can be applied only to ﬁxed-structure systems, that is to systems in which the number of possible interactions is ﬁxed in advance and the coupling parameters are to be considered as constants, determined once and for all, like the electron charge or the gravitational constant. On the other hand, in many physical and nonphysical complex systems we need a generalized description able to take into account the occurrence of a variable number of interactions and of space-time dependent coupling parameters. To bear witness to the interest of physicists in this kind of complex systems, we remind that in recent times, such a description was introduced in Quantum Field Theory under the name “Third Quantization” (Maslov and Shvedov, 1998). Whereas the expression “Second Quantization” refers to a situation in which we allow a nonconservation of particle number, associated to processes of particle creation or destruction, the expression 36 Germano Resconi

“Third Quantization” refers to a more general situation in which even the number and the kind of physical fields (that is of possible interactions) are allowed to vary. The practical implementation, however, of a Third Quantization procedure is very difficult, owing to the fact that we still lack a description of its classical counterpart, that is of a classical situation in which the kinds of physical fields can vary as a consequence of suitable interactions. From a general point of view, this would require a mathematical approach based on functionals instead of ordinary functions; however, even if someone tried to explore such a possibility (cfr. Kataoka and Kaneko, 2000, 2001), it still appears as implying considerable technical difficulties. To this regard, the simplest way to describe the “kind” of a physical field is to focus on its coupling constant with the other fields (e.g. with the substratum field as in the previous section). If this constant were no longer a constant, but a space-time dependent quantity, we had a possibility of introducing a simple description of a kind-variable field. Inspired by these developments and these considerations we then introduced a generalized gauge transformation: ψ’ = ψ e iqϕ where q is the (space-time dependent) “charge” of the gauge field (that is its “coupling constant”) and ϕ is the gauge field itself. In the particular case in which q is a function only of the “compensating field” Aµ we can develop the general compensating field in term of a Taylor series in Aµ, thus obtaining, at the first order, a gauge derivative of the form:

Dµ = ∂µ χµ,ρ Aρ ρ From the expression of such a gaugeXderivative we can obtain a generalized field equation describing the evolution of the compensating field Aµ. Such an equation is based on a number of logical steps (for the details see Mignani, Pessa & Resconi, 1999) which characterize all gauge theories which, in a sense, generalize Maxwell’s theory of electromagnetic field. These steps can shortly be summarized as follows: (1) definition of a generalized gauge derivative and whence of the field potentials; (2) computation of the commutator of the generalized gauge derivatives defined in the previous step; the obtained result lets us find the relationship between the field strength and the field potentials; (3) the Jacobi identity applied to the commutator gives the dynamical equations satisfied by field strength; Gauge Generalized Principle for Complex Systems 37

(4) the commutator between the generalized gauge derivative and the previous commutator (triple commutator) lets us define the field currents and find the connection between field strength and field currents.

When applying this procedure to our previous definition of generalized gauge derivative, we obtained, after a number of calculations, that the step (4) gave rise to the following complicated equation connecting the field potentials with the field currents:

A = J ν − ν where

J = [(∂µχδ )A (∂µχγ )A ] + [χδ (∂µA ) χγ (∂µA )] − ν µ ν δ − ν µ γ ν µ δ − µ ν γ +[2(∂µχδ )(∂ A ) (∂ χγ )(∂µA ) (∂µχγ )(∂ A )]. ν µ δ − ν µ γ − µ ν γ From such an equation it is possible to derive a number of consequences, the most important of which is that, in presence of space-time dependence of the “charge” (or “coupling constant”) of the gauge field, the compensating field undergoes a spontaneous spatial decay in crossing the physical medium in which the field itself is embedded. Such a circumstance follows from the fact that the previous expression of the field current contains terms which are directly proportional to the field potentials themselves. This means that the dynamical equation ruling the behavior of field potentials has solutions decaying to zero. Such an effect is analogous to the Meissner effect in superconductors and shows that in the case of variable-structure systems the variability of the interactions prevents from a complete transmission of the information (carried by the gauge field) within the system itself. In other terms, inner coherence opposes to the propagation of disturbances mediated by the interactions themselves and can preserve the stability of the system itself; related, in turn, just to the variability of the interactions.

5. Gravitational Maxwell Like Physical System and Compensation (Dark Matter and Energy) 5.1. Introduction Cosmic microwave background (CMB) temperature anisotropies have and will continue to revolutionize our understanding of cosmology. With the CMB Wayne Hu and Scott Dodelson [1] With the (CMB) had established a cosmological model for which a critical density universe consisting of 38 Germano Resconi mainly dark matter and dark energy. In this paper we present two complementary non-conservative gravitational theories. For these theories the energy and matter dynamically disappear into the dark matter and the dark energy and vice-versa the dark mater and energy appear as ordinary energy and matter. So the ordinary energy and matter are not conserved but are in communication with the dark matter and energy. The new approach to gravity includes the dark energy and matter as intrinsic part of the theory itself. The dark matter and energy are quantum gravitation physical phenomena described as cosmological constant. In the Universe the quantum wave function or the quantum field are not separate from the gravitational field. The interaction between Gravitation field and Quantum field in the vacuum is a source of the gravitational field. In the Einstein equations, the dark matter and energy are introduced by the cosmological constant as external elements without any theoretical justification. The geometric interpretation in G. Resconi [2] of the dark matter and energy came from a pure geometric extension of the Maxwell equations and of the Einstein equations to the quantum theories. The field approach in Ning Wu [3,4 ] represents the dark matter and dark energy in the ordinary gravitational field in a flat space with interaction with quantum field. The G. Resconi work and the Ning Wu work are gauge theories where the geodetic line is interpreted in two different ways. In G. Resconi the geodetic line is the same as in the Einstein theory. In Ning Wu the geodetic line is the trajectory of a massive body in a flat space as in the Newton theory. Ordinary matter and energy are elements that include ordinary gravitational phenomena and dark matter and energy include quantum phenomena. Quantum field and gravitational field are connected one with the other. Only total matter and energy that include dark and ordinary are conserved. We can experiment that ordinary matter disappear into the dark matter and that dark matter disappear into ordinary matter.

5.2. Maxwell Equation We know that the Maxwell equations in the tensor form are

F + F + F = 0 αβ/γ βγ/α γα/β (12) F = 4πJ αβ/β − α where

∂Aβ ∂Aα ∂Fαβ Fαβ = ( ) and Fαβ/γ = in the ﬂat space-time ∂xα − ∂xβ ∂xγ Gauge Generalized Principle for Complex Systems 39 and Aβ is the electromagnetic potential. Using the covariant derivative notation deﬁned by D = ∂ ieA (13) µ µ − µ we have the classic relation [D , D ] D D D D = ieF . (14) µ ν ≡ µ ν − ν µ − µν For the (14) we have F = F µν − νµ because

Fαβ/βα = F01/10 + F02/20 + F03/30 +

+F10/01 + F12/21 + F13/31 +

+F20/02 + F21/12 + F23/32 +

+F30/03 + F31/13 + F32/23 = 4π(j0/0 + j1/1 + j2/2 + j3/3) = 0.

The divergence of the current is equal to zero. We know that for the tensor form of the Maxwell equations, the Maxwell equations are invariant for every affine transformation. In particular the Maxwell equations are invariant for the Lorenz transformations. We remember that from pure electro- static field with a suitable Lorenz transformation, we can obtain a general electromagnetic field. For more general transformations, the Maxwell equations are not invariant. The aim of this paper is to rewrite the Maxwell equations so as to be invariant for any transformations of the space time.

5.3. Maxwell Scheme To write the extension of the Maxwell equations, we must deﬁne the derivative operator in a general way. If Ω is a general continuous transformation we have −1 Dγ = Ω∂γ Ω or DγΩ = Ω∂γ (15) where Dγ is the covariant derivative for the gauge transformation Ω The expression Dγ Ω = Ω∂γ can be written also in this way −1 Dγ = ∂γ + [Ω, ∂γ ]Ω . (16) 2 ie x A dx When Ω = e x1 µ µ we have for the (16) the ordinary covariant derivative (13). R 40 Germano Resconi

For any type of transformations Ω we assume that the expression (14) is true. In this case we have [D , D ] D D D D = ieF (17) µ ν ≡ µ ν − ν µ − µν where Fµν is the general form for any ﬁeld generated by the transformation Ω. For the commutative property

[Dµ, [Dη, Dν ]]ψ + [Dη, [Dν , Dµ]]ψ + [Dν , [Dµ, Dη]]ψ = 0 (18) the (7) can be written in this way

[Dµ, Fην ]ψ + [Dη, Fνµ]ψ + [Dν , Fµη]ψ = 0 (19) because

−1 −1 [Dγ , Fαβ]ψ = [∂γ + [Ω, ∂γ]Ω , Fαβ ]ψ = [∂γ , Fαβ]ψ + [[Ω, ∂γ ]Ω , Fαβ]ψ

∂Fαβ −1 = ( + [[Ω, ∂γ ]Ω , Fαβ])ψ (20) ∂xγ for ∂Ω = 0 we have [ Ω , ∂ ] ϕ = 0 and ∂xγ ∂xγ

∂Fαβ [Dγ, Fαβ ]ψ = ψ (21) ∂xγ when the covariant derivative is the (13) then

∂Fαβ [Dγ, Fαβ ]ψ = ψ (22) ∂xγ and the (8) is the same ﬁrst equation as in (12). For the (22) the equation

[Dγ , [Dα, Dβ]]ψ = [Dγ, Fαβ ]ψ = χJαβγψ (23) when (21) and (13) are true , χ = -4π and γ = β. The (23) becomes the second equation of the (12). In conclusion the Maxwell Scheme for gauge transformation Ω is [D , F ] + [D , F ] + [D , F ] = 0 γ αβ α βγ β γα (24) [D , F ]ψ = χJ ψ γ αβ αβγ where

−1 −1 Fαβ = [Dα, Dβ] and Dα = ∂α + [Ω, ∂α]Ω or Dα = Ω∂αΩ (25) and Jαβγ are the currents of the particles that generate the Gauge Field F. For the (24) the currents have this conservation rule

Jαβγ + Jβγα + Jγαβ = 0. (26) Gauge Generalized Principle for Complex Systems 41

5.4. Non-Conservative Equation of Gravity by Maxwell Scheme Given the aﬃne transformation Ω xη xµ = xµ(xη). → Given the ﬁeld φ ( Scalar Field ) and xη ∂φ we have → ∂xη ∂φ ∂φ ∂xη ∂φ ∂φ = = Ω(xη) . ∂x → ∂x ∂xµ ∂x ∂x η µ η η η X For the previous gauge transformation Ω we have ∂φ ∂φ DkΩ = Ω∂k ∂xη ∂xη and

∂φ ∂φ − ∂φ ∂φ ∂φ D = ∂ + [Ω, ∂ ]Ω 1 = ∂ + Γµ k ∂xµ k ∂xµ k ∂xµ k ∂xµ kj ∂xj µ where Γkj are the Cristoﬀel symbols and

∂φ j ∂φ Fk,h = [Dk, Dh] = Rihk ∂xi ∂xj j where Rihk is the Riemann tensor. In this case the Maxwell scheme gives us the set of equations

[Dγ, Rαβ] + [Dα, Rβγ] + [Dβ, Rγα] = 0 [D , R ]ψ = χJ ψ γ αβ αβγ that can be written in this way a a a Rbjk + Rbhk + Rbkj = 0 a a a DkRbjk + DjRbhk + DhRbkj = 0  a a  DhRijk Dµφ + RhjkDaDµφ = χJhjkDµφ. Where the firstequation is the well known Riemann tensor symmetry, the second is the Bianchi identity and the last is the equation for the Gravity Field. The current can be given by the form 1 J = D T D T (g D T g D T ) for which DkJ = 0 ijk i jk − j ik − 2 jk i − ik j ijk and Tjkis the energy-momentum tensor. When Dγφ = 0 the equation for h α the Gravity Field is D Rih = χJij for the expression of the current we have 1 R = χ(T g T ) ij ij − 2 ij 42 Germano Resconi that is the Einstein field equations. When Dγφ is different from zero the field equations are

D Ra D φ = χJ D φ Ra D D φ. h ijk a hjk a − hjk a a The new source of gravitational ﬁeld 2 a Daφ Λhij = R − hjk Daφ is a negative source comparable with the cosmological constant. In the general coordinate the cosmological constant is proportional to the second derivative of the quantum scalar ﬁeld φ and is proportional to the curvature of the space-time. For the Kline–Gordon equation we have

αβ 2 αβ 2 η ∂αφ∂βφ = m φ and in the general coordiante η DαφDβφ = m φ so the cosmological constant can be written in this way

2 a m φ Λhij = Rhjk . − ( ∂φ + Γα ∂φ ) ∂xa βγ ∂xα For weak and nearly static gravitational ﬁeld we have ∆G 1 ∂g G R 44 44 = 2 and Γα,44 ( ) with g44 1 2 − c ' 2 ∂xα ' − c and m2φ∆G m2φ∆G S S Λ = Q G ' c2 + ∂G ∂φ ' c2 c2 ∂xµ ∂xη

2 where G is the gravitational ﬁeld, SQ = m φ is the source of the quantum waves and SG are the sources of the scalar gravitational ﬁeld. Because

−11 3 −1 −2 −22 −3 −32 SG GN ρ = 6.673 10 m kg s 3 10 kg m = 1.8 10 , ≈ − − 1/c2 10 17 s2 m 2. ≈ Where ρ is the density of local halo, we have Λ 10−49 [m2φ ] that is ≈ − − comparable with the experimental value of Λ 2.853 10 51 h 2 m2 where ≈ 0 h 0.71, The cosmological constant is a physical effect of interaction 0 ≈ between the gravitational field and the scalar quantum field. The quantum gravity phenomena of the cosmological constant are represented by the black matter and energy. In [3] we show that the derivative coupling with the quantum field introduces an extra-mass term or black matter in the standard Schwarzschild Gauge Generalized Principle for Complex Systems 43 metric. The application of such a result to perihelion shift and light deflec- tion yields results comparable with those obtained in General Relativity. It is also shown that the non conservative theory of gravity implies a cosmological model with a locally varying, non zero black matter and energy or cosmological “constant”.

6. Network of Intelligent Agents and Compensation 6.1. Introduction Any problem solving can be modeled by actions or methods by which, from resources or data, one agent makes an action to obtain a result or arrive at a task. Network of actions can be used as a model of the behavior of the agents. Any sink in the network is a final goal or task. The other tasks are only intermediate tasks. Any source in the network is a primitive resource from which we can begin to obtain results or tasks. Cycles in the network are self-generated resources from the tasks. Now we denote agent at the first order any agent that can make one action or can run a method. Now we argue that there also exist agents at the second and at the more high order. Agents that copy the agents of the first order are agents of the second order. To copy one agent of the first order means to copy all the properties of one agent or part of the properties. This is similar to the offspring for animals. The network of resources, action, tasks in the new agent has the same properties or part of the properties of the original network. In the copy process is possible that the new agent has new properties which are not present in the prototype agent. This is similar to the genetic process of the cross over. Agent of the second order uses the prototype agent as reference to create new agent in which all the properties or part of the properties of the original agent are present. When all properties of the prototype network of agents are copied in the new network of agents, we have a symmetry between the prototype network of agents in the new network. When agents are permuted in the same network, agents can change their type of activities without losing the global properties of the network. The properties are invariant for the copy operation as permutation. We remember that also if two networks of agents have the same properties they are not equal. When in the copy process only part of the properties does not change, and new properties appear, in this case we say that we have a break of symmetry. For example in the animals in the clone process one cell is generated from another. The new cell has the same properties of the old cell. In this case we have symmetry among cells. In fact because any cell is considered as 44 Germano Resconi a network of internal agents (enzymes) two cells are in a symmetric position when the internal network of both the cells has the same properties. With the sexual copy process is possible that we lose properties or we generate new properties. In this case the cellular population assume or lose properties. We break the symmetry in the cellular population. Adaptation process can be considered as a copy process triggered by the environment. For example to play chess is a network of possible actions with resources and tasks. Any player is an agent of the second order which can change the network of the possible actions. The player can copy the schemes or network of actions located in the external environment in his mind. A physicist who makes model of the nature is a second order agent which makes copy of the agents network of actions in a physical nature into the symbolic domain of the mathematical expressions. Agents as ants can share resources from one field generated by other agents or ants. This field is a global memory that is used by the agents. For example ant pheromone field, generated by any ant, is used by all ants. In this way ants are guided to obtain their task (minimum path). In this case the pheromone field is an example of global memory resource. Network of connection among ants and its field is shown in the paper. Agents which take care of copying one network of agents in another network of agents are second order agents. Because we can copy also the network of agents of the second order by agents, these agents are at the third order. In this way agents of any order can control and adapt network of agents at a less order.

6.2. Agents and Symmetry (Coherence) Any agent [10, 11,12 ] at the first order can be considered as an entity which, within the domain of the resources, uses action to obtain a wanted range of tasks. Resources can be any type of information or entities as physical resources, functions, table of data or any type of information that is necessary to make the action and obtain the wanted task. Action is a method or a set of methods or in general any activity by which we solve our problem and obtain our task. In a symbolic way we can represent any simple agent at the first order by the figure 2. Now agent of the first order uses the resources, generates actions and obtains the tasks. Agents of the first order cannot generate other resources, cannot change the actions or have new tasks. Only agents of the second order can change the resources and the tasks and implement the suitable actions. To introduce agents of the second order we must extend the Gauge Generalized Principle for Complex Systems 45

Fig. 2. Resource, Action, Task of the agents or ﬁrst order. traditional paradigm of input/output structure in the ﬁgure 1. To built the second order of agents we use the theory described in the papers [1-8]. In this theory denoted General System Logical Theory, we begin with the graph

Fig. 3. Graph support of the agent coordinate ( coherent ) actions.

On the ﬁgure 3 we locate the resources and the task represented symbolically in this way: Resources Sk, tasks Th. The network of agents of the ﬁrst order, one for each arrow is

Fig. 4. Resources, Actions and tasks for a network of ﬁve collaborative agents of the ﬁrst order, one for any arrow.

In figure 4 agent whose action is Action1,2 and agent whose action Action1,4 share the same resources S1. In this case agents are dependent on the same resource. Again in figure 4 agents whose actions are Action2,3 and Action4,3 46 Germano Resconi collaborate to obtain the same task T3. One agent is also subordinate to another agent when he must wait the action of the other agent. For example the agent whose action is Action2,3 is subordinated to the agent whose action is Action1,2. the graph in figure 3 with the agents of the first order is an holonic system [12] and shows all possible connections among agent to obtain global task obtained by the fixed interconnection of resources and tasks. We remark that tasks for one agent can become resources for other agents. Cycles inside the network of agents are homeostatic systems which regenerate the initial resources after an interval of time. When the graph has no cycle the network of agents at the first order stops its process when obtain all the tasks without regeneration of the resources. Symmetry principle: In a point of the intelligent network the entity is a resource and at the same time can be also a task. So we have a fundamental symmetry principle “The task in one node is the same entity of the resource in the same node”. Agent of the second order is responsible for the allocation of the source, task and actions. Only the agent of the second order can change the tasks and resources inside the network of agents. Now we describe the possible changes obtained by the agent at the second order.. The first change is of this type The agent of the second order changes the tasks and the resources with the same transformation. In this way the new intelligent network preserves the symmetry principle. For example we have this change

Fig. 5. Transformation of the resources, tasks by agent of the second order with the preservation of the symmetry principle.

In figure 5 we show a possible change of tasks and sources where we have the same structure. The new graph obtained by the second order of agents is similar to the previous one but is not equal. Similar means that have the same properties. In fact the first network and the second network in figure 5 Gauge Generalized Principle for Complex Systems 47 have the same graph in common that is an invariant of the transformation. Because the graph is the same, all the properties of the first network are copied in the second network. For the relativistic principle, the new network in figure 5 can also be obtained by changing the graph but without changing the tasks and the resources. In figure 6 we show the same transformation as in figure 5 without changing the tasks and the resources.

Fig. 6. Dual representation of the transformation in ﬁgure 4 where the tasks and resources are in the same position but we change the graph.

Formally the transformation in ﬁgure 5 can be given by the operator P in this way

1 2 3 4 P = andP [Resources] = P [T asks]. 2 3 4 1 The transformation P by the agent of the second order in ﬁgure 5 can be represented in ﬁgure 7.

Fig. 7. Action P of the agent of the second order that copies one network in another.

In a symbolic way the transformation in ﬁgure 6 can be represented in this way. 48 Germano Resconi

Fig. 8. Symbolic representation of the transformation in ﬁgure 6.

The symbolic graph in ﬁgure 8 is denoted elementary logical system of the order two. See papers [1- 8]. Now the agent of the second order can generate more complex transformations of the resources and tasks. In fact the agent of the second order can collect tasks in clusters and consider all the tasks in the same cluster as equal to the same for the resources. In this case we have the transformation.

Fig. 9. Agent of the second order makes clusters of tasks and resources so as to have the same structure but with repetition.

In ﬁgure 9 the cluster approach generates a new graph which appears equal to the initial graph but with the property that the parts of the graph inside the clusters are equal, we cannot make any distinction (granulation).

Fig. 10. The agent on the same structure creates clusters of equal tasks and sources.

Formally the transformation in ﬁgure 10 can be given by the operator P in this way 1 2 3 4 P = and P [Resources] = P [T asks]. 1 1 3 3 The agent of the second order has another possibility to change the network of agents of the ﬁrst order. The possibility is to breach the symmetry between the resources and the tasks. Gauge Generalized Principle for Complex Systems 49

In this case formally we have one transformation for the tasks and another diﬀerent transformation for the resources. For example we have 1 2 3 4 1 2 3 4 P [Resources] = and Q[T asks] = . 2 3 4 1 4 1 2 3 So we have the transformation represented in ﬁgure 11.

Fig. 11. Agent of the second order allocates the sources and the tasks, in the same graph, but with diﬀerent operations P and Q.

We remark that in the same node of the graph the resources and the tasks have not the same index. In this case we break the fundamental principle of symmetry. The entity in the node is not one entity but there are two entities: one for the resources and the other for the tasks. In this way we lose the correspondence between node and entities. The break of symmetry symbolically is represented by the elementary logical system in ﬁgure 12.

Fig. 12. Symbolic representation of the transformations P and Q in ﬁgure 11.

We remark that in the transformation of the ﬁgure 10 when we locate only one entity in one node we have always a degree of conﬂict among entities (Resources, tasks). In fact for example when in the nodes we put the entity whose index is equal to the index of the resources we have the network. 50 Germano Resconi

Fig. 13. At any node we assign one entity whose index is equal to the index of the resources.

From figure 12 we see that we have for any node a degree of conflict that cab be shown in this way node node 1 node 2 node 3 node 4 . degree of conflict 1 2 1 1 2 3 2 3 To solve the conflict we have different possibilities. The first possibility is to define in the same node two entities and separate one from the other as we show in figure 14. When we separate the two entities (one for the tasks and the other for the resources) inside the same node in two different nodes as we show for example in figure 13.

Fig. 14. Separation of the entities in the transformation of the tasks and resources.

When we separate the entities in the same node, the network that we obtain is shown in figure 14 where the conflict is completely eliminated. Another way to solve the conflict without changing the original graph is to define a compensatory network whose arrows are inside the node in this way. The internal transformation moving from one entity to another inside any node is a compensation transformation which eliminates the original conflict. Symbolically represented in figures 16-17. Gauge Generalized Principle for Complex Systems 51

Fig. 15. Entities encapsulated in the same node are separate so as to restore the primitive symmetry between tasks and resources and eliminate any conﬂict.

Fig. 16. Separation of the entities and internal connection to solve the conﬂict.

6.3. Natural Example of the Coherence in Ant Minimum Path In this part we describe examples of Bio-inspired Algorithm by agents [15]. Given a fully connected graph G where n is the set of points and e is the set of connections between the points (a fully connected graph in the Euclidean space) see ﬁgure 18. The problem is ﬁnding a minimal length closed tour that visits each point at a time. The length of the path between points i and j

d = (x x )2 + (y y )2. i,j i − j i − j q 52 Germano Resconi

Fig. 17. Compensation transformation to solve the conﬂicts.

Fig. 18. Fully connected graph.

Initially all m ants are uniformly distributed in all points. Each ant is a simple agent with the following characteristics: (1) It chooses the point to go to with a probability that is function of the point distance and of the amount of trail present on the connecting edge; (2) When it completes a tour, it lays a substance called trail on each edge (i,j) visited. Let in document be the intensity of trail on edge (i,j) at time t. Each ant at time t chooses the next point, where it will be at time t+1. When time t + n each ant has completed a tour. Then the trail intensity is updated according to the following formula.

τi,j (t + n) = ρτi,j (t) + ∆τi,j (27) ρ is a coefficient such that ( 1 - ρ ) represents the evaporation of trail between time t and t + n, m k ∆τi,j = ∆τi,j (28) kX=1 k where ∆τi,j is the quantity per unit of length of trail substance (pheromone Gauge Generalized Principle for Complex Systems 53 in real ants) laid on edge (i,j) by the k-th ant between time t and t+n. The coefficient ρ must be set to a value <1 to avoid unlimited accumulation of trail and set initial intensity to be small positive constant. The field of trial substance can be represented in a space of the relations. Any point in this space is a connection edge. At any connection edge we assign the intensity of the trial substance generated by all the ants in the movement inside the graph.

Fig. 19. Field of trial substance inside the fully connected graph

The source, task and actions of the agent ants to generate the ﬁeld F is given by the simple graph in ﬁgure 17.

Fig. 20. Graph sources and tasks to generate the ﬁeld S.

In figure 19 we have the feedback by which any ant generates the field of the trial substance. The graph is repeated in the same way for all the edge of the full connected graph. In the graph 19 at the first node we have the source S1 that is the value of the trial substance that any ant in a particular edge can measure. With the value of S1 the ant p has the task 54 Germano Resconi

Tp to decide to move inside the edge. The task of the decision of the ants is given in this way. Given the visibility η = 1 . This quantity is not modiﬁed during the run of the i,j di,j Ant System, as opposed to the trail that instead changes according the previous formula (24). The normalized decision index from point i to point j for the k=th ant is

α β Di,j (t) = τi,j (t)ηi.j (29) where α and β are parameters which control the relative importance of trail versus visibility. When the ant decides to generate the Sp trial with the task T to change the field S from S the task T1 is to be maintain the field S in time in a way to be used from the other ants. The coherence of the ant system consists in the invariance of the graph in figure 19 with different assignment of the sources and task as we had discussed. Therefore the decision index is a trade-off between visibility (which says that close points should be chosen with high decision index, thus implementing greedy constructive heuristic) and trail intensity at time t (that says that if on edge (i,j) there has been a lot of traffic then it is highly desirable). An ant algorithm presents the following characteristics:

(1) It is a natural algorithm since it is based on the behavior of ants in establishing paths from their colony to feeding sources and back; (2) It is parallel and distributed since it concerns a population of agents moving simultaneously, independently and without a supervisor; (3) It is cooperative since each agent chooses a path on the basis of the information, pheromone trails, laid by the other agents which have previously selected the same path; this cooperative behavior is also autocatalytic, i.e. it provides a positive feedback, since the probability of an agent choosing a path increases with the number of agents that previously chose that path; (4) It is versatile that it can be applied to similar versions the same problem; for example, there is a straightforward extension from the traveling salesman problem (TSP) to the asymmetric traveling salesman problem (ATSP); (5) It is robust that it can be applied with minimal changes to other combinatorial optimization problems such as quadratic assignment problem (QAP) and the job-shop scheduling problem (JSP).

In the discrete case we have a graph of actions that connect diﬀerent entities. Gauge Generalized Principle for Complex Systems 55

Now for a continuous set of entities we have a ﬁeld of entities. The action in this case is an operator that connect the resource and the task ﬁelds. Immune System equation [13, 14].

df(ak) dt = mj,kaj ak + pkak j,k (30) f(x) = 1P ln( 1−x ) 2 − x See the behaviour of function f in ﬁgure 21.

Fig. 21. Behavior of the function f.

where gj,k can be positive or negative the same for pk. The f(a)k is the stimulus of the antibody and the ak are the concentration of the antibodies. Because we have

T T mj,kaj ak + pkak = A MA + P A . (31) Xj,k For the transformation A = B + C and (32) ak = bk + ck we have AT MA + P AT = (B + C)T M(B + C) + (B + C)T P (33) = BT MB + CT MB + BT MC + CT MC + BT P + CT P. Now for CT MB + BT MC + BT P = BT M T C + BT MC + BT P (34) = BT [(M T + M)C + P ] = 0 we have − C = (M T + M) 1P (35) − 56 Germano Resconi and the (3) becomes AT MA + P AT = BT MB + CT MC + CT P (36) = BT MB + CT MC + CT P = BT MB + CT (MC + P ). In this case we eliminate in the quadratic form the term of the ﬁrst degree. So we have

T T mj,kaj ak + pkak = B MB + C (MC + P ) = mj,kbj bk + hk. (37) Xj,k Xj,k So the (1) becomes df(b + c ) k k = m b b + h . (38) dt j,k j k k Xj,k Now the matrix M can be represented by a network among the cellular concentration bk.

Fig. 22. Relation among four diﬀerent types of antibody cells where 1, 2 , 3, 4 are the concentrations b1, b2 , b3, b4.

All the permutations of the types of the antibody give a similar network. The network 20 can be interpretd in this way. One antibody is considered the source of the interaction with another antibody or task antibody with an assigned affinity m. The same antibody can have two different functions. One antibody is the cause (source) of the change of the other antibody (task) but at the same time the same antibody can be the task for other antibody cells (sources). Every Immunity System can change its structure so as to have similar behavior given by the (9).The structure in figure 21 is the copy of the structure in figure 17. In the copy process we obtain two different structures for which the interaction among antibodies has the same properties even if the affinity m (relationship) among the antibody cells is different. Gauge Generalized Principle for Complex Systems 57

Fig. 23. We break the symmetry between the antibody cells. When one type of cells (task, in the first box) is under the influence of another antibody type (source), the same type (task) is not the source but we move to another antibody type (second box) to obtain the source of the positive or negative influence.

In figure 23 we break the ordinary symmetry for which the same antibody cell is under the action of other antibody and at the same time acts as source of influence on other antibody. In fact in figure 22 for the same structure one type of antibody cells are the task and another type of the antibody is the source of the positive or negative influence. So we can have the same structure but with change of the antibody type in the two function as task or source. We know that any parent has cell with half genome. Now in figure 24 we show four parts of the genome or genes connected one with the other by a network. We assume that in the genome the genes are not independent one from the other but there is a correlation between one and the other. The operator P is the ordinary mutation process for which we change the genes.

Fig. 24. Change of the genes (1, 2, 3, 4) of the genome in the source parent.

We fuse the source and the task parents and we have the children in ﬁgures 24-25 58 Germano Resconi

Fig. 25. Change of the genes (1, 2, 3, 4) in the genome of the task parent.

Fig. 26. Genome of the children obtained by the fusion or crossover of the source parent and task parent. We remark that even if the network is the same both for children and parents, due to symmetry breaking the children are diﬀerent from the parents.

7. Conclusion We show that agents are connected in an intelligent network where at any arrow we associate one agent at the first order whose work is to transform, by action, one resource in a goal or task. When the task is to change the network of agents of the first order, we introduce the agents of the second order. Agents at the second order see the agents of the first order in an holonic way [12, 16] as a totality. The agents of the first order can collaborate, can share common resources or can be subordinated one to the others. Agents of the second order have many different types of transformation so as to adapt the intelligent network to the global different tasks. A deeper connection with natural agents as Ant or Immune System or Genetic System is represented by the adaptation of the intelligent network by agents of the second orders. We can also define networks of agents of the second order whose tasks are networks and resources are networks. The adaptation of network of agents at the second order is realised by agents of the third order. We can continue in this rank of orders of agents to elaborate very high abstraction in the realisation of tasks by resources. Gauge Generalized Principle for Complex Systems 59

References 1. Penna M.P.,Eliano Pessa,G.Resconi,(1996,October) General System Logical Theory and its role in cognitive psychology, Paper presented at the Third European Congress on Systems Science Rome. 2. Resconi G.,G.V.Tzvetkova (1991, June) Simulation of Dynamic Behaviour of robot manipulators by General System Logical Theory, Presented at the 14-th International Symposium “Manufacturing and Robots”. 3. Resconi Germano, Gillian Hill (1996, October) The Language of General Systems Logical Theory: a Categorical View, Presented at the European Congress on Systems Science Rome. 4. C.Rattray,G.Resconi,G.Hill (1997, March) , GSLT and software Develop- ment Process, Presented at the Eleventh International Conference on Math- ematical and Computer Modelling and Scientific Computing, Georgetown University, Washington D.C. 5. G.A.Kazakov and G.Resconi (1994) Influenced Markovian Checking Pro- cesses ,General System Logical Theory, International Journal General Sys- tem 22, pp 277–296. 6. Saunders Mac Lane (1971) Categories for Working Mathematician, Springer-New York Heidelberg Berlino. 7. M.D..Mesarovich and Y.Takahara (1989) Abstract System Theory, In Lec- ture Notes in Control and Information Science 116 (Springer Verlag). 8. Resconi G. M.Jessel (1986) A General System Logical Theory, International Journal of General Systems 12, pp 159–182. 9. G.J. Marshall , A. Behrooz (1995)Adaptation Channels , Cybernetics and Systems 26, (3), pp 349–365. 10. Jacques Ferber (1999) , Multi – Agent Systems, Addison – Wesley. 11. Germano Resconi, Lakhmi C. Jain (2004) Intelligent Agents, Sprinter. 12. S.M.Deen (2003) Agent Based Manufacturing (Advance in Holonic Ap- proach), Springer. 13. P.M. Frank, B. Koppen-Seliger (1997) New developments using AI in fault diagnosis, Engineer Application Artificial Intelligence. 10 (1) pp 3–14. 14. G-C Luh, W.C. Cheng (2005) Immune model-based fault diagnosis, Math- ematics and Computing in simulation pp 515–539. 15. Stephan Olariu , Albert Zomaya (2006) Handbook of Bioinspired Algorithm and Application, Chapman & Hall/CRC. 16. V. Mahadevan , R.Braun and Z. Chaczko (2004) An exploratory study of network performance evaluation of telecollaboration environments, Science SCI Florida. 17. Wayne Hu and Scott Dodelson (2002) Cosmic Microwave Background Anisotropies, Annu.Rev.Astron.and Astrophys, pp 1–50. 18. R.Mignani , E.Pessa , and G.Resconi (1997) Non-Conservative Gravitational Equation, General Relativity and gravitation, 29, (8), pp 1049–1073. 19. Ning Wu, (2001) Commun.Theor.Phys. 36, pp 169–172. 20. Ning Wu,(2001) Gauge Theory of Gravity, September 21. 21. Frampton, P.H. (1987) Gauge Field Theories. Benjamin-Cummings, Menlo Park, CO. 60 Germano Resconi

22. Kataoka, N., and Kaneko, K. (2000) Functional Dynamics. I: Articulation Process, Physica D 138, pp 225–250. 23. Kataoka, N., and Kaneko, K. (2001) Functional Dynamics. II: Syntactic Structure, Physica D 149, pp 174–196. 24. Maslov, V. P., and Shvedov, O. Yu. (1998)Large-N Expansion as Semi- classical Approximation to the Third-Quantized Theory. Preprint hep- th/9805089. 25. Mignani, R., Pessa, E., and Resconi, G. (1999) Electromagnetic-Like Gen- eration of Uniﬁed-Gauge Theories, Physics Essays 12, pp 61–79. 61

Undoing Quantum Measurement: Novel Twists to the Physical Account of Time

Avshalom C. Elitzura and Shahar Dolevb a ISEM, Institute for Scientiﬁc Methodology, Palermo, Italy b Philosophy Department, Haifa University, Haifa, Israel a [email protected]

Since quantum measurement remains so ill-understood, its time-reversed version is bound to oﬀer new insights. If undoing a quantum measurement is impossible in principle, it bears on the old problem concerning the origin of irreversibility. If it is possible, it gives new twists to basic notions like indeterminism and the uncertainty principle. We demonstrate this approach on the time-reversed version of the EPR experiment, where two entangled particles undergo measurements followed by ”unmeasurements.” Whereas Bell’s theorem rules out local hidden variables, our experiment yields a complementary proof that even nonlocal hidden variables lead to observed violations of relativity. We are therefore left with either genuine randomness or a subtle time-asymmetry at the basis of quantum phenomena. Both options render the relativistic account of time incomplete. Genuine emergence and novelty turn out to be the most fundamental characteristics of physical reality.

Keywords: Unmeasurement, Quantum Reversibility; Time-Reversal; Quantum Erasure; Foundations Of Quantum Mechanics; Quantum Measurement Theory; Entanglement and Quantum Nonlocality PACS(2006): 01.70.+w; 03.65.Ta; 42.50.Xa; 03.65.Ud; 89.75.k

1. Introduction Quantum measurement, after more than hundred years of research, remains the main obstacle to a better understanding of quantum theory. What about the time-reversed process? Is it possible in the ﬁrst place? And whether the answer is positive or negative, what bearing has it on all the paradoxes besetting quantum mechanics? Merely raising the question makes us aware that quantum measurement is deeply related to a broader physical riddle, namely, irreversibility and time-symmetry in general. This gives further incentive to study the issue. This article communicates a work in progress. Some of its conclusions are 62 Avshalom C. Elitzur and Shahar Dolev tentative, hopefully to be made more rigorous in the future and stimulate further research.

2. The Physics of Time-Reversal Consider an elastic collision between two particles. It is possible to reverse the process by reversing the momenta of the particles after the collision, such that the entire process would evolve backwards, retrieving the initial state from the ﬁnal one. What about larger, macroscopic processes, involving myriad of particles? Mainstream physics holds that the question is merely technical. Given a suitable technology, so goes this majority opinion, every process can be reversed. No rigorous theoretical limit to reversibility has been pointed out so far.

3. Introducing ‘Unmeasurement’ The quantum version of the reversibility question is even more intriguing. Can a quantum measurement be time-reversed such that a measured particle will resume its initial superposed state? While opinions are more divided on this issue, it is obviously akin to the issue of time-reversal in general. A few interpretations of quantum mechanics invoke the notion of ”wave- function collapse,” an abrupt change of the measured particle’s state that is irreversible (see below). Other interpretations, however, avoid collapse, regarding measurement as reversible as any other process. The issue is not entirely divorced from present-day technology. A procedure known as ”quantum erasure” (Scully, Englert & Walther, 1991) pur- ports to time-reverse quantum measurement. In the commonest version of this method, a particle undergoes a double-slit experiment, but while it passes through the slits it is entangled with another particle. Measure- ment of the latter particle can reveal which path was taken by the former. Consequently, the original particle’s interference is destroyed. However, by subjecting the other particle to an appropriate measurement analogous to interference, each particle gives a ”key” that enables observing the other’s interference. A closer inspection, however, reveals that no real undoing of measurement takes place in this method because no real measurement was made in the first place. Rather, entanglement was employed, later to be countered (but not undone) by a more subtle measurement. Elitzur and Dolev (2001) proposed a different method of unmeasurement based on interaction-free measurement. They showed, for the first Undoing Quantum Measurement: Novel Twists to the Physical Account of Time 63 time, that just like measurement, unmeasurement too exerts a non-local effect in the EPR experiment. Another advance of their technique is that it is not subject to the No-Trace Restriction (see Section 5 below). However, the method works only for partial measurements, and its success becomes increasingly improbable as the measurement’s outcome approaches certainty. Other methods, still far from technical feasibility, have been studied (Peres, 1980; Greenberger & YaSin, 1989). They indicate that, just like earlier famous gedankenexperiments, deriving novel predictions gives an incentive to experimentalists to turn them into real experiments. Unmeasurement, then, is of enormous importance for theoretical physics. If quantum measurement turns out to be irreversible for some basic physical reason, then the age-old problem of the origin of irreversibility and time-asymmetry (Savitt, 1995) will get a fresh answer. If, on the other hand, quantum measurement turns out to be reversible, several surprising results would follow, bearing on other fundamental questions.

4. Collapse entails Fundamental Irreversibility What is measurement in the first please? Rather than implicating human observers, consciousness, etc., let us adopt Bohr’s simple definition ‘an irreversible act of amplification.’ The meaning of ‘irreversible’ naturally, will be the bone of contention. Next, a closely related term needs to be considered. ‘Collapse of the wave-function’ is a process that several interpretations of quantum mechanics invoke in order to explain the strange nature of measurement. It is important to realize that this collapse, if it indeed occurs, is irreversible in principle, regardless of technology. Consider a wave-function of a single photon being split by a beam- splitter, each half going to another detector. One detector clicks will the other remains silent. In other words, the particle is detected in one of its two possible locations and turns out not to be in the other. The collapse interpretation holds that, while the wave-function initially went to both locations, measurement has made it materialize in one and totally vanished from the other location. Can the entire process be time-reversed? As long as we are talking at the gedanken level, it is possible to make the detector that clicked eject back the particle it has detected. But what about the other detector? Obviously, it cannot eject back the ‘nothing’ that it did not absorb. Consequently, the wave-function will not return to its original state, failing, for example, to exhibit interference. Wave-function collapse, then, 64 Avshalom C. Elitzur and Shahar Dolev entails a fundamental form of irreversibility, perhaps the most fundamental one. So, does collapse occur? Our own belief is that it does, thereby providing the microscopic basis to all macroscopic manifestations of irreversibility. But of course, ‘belief’ is no substitute for a rigorous proof. Hence we shall henceforth study quantum measurement as if collapse does not occur. Rather, we shall follow hidden-variables theories and assume that some ‘guide wave’ always remains where the particle turns out not to be. It is such hidden variables that make the retrieval of the entire wave-function possible.

5. The Sacriﬁce: No Record of the Measurement’s Outcome must be Left Annoyingly, undoing a quantum measurement carries a peculiar price: Un- measurement necessitates complete elimination of any record of the measurement’s outcome, including the observer’s memory and knowledge. We must remain oblivious of the outcome that has been erased, as even the smallest trace of a measurement left unreversed would ruin the entire procedure. Is there any point, then, to considering quantum unmeasurement if the outcome to be undone must never be known? Fortunately, the answer is a resounding yes in the EPR case. We do not need to know the outcomes of the measurements that we seek to undo. Suﬃce it that we can see whether, following the unmeasurement, the two particles are entangled again.

6. Unmeasurement in the Context of the EPR Experiment The EPR experiment (Einstein, Podolsky & Rosen, 1935) gave rise to one of physics’ most famous paradoxes, in that it helped prove that quantum mechanics involves interactions that violate, albeit indirectly, the relativistic prohibition on velocities greater than light. Consider an atom with spin 0 ejecting two spin-1/2 particles that ﬂy far apart towards two measuring devices that measure whether each particle’s spin is 1/2 (“spin up,” ↑) or –1/2 (“spin down,” ↓). Conservation laws oblige the two spins to be opposite: ↑↓ or ↓↑. Is the correlation between the spins pre-established, having been ﬁxed when the particles left the atom? A celebrated proof by Bell (1964) ruled out just that: The correlation can be established only upon the particles undergoing measurement. Hence, each particle must somehow “know” Undoing Quantum Measurement: Novel Twists to the Physical Account of Time 65 which measurement the other particle is undergoing and what the outcome is, in order to maintain the appropriate correlations even when the two measurements are orthogonal. Now let unmeasurement be considered in this case. Straightforwardly:

Let two EPR particles, A and B, undergo measurements. Then let the measurements be time-reversed. Would the two particles resume their entangled state, being capable of manifesting again Bell-inequality violations upon consecutive measurements?

This, of course, is only a gedankenexperiment, as was initially the EPR itself. Present-day technology is far from capable of time-reversing any large-scale process, let alone the ill-understood quantum measurement. None of this should concern us here. We envisage an idealized, very small and perfectly isolated laboratory, within which the particle pair undergoes a pair of measurements and then the entire process is “undone” without inspecting the measurements’ outcome (see 4 below for the reason for this avoidance). Then let the two particles come out of the laboratory and be subjected to yet another pair of new measurements, this time to be fully inspected. Would these later measurements manifest Bell-inequality violations? The stakes, no doubt, are high. A negative answer would be nothing short of a theoretical proof that quantum measurement is inherently irreversible, likely the source of all larger-scale irreversibilities. What about a positive answer? Here, two assumptions need to be made in order to allow unmeasurement, and the result is far-reaching.

7. First Restriction: Reversibility entails Determinism It is almost a tautology that, for a process to be reversible, it must be strictly deterministic. To retrieve the process’s ﬁnal conditions from the initial ones, each and every one of the intermediate stages must strictly determine the next one. And when the process involves numerous molecules, the slightest deviation from determinism will result in the time-reversal’s failure (Elitzur & Dolev, 1999). For the EPR experiment to be reversible, therefore, we must assume, in line with the “hidden variable” theories, that the spin values yielded by the two measurements are not genuinely random but rather determined by causal factors, merely unknown yet objectively existent. 66 Avshalom C. Elitzur and Shahar Dolev

Apparently, our quest should end at this point with the desired conclusion. Has Bell’s theorem not ruled out hidden variables? Experimental tests of the theorem show that the two particles could not have their spins ﬁxed when they were emitted from the atom. Q.E.D.? No, because Bell’s theorem ruled out only local hidden variables, leaving it possible that nonlocal variables play a role (this was Bell’s own belief). In other words, the process may be deterministic as long as an event can exert its causal eﬀect on another spacelike-separated event, i.e. instantaneously. The stakes, then, are even higher than a quantum origin of irreversibility: If we rule out the exotic possibility, never challenged so far, that the particles’ spins are determined by nonlocal deterministic causes, we shall banish determinism even from its last resort in quantum mechanics.

8. Second Restriction: Quantum Determinism Entails Absolute Time We assume, then, that for quantum mechanics to be reversible it must obey some hidden determinism, albeit of a nonlocal type. This leads to the second and last restriction: Nonlocal hidden variables entail an absolute time parameter. Consider the presumed hidden factor, F↑ or F↓, that determines the spin direction. Where can such a factor reside? Bell’s theorem has banished it from its natural place, namely, the common source of the particles, from which it could locally determine the particles’ spins. It can therefore be present only in the measuring apparatus. Presumably, then, the presence of the factor F↑ or F↓ in the measuring advice causes the spin to be ↑ or ↓, respectively. Now, in the EPR case, the same factor may often happen to be present in both measurements of the two particles A and B. By angular momentum conservation, it is impossible for both spins to be ↑. We must therefore assume that the causal priority goes to the “earlier” measurement, the “later” being able only to amplify the “already” determined spin.a The quotes around “earlier,” “already” etc., remind us that the two EPR measurements are space-like separated, hence any temporal relations between aA bolder assumption, namely that one EPR measurement can causally affect the earlier one, would immediately give rise to causal loops. Indeed, the retrocausal models, (Sheehan, 2006 and references therein) assume that measurement affects the particle’s entire history backwards down to its emission, but these models are understandably very careful not to invoke any deterministic mechanism that would lead to such causal loops. Undoing Quantum Measurement: Novel Twists to the Physical Account of Time 67 them can exist only within an absolute reference frame. Indeed, an absolute time parameter (say, that of the entire Universe’s rest frame) is the inevitable consequence of any nonrelativistic theory that allows effects to propagate faster than light. With this minimum of two restrictions, let us proceed to the unmeasurement experiment.

9. Time-Reversing Nonlocal Hidden Variables enables Distinguishing Absolute Motion Let, then, two EPR particles A and B undergo spacelike separated measurements M1 and M2, respectively, with a small time interval between them (in the absolute time frame), such that M1 occurs slightly “before” M2. Suppose now that the presumed spin-determining factor F↑ happens to be present in both M1 and M2. As M1 is slightly “earlier” than M2, only M1 aﬀects A’s spin to be ↑, thereby forcing B to be ↓, in compliance with conservation laws. Consequently, M2 can only amplify the ↓ result imposed by M1, despite the presence of F↑ in M2 too. Next, let the entire experiment be time-reversed. Let two unmeasurements M 1 and M 2 be performed on the two particles. By our above second Restriction of Determinism, we assume that the spin-determining factor F↑ that resides in both measuring devices also takes part in the two unmeasurements. As M1 occurred before M2, M 2 must now occur before M 1. This, by deﬁnition, is true for every deterministic time-reversal. The proof for this requirement is simple, given in Appendix (1), Fig. 1 depicts the experiment. So far so good. Things do not go well, however, when the frame is a moving one, within which the experimental setup must obey Lorentz transformations. These transformations, in turn, clash with the absolute time- parameter which, by the above Absolute Time Restriction, is supposed to govern the nonlocal correlations. The clash arises as follows. Let the entire experiment – the spin measurements and their time-reversal – be carried out in some inertial frame moving at near-c velocity with respect to the observer. Fig. 2 reveals the crucial twist brought by this motion: The simultaneity plane t0 of the moving frame becomes slanted proportionately to the frame’s velocity, deviating from the absolute simultaneity plane t. Consequently, the time-interval between the spacelike-separated M1 and M2 merely dilates, but the M 2-M 1 interval is reversed. This will ruin the overall time-reversal of the EPR experiment. 68 Avshalom C. Elitzur and Shahar Dolev

Fig. 1. An EPR experiment undergoing time-reversal. On each particle’s world- line, the dotted segment represents the time interval during which the particle is superposed whereas the solid segment depicts its having a deﬁnite spin.

Conversely, if the reference frame moves to the opposite direction, the “unmeasurements” will maintain the same absolute time order as in the absolute time, but the measurements themselves will suffer the same order- exchange, leading again to the failure of time-reversal. One escape hypothesis, albeit far-fetched, may be proposed here instead of the Absolute Time Restriction: Perhaps, in a moving frame, the nonlocal hidden variables do not operate in absolute time but obey the Lorentz transformations too? This possibility has been considered long ago (Elitzur, 1996) and shown to clash with conservation laws (see Appendix 2). The clash arises when the moving frame reverses its motion during the time interval between the two EPR measurements: The presumed cause-effect relations between the measurements must reverse, enabling anomalous cases of ‘↑↑’ or ‘↓↓’ spins to occur, in defiance of angular momentum conservation.

10. The Alternative Left: Genuine Randomness What, then, have we achieved in the above analysis? Bell’s theorem ruled out local hidden variables. We have now proved that nonlocal hidden variables are equally impossible. One way out of the dilemma is oﬀered by the collapse theories: If there is something inherently irreversible to measurement, no distinction between Undoing Quantum Measurement: Novel Twists to the Physical Account of Time 69

Fig. 2. The same experiment in a moving frame.

rest and motion will be possible in the first place. In other words, perhaps unmeasurement, the very subject-matter of this paper, is simply impossible. But, once again, we have no decisive proof for this option, although we favor it. Let us, then, consider the alternative and show that it has equally intriguing consequences. We assume, then, that measurement is reversible. In this case, a measurement’s outcome is genuinely random, but a subtle time-asymmetry emerges. Quantum measurement is indeterministic in one direction and deterministic in the other: The transition from superposition to a definite value can lead to one out of several outcomes, but time-reversing each of these outcomes must lead to the same state of superposition. As far as we know, no observable reversibility can be derived from this asymmetry. Consider a process that contains some quantum spin measurements. Time-reversing the process ensures that the initial superposition states will be retrieved. Then, re-running the process must (by our above proof against nonlocal hidden variables) give rise to new spins, different from the earlier ones. Yet, by the No Record Restriction (Section 4), it is impossible to know what the initial spins were. A slight change, however, enables us to derive a surprising prediction that is certainly observable. 70 Avshalom C. Elitzur and Shahar Dolev

11. The Consequence: Unmeasurements are Immune to Non-Commutation Laws We have pointed out that genuine quantum randomness gives rise to a subtle asymmetry, in that measurement turns superposition into one out of many possible states randomly, while unmeasurement turns every possible state into superposition with certainty. Can this asymmetry be observed? Yes. All we have to do in order to observe it is to make the two EPR measurements noncommuting, say, subject one particle to measurement of its spin along the x direction while the other’s spin is measured along the y direction. Here, time plays an observable role: While the measurements of the spin directions are non-commuting, the unmeasurements do commute. The reason is simple. Consider a pair of EPR spin measurements, one of the x direction and one of the y direction. Bell’s theorem proves essentially that the result of each particle’s measurement depends on the choice of measurement to be performed on the other particle and its result. Hence, if x is performed before y, the following counterfactual holds: The outcomes would be diﬀerent have the measurements been made in a reverse order. This is a direct consequence of Bell’s theorem. It is equally clear, however, that the same cannot be said about the unmeasurements of x and y, otherwise it will be possible to distinguish between motion and rest as in the case of nonlocal hidden variables in the previous section. A subtle asymmetry between measurements and unmeasurements arises, which warrants an experimental test: If unmeasurements turn out to be possible, they will be immune to non-commutation relations.

12. Hawking’s Information-Loss Conjecture Supported Next we wish to point out a surprising consistency of our Genuine Ran- domness Conjecture with one of the most intriguing issues in present-day physics, namely, Hawking’s (1976, see Preskill, 1993) information-loss paradox. True, Hawking himself has eventually retracted from his claim, but several physicists, we among them, remain unconvinced. In essence, Hawking has shown that when a black hole swallows matter, the information associated with that matter is lost, and when the black hole evaporates, new information is created at its event horizon. This information seems to be genuinely new in that it appears to be unrelated to the matter swallowed earlier. Undoing Quantum Measurement: Novel Twists to the Physical Account of Time 71

But this is precisely what the Genuine Randomness Conjecture holds to take place every time a measurement is made! If a particle is ﬁrst measured, then unmeasured and then measured again, the outcome of the last measurement must not be determined by the ﬁrst one. This is uncannily similar to the process in which empty space gives rise to virtual particle- pairs that form and then mutually annihilate. When such a virtual pair becomes real,as in black hole evaporation, new information is created, not determined by any previous process. In mainstream physics today this result is regarded as disturbing and often dismissed as erroneous. Our analysis, in contrast, renders this genuine novelty, natural: New information is generated with every quantum measurement.

13. A New Twist to the Transactional Interpretation Finally, let us assess our ﬁndings in terms of one of the most consistent and appealing interpretations of quantum mechanics, namely, the transactional interpretation (Cramer, 1986). It has gained a widening acceptance during the last decade, not the least because of its simplicity, elegance and explanatory power (see, e.g. Sheehan, 2006). In the transactional interpretation, any quantum interaction is the result of two wave-functions, one retarded and the other advanced, going back and forth in time between the source and the absorber, eventually giving rise to a complete quantum exchange of a particle. EPR-Like eﬀects get a very fresh explanation this way: The back-and-forth wave-functions form a spacetime zigzag, connecting the two particles in the present through their common origin in the past. How, then, should an EPR experiment look like in this framework when the two particles undergo a pair of measurements and then a pair of unmeasurements? Again, the transactional interpretation faces the same choice as other interpretations: If it holds that unmeasurement is impossible, then it is pointing out the microscopic source of irreversibility. If, on the other hand, it allows unmeasurements, the resulting picture is very peculiar: An entire spacetime segment undergoes ‘revision’, such that, the entire history of the EPR pair is ‘re-written’. We have earlier (Elitzur & Dolev, 2006) derived a similar conclusion from the transactional interpretation with the aid of novel gedankenexperiments indicating that spacetime itself is subject to a subtler form of evolution, not yet accounted for in the present-day relativistic framework. 72 Avshalom C. Elitzur and Shahar Dolev

14. Summary What, then, has unmeasurement added to our understanding of quantum mechanics? With due modesty, quite a lot. Our analysis indicates that measurement may be fundamentally irreversible, hence unmeasurement probably cannot take place. Quantum mechanics, presently held to be a time- symmetric theory, may turn out to be the source of all asymmetries. Conversely, if unmeasurement is possible, measurement must be genuinely random. In order for unmeasurement not to lead to results that violate special relativity, the outcome of each and every measurement must not be determined by local hidden variables neither by non-local ones. The latter restriction is a novel result, making quantum randomness absolute. Every event of measurement must introduce a genuinely new bit of information into the universe. Which of the two alternatives eventually turns out to be correct? We do not know yet, but a more decisive argument will hopefully be found. As for the interpretation of quantum mechanics, we have shown that, within the transactional interpretation, unmeasurement indicates that two or more histories must be ‘written’ and then ‘re-written’ on top of one another on the same time segment. Spacetime itself must therefore be subject to evolution in some higher time parameter. This is very likely the time of Becoming (Elitzur & Dolev, 2005a, 2005b), so missed in present-day physics (Davies, 1995). It seems safe to conclude, therefore, that the famous ‘uneasy truce’ between relativity and quantum mechanics has never been uneasier. If there are hidden variables beneath the quantum level, then, by an earlier proof of ours (Elitzur & Dolev, 2005a), they must be ‘forever-hidden variables’ in order to never give rise to violations of relativity. But then, by the same reasoning that has lead Einstein to abolish the aether, they probably do not exist. Indeed, it has been proved long ago (Elitzur, 1992) that God must play dice in order to preserve relativistic locality. Hence, randomness, novelty and emergence, which for luminaries like Parmenides, Spinoza and Einstein were mere epiphenomena to be explained away, are likely the Uni- verse’s very mode of existence.

Appendix 1: The Order of the “Unmeasurements” must be Precisely Opposite to that of the Measurements The world-lines of particles A and B are as in Fig. 1: The dotted segments represent superposition while the solid segments represent definite Undoing Quantum Measurement: Novel Twists to the Physical Account of Time 73 spins. Each double-line represents the world-line of the measuring instrument that carries first the measurement, and then its undoing, on the same particle. On the left is a correct time reversal of the two measurements. Al- though the deterministic F↑ factor is present in both measurements, M2 fails to force particle B to have the ↑spin because of the priority of M1. During the time-reversal, whatever causal influence M2 had, it is undone by the time M1 occurs. Now read the whole experiment backwards in time: Every unmeasurement becomes a measurement and vice versa. Everything seems equally consistent. On the right is the same procedure without being strict with reversing the order of the unmeasurements. Here, the backward-reading of the experiment is clearly odd: Why should M 2, now turned into a measurement, fail to turn the superposed particle into a ↑, getting ↓ instead? The rest of the evolution is equally inconsistent.

Fig. 3.

Appendix 2: Relativistic Nonlocal Effects Lead to Conservation Laws Violations Consider the following hypothesis: The nonlocal effects observed in the EPR experiment are exerted in accordance with special relativity, that is, the simultaneity plane changes in accordance with the system’s motion. Next, consider an EPR experiment carried out in a frame that moves 74 Avshalom C. Elitzur and Shahar Dolev at a near-c velocity in one direction and then in the opposite direction. Let the two measurements be carried out with a slight time-interval, one before and one after the velocity reversal. In this case, a measurement that resided in the other measurement’s ‘future’ prior to the reversal will suddenly shift to the ‘past’. The presumed nonlocal effects would therefore fail to occur, giving rise to spin conservation violations.

Fig. 4.

References 1. Bell, J. S. (1964) On the Einstein-Podolsky-Rosen Paradox. Physics, 1, pp 195–200. 2. Cramer, J. G. (1986) The transactional interpretation of quantum mechanics. Rev. Mod. Phys. 58, pp 647–687. 3. Davies, P. C. W. (1995) About Time: Einstein’s Unﬁnished Revolution. New York: Simon & Schuster. 4. Einstein, A., Podolsky, B., & Rosen, N. (1935) Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777– 780. 5. Elitzur, A. C. (1992) Locality and indeterminism preserve the second law, Physics Letters A, 167, 335-340. 6. – (1996) Time anisotropy and quantum measurement: Clues for transcending the geometric picture of time, Astrophysics and Space Sciences (special issue), 244, 313-319. 7. Elitzur, A.C., & Dolev, S. (1999) Black hole evaporation entails an objective passage of time, Foundations of Physics Letters,12, pp 309–323. Undoing Quantum Measurement: Novel Twists to the Physical Account of Time 75

8. Elitzur, A.C., & Dolev, S. (2001) Nonlocal effects of partial measurements and quantum erasure, Phys. Rev. A 63, 062109-1—062109-14. 9. Elitzur, A.C., & Dolev, S. (2005a)Quantum phenomena within a new theory of time, In Elitzur, A.C., Dolev, S., & Kolenda, N. [Eds.] Quo Vadis Quantum Mechanics? , pp 325–350. New York: Springer. 10. Elitzur, A.C., & Dolev, S. (2005b) Becoming as a bridge between quantum mechanics and relativity, In Saniga, M., Buccheri, R., and Elitzur, A.C. (Editors) Endophysics, Time, Quantum and the Subjective, pp 589– 606. Singapore: World Scientific. 11. Elitzur, A.C., & Dolev, S. (2006) Multiple interaction-free measurement as a challenge to the transactional interpretation of quantum mechanics, In Sheehan, D. [Ed.] Frontiers of Time: Retrocausation – Experiment and Theory. AIP Conference Proceedings, 863, pp 27–44. 12. Greenberger, D. M. & YaSin A. (1989) “Haunted” measurements in quantum theory, Foundation of Physics, 19, pp 679–704. 13. Hawking, S. W. (1976) Breakdown of Predictability in Gravitational Col- lapse, Physical Review D, 14, pp 2460–2473. 14. Preskill, J. (1993) Do black hole destroy information?, International symposium on black holes, membranes, wormholes and superstrings. World Sci- entific, River Edge, NJ. 15. Peres, A. (1980) Can we undo quantum measurements?, Phys. Rev. D 22, pp 879–883. 16. Savitt, S. (1995) Time’s Arrow Today, Cambridge: Cambridge University Press. 17. Scully, M. O., Englert, B.-G., and Walther, H. (1991) Quantum optical tests of complementarity, Nature 351, 111–116. 18. Sheehan, D. [Ed.] (2006) Frontiers of Time: Retrocausation – Experiment and Theory, AIP Conference Proceedings , 863. This page intentionally left blank 77

Process Physics: Quantum Theories as Models of Complexity

Kirsty Kitto The School of Chemistry, Physics and Earth Sciences Flinders University, Adelaide, Australia kirsty.kitto@ﬂinders.edu.au

Originally based upon a pregeometric model of the Universe, Process Physics has now been formulated as far more general modeling paradigm that is capable of generating complex emergent behavior. This article discusses the original relational model of Process Physics and the emergent hierarchical structure that it generates, linking the reason for this emergence to the historical basis of the model in quantum ﬁeld theory. This historical connection is used to motivate a new interpretation of the general class of quantum theories as providing models of certain aspects of complex behavior. A summary of this new realistic interpretation of quantum theory is presented and some applications of this viewpoint to the description of complex emergent behavior are sketched out.

Keywords: Quantum Models; Quantum Information; Process Physics; Global Colour Model; Complex Systems PACS(2006): 03.65.w; 03.67.a; 03.70.1k; 89.75.k; 89.75.Fb

1. Process Physics The dominant modeling methodology of physics is static. We model the universe as a 3 + 1 dimensional pre-existing structure, where time shares a very similar status to that of space, despite our everyday experiences of the two phenomena as being remarkably different. Process Physics sprang out of an initial desire to explain from where such a structure could have emerged (i.e. to form a pregeometric model of the Universe). A particular aim was to explain the relationships between space, time and matter, especially with respect to how such phenomena could dynamically emerge from an underlying model. The original pregeometric modeling has proven to be, at least in its initial stages, rather successful,1 and the reader is encouraged to consult that reference for more details. This paper will take a different approach. It 78 Kirsty Kitto is becoming apparent that the dominant modeling methodology of physics is not isolated and this has led to a search for new approaches to modeling in general.2 This paper will discuss some of the work that has been performed during this search for a more process oriented approach to modeling. The standard modeling paradigm of science is dominated by reductionism. This technique, which involves breaking an apparently complex problem into smaller, more manageable pieces, reduction and then combining the solutions obtained from these smaller problems into a larger solution which represents the original system, synthesis, has been remarkably successful in science. However it appears likely that this technique is approaching the end of its general scientific validity.2–4 Many of the systems in which we are now interested cannot be so simply approached, a problem which has led to a general resurgence of holistic and systems oriented approaches.∗ There is a general pattern emerging from this work; most of these problems of analysis tend to appear when an attempt is made to apply reductive techniques inappropriately to systems that are often classified as complex in some manner that is yet to be adequately defined.5 Despite this problem of definition, a number of characteristics shared by such systems can be identified which make the appropriateness of their analysis by standard reductive approaches somewhat dubious. Such characteristics include: (1) Dynamic contingent behaviour indicative of a constant processing of information, energy, matter etc. This is in contrast to the generally very static models currently used where temporal processes are represented by mapping them to a real number line. Such a geometric approach to time tends to lose some of the most commonly experienced temporal phenomena, and has led to a number of well known problems in physics such as the lack of an arrow of time in our models, and the denial of a present moment.6 (2) Contextual responses to environmental perturbations, which mean that the system cannot be straightforwardly separated into a subsystem of interest and an environment that can be ignored.2 (3) Hierarchical organisation where generally stable subcomponents functioning on relatively fast timescales are incorporated (often nonaddi- tively) into larger structures operating more slowly.7,8 A process of feedback is often manifested too, where the higher level of the hierarchical structure so formed can exert causal control over the lower level components.9

∗See for example the list of references at http://pespmc1.vub.ac.be/EVOCOPUB.html Process Physics: Quantum Theories as Models of Complexity 79

(4) Far from equilibrium and dissipative behaviour,10 which means that traditional models based upon static equilibria and other standard thermodynamic concepts are not necessarily applicable. (5) The exhibition of long range order, which is stable in the face of perturbations.† This is an unusual feature, perturbations have traditionally been thought to disrupt long range order, and this is generally the case for classical systems, but not all systems are so unstable in such situations. For example, living organisms are not signiﬁcantly disrupted by incoming energy, instead they convert such energy into chemical form and transport it over far greater distances than are predicted by linear models, leading to the so-called “bioenergetic crisis”.11 (6) Relational structure and evolution. The traditional modeling of such systems as objects exhibiting a number of interactions between themselves tends to limit the emergent behaviour that can be exhibited by them.12 For example, in attempting to model the Universe using a number of fundamental objects we lose our ability to describe those very objects.

Interestingly, an analysis of the low level relational model of Process Physics (which will be discussed in the next section) reveals that it shares a number of these characteristics,6 a result that led to an in-depth analysis of the reasons why this might be the case.2 This paper will discuss some aspects of this journey of analysis. It will start with a quick review of the original low level relational model of Process Physics, illustrating the way in which it appears to be capable of generating interesting emergent behaviour. An analysis of the structures emergent from within this model will reveal that it is capable of dynamically generating apparently complex hierarchical behaviour. This will lead us to an examination of the historical roots of the model, which lie in quantum ﬁeld theory (QFT), and to the hypothesis that quantum theories are capable of describing a far wider class of system than is generally considered to be the case. In particular, the claim will be made that QFT is capable of describing complex emergent behaviour. The remainder of this work will consist of a brief survey of some future promising avenues of research that follow from such a conclusion.

†At least within the limits of the systems tolerances. Outside this range catastrophic failure generally results. 80 Kirsty Kitto

1.1. The Low Level Process Physics Model The initial low level model constructed within the Process Physics paradigm attempts to work around the static object driven methodology of reductionism by adopting a more relational approach. This is achieved by starting with a set of nodes for the sake of analysis, but instead of focussing upon the objects themselves, the connections between individual nodes are analysed. Thus, considering a set of N nodes, we assume that they are connected in some way, with a connection strength between node i and node j given by the real value Bij (see ﬁgure 1). We represent a set of these relational val-

node 1 2 3 4 5

1 1 0 1 1 0 0 B B12 13 2 −1 0 0 0 1 4 3 3 −1 0 0 0 1 2 = [Bij ] = B B25 4 0 0 −1 0 1 5 5 0 −1 1 −1 0

(a) (b)

Fig. 1. A simple relational structure of nodes and their connections. (a)Nodes i and j are considered connected if they have a non-zero Bij value. Arrows indicate the sign of the Bij value. (b) This structure of nodes and their connections can be represented in an antisymmetric matrix, zero values represent no connection between nodes i and j, while a nonzero value indicating a connection is represented here as ±1 depending upon the direction of the arrow drawn in (a). ues as a square antisymmetric‡ matrix B. Notice that while the pitfalls of object driven methodologies have been explicitly recognised above, we are still driven to talk in terms of nodes; it is very diﬃcult to leave an object- based methodology behind. This is due to the very nature of our analytic techniques which have always been based upon the reductionist methodology. However, any system simply positing such a priori objects cannot be considered fundamental when modeling the Universe, as the explanation of these objects must lie outside the system being modeled. A partial solution to this dilemma arises if we recognise that the nodes can in turn be deﬁned

‡ Antisymmetry ensures that Bii = 0 thus avoiding explicit node self connection. The internal structure of nodes will be incorporated shortly. Process Physics: Quantum Theories as Models of Complexity 81 for the purposes of the model as a system of nodes.§ It becomes apparent that all nodes can be thought of as composed of collections of nodes in turn, and in particular that the start up nodes can be viewed as names for subnetworks of relations. This result is ensured if the constructed system exhibits self-organised criticality (SOC), which enforces a fractal structure on the system.13,14 The SOC requirement is intrinsically linked with a new processing notion of time in this system. This is because the relational fractal structure is generated by a noisy non linear iterative map displaying SOC behavior. Thus in attempting to construct a model that does not postulate a priori fundamental objects, we ﬁnd the need to introduce a time-like process. In contrast, standard physics with its use of a priori objects is linked with the standard geometrical model of time and the associated problems mentioned above. The particular map used was suggested by the Global Colour Model of quark physics.15 Details of the history of this map will be discussed in section 4, but for now we simply assume that the B matrix representing this system of nodes updates discreetly according to the following iterative process:

−1 Bij → Bij − α(B + B )ij + ωij, (1) where i, j = 1, 2 ...,N and N → ∞. The constant α is an arbitrary tuning 16 parameter, while the term ωij represents an additive noise term, which provides a sense of openness to the system. At each iteration, the action of the noise term leads to the creation of new Bij links, incorporating a sense of innovation and contingency into the system. The noise term, when used iteratively in equation (1) is responsible for the notion of time that arises in the model. The dynamics are irreversible, with one particular past, which can be recorded as a history, but not relived. Future states of the system cannot be known. However certain sets of ensemble predictions can be made. In this sense a dynamical notion of time is captured by the system, with a markedly diﬀerent ontology from the static four dimensional spacetime of standard physics. This leads to the identiﬁcation of this system and its associated modeling techniques under the broad categorisation of Process Physics. We note for now that this modeling of time is far more appropriate in the modeling of living systems, providing a sense of contingency and dynamism that is

§B is assumed to be a very large (→ ∞) matrix. 82 Kirsty Kitto generally lacking from more standard approaches. The nonlinear matrix inversion term also performs a critical role in the system. It causes separate structures brought into existence by the noise term to link up, modeling a process of self-assembly. It is interesting to examine the dynamics of this process in detail. The system can be started with B ≈ 0 which represents the absence of any signiﬁcant relational information. Under successive iterations of equation (1) the B matrix assumes a sparse structure that can be organised into a block diagonal form. This suggests that there is a tendency for modes to organise into some sort of modular structure, a point that we shall return to in Section 2.1.

t iT D0 ≡ 1 T T T T T tT t D1 = 2 T T T T T t t tT t T D2 = 4 T T Tt D3 = 1 Fig. 2. A N = 8 spanning tree for a random graph (not shown) with L = 3. The distance distribution Dk is indicated for node i.

Assuming that the large ωij arise with ﬁxed but very small probability p, the geometry of the structures formed can be revealed by studying the probability distribution of minimal spanning graphs with Dk nodes and k links from an arbitrary node i where D0 ≡ 1 (see ﬁgure 2). This probability distribution is given by17

L−1 D1 Y P p i=1 D D D D P[D, L, N] ∝ (q j=0 j ) i+1 (1 − q i ) i+1 (2) D !D ! ...D ! 1 2 L i=1 where q = 1−p, N is the number of nodes and L is the maximum depth from node i. The most likely pattern can be found by numerically maximising P[D, L, N] for ﬁxed N with respect to L and Dk. This procedure has been performed,18 and the following results are from that analysis. Process Physics: Quantum Theories as Models of Complexity 83

Figure 3 shows the set of Dk (distance distribution) values obtained from one of these numerical experiments, where the log of the probability of a large noise value is set at log10 p = −6, and the number of nodes is ﬁxed at N = 5000. Also shown in the ﬁgure is a curve

d−1 Dk ∝ sin (πk/L) , (3) with best fit to the data when L = 40 and when the dimensionality of the fit, d = 3.16. This same curve is obtained from the surface area of a n- dimensional sphere. Figure 4 shows the range in d for fixed N = 5000 and varying p values. We see that for p below some critical value log10 p < −5, d ≈ 3.

250

200

150 k D 100

0 0 10 20 30 40 k

Fig. 3. The distance distribution Dk between nodes for a typical emergent structure under equation (1). This is obtained by numerically maximising P[D, L, N] where N = d−1 5000 and log10 p = −6. A curve is also drawn showing that Dk ∝ sin (πk/L), with best ﬁt d = 3.16 and L = 40.

This indicates that the connected nodes have a natural embedding in a S3 hypersphere, which is very suggestive of the 3-dimensionality of space. Thus this model goes some way towards predicting the observed three 84 Kirsty Kitto

4 d

-8 -7 -6 -5 -4 Log10p

Fig. 4. Range of d values for N = 5000 as a function of the probability p of large noise values. dimensional structure of space as an emergent phenomenon, a feature that is usually assumed in physics. Notice that the nodes are not exactly embeddable (which would require d = 3), there is a proportion of extra links. This is a key observation which forms the basis of a theory of matter as a quantum foam that is currently under development.1 We shall discuss the underlying ideas of this theory shortly. While they are not the only structures generated by the system, their maximum likelihood makes the hypersphere structures the most common. Splitting the large B matrix into its constituent independent submatrices Bsub we realise that each is almost singular (det(Btree) ≈ 0) but that the noise term ensures extra Bij terms which lead to a small valued determinant. This small valued determinant implies that the next iterative step of the system will lead to new large valued Bij entries upon inversion of the B matrix. Hence tree structures are sticky, at each iteration cross-links form between the structures which act to join them, and to produce larger structures. This behaviour has been examined in detail in a recent PhD thesis.16 Thus, under the influence of the iterative equation (1) the system can be seen to ‘grow’ with a steady increase in relational structure. As the Process Physics: Quantum Theories as Models of Complexity 85 tree structures stick together, they become less easily embeddable in a S3 structure. This phenomenon can be seen in figure 4, where beyond the critical value of d ≈ 3 the dimension of the structures quickly becomes very high; they are no longer embeddable in S3 and should instead be thought as defects in the system. These defect graphs gradually lose the ability to form new links, which tend to arise only at the level of ‘leaves’, or the endpoints of the graphs. As a result the defects become unable to sustain themselves and eventually they fade away from the system, or ‘die’. However, the system itself generally grows faster than it loses such structures (see Ref. 16 for more details of these dynamics). The nonlinear term in (1) is self-referencing; all elements of B that arise in a previous iterative step are required in the computation of the next value of each Bij element. Thus, this term can be seen to incorporate a weak notion of internal self-observation into the system. In particular, any node has the capacity to profoundly affect the rest of the system if it randomly receives a large ωij value at the next iteration of the map. The fact that one node is not ‘close’ to another in a particular S3 embedding does not mean that it cannot be affected strongly by it. This leads to a very strong form of contextual behavior in the system, no one element can be considered as isolated, in fact, at the next iterative step it may become strongly linked to a node which was previously not considered important to its dynamics.

2. Emergent Structures within the Process Physics Paradigm There is a sense of emergence in this system. Both the embeddable and the associated non embeddable defect structures which appear to be forming are not explicitly present in either the update equation (1) or the original very simple relational structure of nodes and connections. An examination of this claim within the framework of hierarchy theory7,8,19–23 helps to clarify the form of emergent behavior realized by this system.

2.1. The Hierarchical Nature of the Model An immediately apparent characteristic of this system is the block diagonal form of the matrix B which arises under the influence of the update equation (1). This block diagonal form also arises in Simon’s near-decomposable systems,22 making it possible to form an understanding of this system as a set of nearly-decomposable modules. This result makes sense. If nodes are 86 Kirsty Kitto seen as themselves made up of structures of connected nodes, then clearly some sort of modular hierarchical system has been created. It is also possible to understand the relational model as forming at least a 2nd order hyperstructure according to a theory of emergent hierarchical structure proposed by Baas.20 This framework depends fundamentally upon the notion of observation used in the identification of new, higher order structures. Thus, in order to register the emergence of new structures or entities within a system it is necessary to identify a mechanism capable of observing those entities. Given that this system is aimed at modeling the Universe, we might ask what such a mechanism could be. How could an observational mechanism external to the Universe be identified? Such a mechanism can be found through the adoption of what might be regarded as an external perspective, but this does not imply that an entity exists externally to the Universe, merely that observation generally has to be implemented at a different modeling level. This point has been recognised by a number of different researchers,24–26 many of whom claim that it is necessary to develop internal perspectives in order to understand complex phenomena such as the mind and life. The term endophysics24,27,28 has been coined to refer to the study of systems which have observers enclosed within them. Such a view is participatory; how we look determines what we see, and with the development of new internal perspectives we might begin to understand the way in which the context of such an observer can affect the observations they make of the system itself.2 Through a proper treatment of context we lose much of the confusion that often surrounds an examination of the Universe; an observer of the Universe can exist within that Universe. However, it will be necessary to adopt more than one model of the Universe in order to explain its complex emergent behaviour. We shall return to this point in Section 3. To construct a Baas hierarchy, we consider the original nodes used in the construction of the relational structure to be a very simple family of N 1 1 first-order structures Ξ = {Ξr : r = 1, 2,...N}, where a first-order observational mechanism O1 is defined as membership in the set Ξ1 . Now, under the influence of equation (1) which we represent as an update functional R, we find that these nodes join up, forming a second order structure:

2 1 1 Ξ = R(Ξi ,O ,Bij) i, j = 1, 2,...,N (4)

2 where Bij is the connection strength between nodes and Ξj designates the structure that appears in the B matrix in the previous sections. The family of second order structures consists of the maximum likelihood structures Process Physics: Quantum Theories as Models of Complexity 87 discussed above that are embeddable in the three-dimensional hypersphere

(ES3 ), along with the non-embeddable defect structures. Considering the space-type structures alone, we can deﬁne the property of being embeddable 3 in S ,(ES3 ) as a second order observationally emergent property,

P ∈ O2(Ξ2). (5) ES3

That is, this property is not present in the simple set of nodes that we started with,

P ∈/ O2(Ξ1), (6) ES3 rather, it becomes apparent under the influence of the update functional (1). The analysis of the remaining structure of this system is very complex and only in the preliminary stages, but it is expected that stable patterns identifiable as third order structures will emerge in the system. The full theory of this process is currently under development, but the general argument can be sketched out. We can somewhat artificially classify the structures in the system 3 as being either exactly embeddable in S (the ES3 structures), or non- embeddable structures which we will term defects (D). It is expected that the system contains stable defects, which we shall term Topological Defects (TD). The stability of these structures will be structural, and constitutes a new third order observation mechanism, O3, within the system. Thus it is expected that the stable TD’s will form a set of K new emergent struc- 3 3 tures Ξ = {Ξv : v = 1, 2,...K} within the system. Equation (1) ensures a constant updating of their components — rather like the units of an organism which are being constantly regenerated. These TD’s (if rigorously identified) would be third order structures in the hierarchy. That is

3 3 3 2 PTD ∈ O (Ξv), and PTD ∈/ O (Ξr) (7) because their emergent stability within the system is deﬁned by their structural stability, i.e. it constitutes a new observation mechanism. Thus, there is a sense of non-computational, or rather observational emergence (as deﬁned by Baas20) in this system. More than one mode of analysis is required in order to understand its behaviour. This suggests that this system is truly exhibiting some sort of very complex behaviour,12 a concept that will be discussed more in the following section. 88 Kirsty Kitto

So far, a third order hyperstructure has been identiﬁed (but not rigorously at the third order).

nodes > S3 + D > stable T D > . . . . (8) | {z } | {z } O2(Ξ2) O3(Ξ3)

It is expected that more structure will emerge from the system. This is because the identified stable structures of this model have a deep connection with the more standard object driven methodologies of particle physics. In Ref. 29 a link to the theory of preons¶ was discovered, and therefore it appears that these systems can recover much of the behavior of the Standard Model. Despite this promise, the analysis of this system from this bottom up level is extremely challenging, both computationally and analytically. This problem has made it necessary to attempt a more top down understanding of this system2 ; why does it generate such interesting emergent behavior? One set of clues as to why this simple model is behaving in such a manner can be found by referring to the discussion from section 1. Summa- rizing that discussion: this system was designed to be more relational than is normally the case, thus it avoids some of the most obvious traps of an object driven modeling approach; it exhibits hierarchical behavior; it has been designed to be far from equilibrium and dissipative; SOC behaviour is intrinsic to the model; there is a sense of long range order, even though most strongly connected nodes will tend to associate closely (i.e. embed well in a 3D hypersphere) the defects associated with this model allow for long range connections between nodes, and some of these are expected to exhibit stability due to their topological nature. Perhaps most importantly, this system exhibits contextual responses; the dynamical, process-type understanding of time in this model means that different runs of the system, while exhibiting similar structure formation, will always exhibit contingent “once-off” events that will depend strongly upon both the random noise terms that arise at each iteration, as well as upon the pre-existing values of the Bij. Thus, there is every reason to suppose that this system is exhibiting some sort of complex emergent behaviour.

¶Preons are one of the proposed replacement ‘fundamental particles’ which in bound states form quarks.30,31 Process Physics: Quantum Theories as Models of Complexity 89

3. High End Complexity The complex emergent behavior realized by this simple relational system is not generally displayed by models of complex systems. For example, it is generally accepted that models capable of generating open ended evolution, or more than two levels of hierarchical behavior have yet to be realized32 . Yet there is every reason to suppose that such behavior is being exhibited by this model (although this is very difficult to verify). Is the behavior of this model complex? We must be careful with our conclusions. There is an ongoing debate in the complex systems science community about what exactly complex behavior entails, and the search for a single unifying understanding of complexity may not in fact lead to one obvious definition.5,33 It has recently been proposed that complexity should itself be regarded as a complex concept, with complex systems exhibiting a range of different behaviors and being appropriately modeled by a number of different techniques and concepts12 , more details can be found there. In essence that discussion introduces a notion of high end complexity to describe those systems that cannot be modeled using our more standard reductive techniques; systems exhibiting high end complexity tend to require more than one mode of analysis for a complete understanding2,12 . Such systems also display highly contextual responses to perturbations, which often means that how we analyze them can determine what we see. This suggests that the scientific ideal of an objective reality immune from our observations does not exist. However, it does not imply that reality is subjective. Generally, the contextual responses of such systems make existing models of their generation from presumably simpler subcomponents unsatisfactory, and their dynamics and evolution once created is not usually well understood either. We require new theoretical tools capable of properly incorporating such contextual dependency into our models. Is it possible that such tools might result from the Process Physics approach? As an example of a system exhibiting high end complexity we might consider the process whereby a fertilised egg can divide, differentiate and eventually develop into a complete organism, specified by the information encoded in its DNA. It is important to realise that this specification is not sufficient for a complete description of the finished organism. Our understanding of nuclear transfer cloning illustrates the fact that DNA alone will not undergo the complex molecular processes associated with development, it must be placed in the context of a cell. Also, the DNA cannot be placed in any cell, it must be placed in an appropriate, as well as viable cell in order to begin replication. Thus, it is not possible to separate genetic content 90 Kirsty Kitto from its surrounds and retain any meaningful sense of a functional system. Computational models in, for example, the field of Artificial Life tend to emphasise DNA alone, and the systematic interdepencies of this process of development are not ususally considered. This tends to result in a model that is capable of exhibiting only very simplistic behavior. A full model of such a process would have to be capable of exhibiting contextual responses like phenotypic plasticity where organisms with the same genotype may, if placed in a different environment, reveal significantly different phenotypes, to the extent that they may even be identified as different species34 . Additional examples of high end complexity include:

(1) The formation and modeling of ecosystems,8 where, for example, differing understandings of the same system can be generated from an analysis of it at varying time and spatial scales. (2) Language evolution and structure is difficult to correctly model, despite what is often a very detailed knowledge of its syntax. For example, the process whereby meaning is generated from initially meaningless utterances is very difficult to understand,35 and postmodern and de- constructive theories suggest that we should be careful in developing theories of meaning that are too simplistic.36 (3) The impact that culture has upon mental well-being appears to be significant. For example, culture mediates nearly every aspect of schizophrenia, from its identification and definition to symptom formation, and even to the eventual course and outcome of disorders.37 This suggests that such conditions cannot be analysed reductively with an emphasis on biological factors or upon social factors alone. Rather, the complex interplay of these factors must be incorporated. This is a remarkably common problem, consider for example the nature versus nurture debate.38 (4) A more modern problem is the incorporation of a user into computing models, where different services might be required by different users from what is in essence the same system. Often, solutions are too hard- wired, and do not lend themselves to extension or adaption to new systems as they emerge.

It is interesting to notice that many of the systems exhibiting high end complexity have an evolutionary form. While we have a general understanding of the effect of the process of evolution in the biological realm, our formal and computational models of this process are sadly lacking. The fact that the more traditional modeling approaches have been unable to Process Physics: Quantum Theories as Models of Complexity 91 generate open ended evolution2 suggests that our current methodologies are somehow flawed when it comes to the generation of interesting complex structure and behavior. What is wrong with our models? A general model capable of generating novel hierarchical structure with different interactions and components at the different levels of understanding, as well as with a possibility for the open ended generation of emergent behavior would go a long way to answering this question. Apparently the simple low level relational model of Process Physics offers a first step in that direction, but how does it differ from the more standard modeling methodologies? A number of problems with our current modeling methodologies have been identified2,12 which, while not problematic when we consider some of the more traditional systems analysed by science, become almost impossible to circumvent when it comes to the modeling and the generation of systems exhibiting high end complexity.

These include a tendency to associate emergence with epistemology, or claim that new behavior is only evident with respect to an observer. This generates a feeling among a number of complex systems science workers that emergence is not a phenomenon per se, rather that it exists only in the mind of an observer. Although there is undeniably an aspect of observer dependence in our modeling of high end complexity, this does not imply that the phenomenon of emergence is necessarily dependent upon the observer; our models might depend upon the observer, but the system itself may exhibit genuine emergence. For example, according to the most generally accepted theories of, for example, biological evolution, the Earth did not originally house life of any form, but gradually living forms arose, mutated and changed, forming new organisms, species, phylogenies etc. This is a process of the dynamical emergence of complex behavior. While there are some borderline cases of species identification, it is generally agreed that species now exist which did not do so in the past, and this has a noticeable effect not just in our understanding of organisms, but upon the behavior of the biotic components of the Earth’s environment. Similar processes can be found in the evolution of linguistic ability, religion, and the generation of new technology. We require models capable of describing this dynamical process whereby new structures and interactions can come into existence. The barrier of objects consists of the assumption that our models must necessarily describe things, which effectively rules out our ability to 92 Kirsty Kitto

generate novelty. For example, in the standard agent based approaches there is a tendency to define an agent as an object capable of undergoing some number of interactions with its environment. However, this means that rather than generating novelty as agents move around some already defined environment, their modes of interaction are effectively used up, resulting in a point where no new behavior is forthcoming. The barrier of objectivity arises when we assume that objectivity is a core criterion of science despite a remarkable number of indications to the contrary which have been arising, often for centuries, in the generally ‘softer’ sciences, such as sociology, anthropology and history. Indeed, that last bastion of reductionism and objective measurements, physics, seems to be falling under a similar curse as quantum systems appear to be rather systematically revealing a tendency to display different outcomes, each depending upon different choices of experimental arrangement.

All of these barriers are discussed more fully in a recent paper,12 which examines the way in which they impact upon our ability to understand and to generate high end complexity in our models, before proposing some possible resolutions to such barriers of modeling. When we reconsider the model discussed in Section 1.1, we see that it appears to be circumventing some of these problems: the Universe is taken to be real; a shift is attempted from more traditional object-based approaches to a more relational framework; and all analyses of higher level structures appear to be dependent in some way upon an observer (who sets thresholds in the low level relational model and then makes secondary models in order to characterise emergent behaviour within the system). However, none of these characteristics was actively sought when the initial model was investigated39 , rather they have been identiﬁed later when an attempt was made to understand the reasons for the success of the model2 . Thus, we might ask at this stage how the historical basis of Process Physics diﬀers from that of the more traditional reductive approaches. Can this basis form a foundation for a new modeling methodology capable of both modeling and generating complex emergent behavior?

4. The Historical Roots of Process Physics The fact that the very minimal relational system discussed in Section 1.1 both exhibits observational emergence and generates a contextually dependent hierarchical structure suggests that it may have some sort of Process Physics: Quantum Theories as Models of Complexity 93 fundamental characteristic that could be used in a more general modeling of complex emergent behavior. In order to fully explore this possibility we will look at the historical roots of the model. The iterative map (1) was suggested by the Global Colour Model (GCM) of quark physics.15,40 This model uses the functional integral method, and approximates low energy hadronic behaviour from the underlying QFT of quark-gluon behaviour, quantum chromodynamics (QCD). This section will discuss the way in which the iterative map (1) was extracted from from the GCM.

4.1. The Global Colour Model (GCM) and the Functional Integral Calculus (FIC) QCD is the quantum field theoretic model of quarks and gluons. Starting from six quark fields,q ¯, six antiquark fields, q, and eight gluon fields, A, a classical action can be defined (in the Euclidean metric) as Z µ 1 1 S [¯q, q, Aa ] = d4x F a F αβ+ (∂ Aa )2 QCD µ 4 αβ a 2ξ µ µ ¶ ³ λa ´ +¯q γ (∂ − ig Aa ) + M q (9) µ µ 2 µ

a a a abc b c where Fµν = ∂µAν −∂ν Aµ +gf AµAν describes the self-interaction of the gluons (f abc are the QCD structure constants, and g is the colour charge), λa 2 are the eight SU(3) colour generators in the Gell–Mann representation, γµ are the Dirac matrices, and M = {mu, md,... } are the quark current masses. This action can be quantized via a QCD generating functional Z

Z = Dq¯DqDA exp (−SQCD[A, q,¯ q]) (10) from which correlators of the general form R DqDq¯DA . . . x . . . e−SQCD [A,q,q¯ ] G(. . . , x, . . . ) = R (11) DqDq¯DAe−SQCD [A,q,q¯ ] can be extracted. These are then used to derive the predictions of QCD, however this procedure is very complicated and for this reason a number of approximations have been developed. The GCM15,40 is a low energy approximation which is used to extract the behaviour of hadrons from QCD. The model is motivated by the observation that at low energy (or long wavelength) we observe only those degrees of freedom associated with hadron behaviour; individual quarks 94 Kirsty Kitto and gluons are not directly observable. This suggests that a reasonable approximation to low energy behaviour could be achieved by choosing a set of variables which reﬂect only the behaviour that we can observe in an experimental setting. The choice of new variables is not arbitrary, it is achieved using what are termed Functional Integral Calculus techniques41–43 which change variables in a dynamically determined way using analogues of the various techniques used in ordinary integral calculus. This process, which has been termed action sequencing39 changes the variables of the QCD generating functional from the very simple expression in (10) to one that describes the hadron via a sequence of steps: Z

Z = Dq¯DqDAexp (−SQCD[A, q,¯ q]) (QCD) (12) Z

≈ Dq¯DqDAexp (−SGCM [A, q,¯ q]) (GCM) (13) Z ∗ ∗ = DBDDDD exp (−Sbl[B,D,D ]) (bilocal ﬁelds) (14) Z ¡ ¢ = DN¯DN... DπDρDω..exp −Shad[N,¯ N, .., π, ρ, ω, ..] . (15)

We might consider FIC as the ﬁrst documented implementation of action sequencing. The details of this procedure are very complicated, the interested reader is referred to15,40 for further information. As an illustration of the power of this approach, we can examine the hadronic action obtained at the end of the complete action sequencing process (to low order) as,

Shad [ N,¯ N, . . . , π, ρ, ω, . . . ] = Z n ³ √ ´ o 4 a a d xT r N¯ γ.∂ + m0 + ∆m0 − m0 2iγ5π T + ... N Z · f 2 f 2 + d4x π [(∂ π)2 + m2 π2] + ρ [−ρ (−∂2)ρ + (∂ ρ )2 + m2ρ2 ] 2 µ π 2 µ µ µ µ ρ µ f 2 + ω [ρ → ω] − f f 2g ρ .π × ∂ π − if f 3² ω ∂ π.∂ π × ∂ π 2 ρ π ρππ µ µ ω π µνστ µ ν σ τ −if f f G ² ω ∂ ρ .∂ π ω ρ π ωρπ µνστ µ ν σ τ ¸ λi + ² T r (π.F ∂ π.F ∂ π.F ∂ π.F ∂ π.F ) + ... . (16) 80π2 µνστ µ ν σ τ We shall not list the new notation used in this action; details can be found in the review article15 or PhD thesis.40 However, even without the details of the notation, when we compare this expression to that of the QCD action (9) we see a vast diﬀerence in the complexity of the two equations. Equation (16), which is is only a low order expansion suggests that there is Process Physics: Quantum Theories as Models of Complexity 95 a very rich set of behavior evident in the behavior of the nucleon; it is an extremely complex system, but its description is well approximated by this theory. Thus, some very complicated emergent behavior has been extracted using this technique. Is it possible that this technique could be generalised in some way? A ﬁrst step towards answering this question was taken when the iterative equation (1) was extracted from this model.

4.2. The Extraction of the Iterator Equation The richness of the behavior exhibited by this model of hadrons led to the hypothesis that it may be possible to regain much of the dynamics in physics without incorporating it into our models axiomatically. We take as our starting point an action that arises in the action sequencing of the generating functional (in equation (14)). This action describes the dynamics of bilocal fields B (which can be considered very similar to the connections between nodes in (1)) and appeared in an earlyk GCM paper41 · ¸ Z M θ B(x, y)B(y, x) S[B] = −T rLn /∂δ(x − y) + Bθ(x − y) + d4xd4y , 2 2g2D(x − y) (17) where /∂δ(x − y) represents an essentially localized momentum term, the M θ term is due to a bilocal meson field, Bθ(x − y) is a hermitian bilocal field and D(x − y) is a complex bilocal field. See the early paper for details, or the more modern and very comprehensive PhD thesis.40 The stochastic quantisation (SQ) procedure of Parisi and Wu44,45 is used to quantize this action. SQ consists of introducing a 5th time τ in addition to the usual 4 space-time points xµ and postulating that the dynamics of some field represented by φ in this extra time τ is given by the Langevin equation: ∂φ(x, τ) δS[φ] = − + η(x, τ), (18) ∂τ δφ where η is a Gaussian random variable. The stochastic average of all fields φn satisfying equation (18) is calculated where τ1 = τ2 = ··· = τl, and kThis paper was an early attempt at using FIC to bosonize QCD but made use of a less appropriate change of variables than the later work utilising this technique. The change of variables used here was to 1c and 8c bilocalqq ¯ variables which worked well in the extraction of meson observables but was less physical, due to the 8c fields which are repulsive for forqq ¯ states. The later work was based upon 1c meson variables coupled 42 with 3¯c and 3c diquark variables. The less physical equation was used as a basis for the derivation because it is slightly simpler than its equivalent in the later work but still results in rich behaviour. 96 Kirsty Kitto

finally the limit τ1 → ∞ is taken. Parisi and Wu proved perturbatively that in this limit the average is equal to the correlation function of the field of interest, i.e., an l-point correlation can be defined where R Dφ φ(x )φ(x )..φ(x )e−S[φ] lim hφ (x , τ )φ (x , τ )..φ (x , τ )i = R1 2 l . η 1 1 η 2 1 η l 1 η −S[φ] τ1→∞ Dφe (19) Thus, at least in the perturbative sense, SQ provides an alternative route to quantization, and in particular to the correlation functions of QFT. We make use of this procedure to quantize the action (17), substituting it into the Langevin equation (18) to obtain ∂B(xµ, τ) δS[B] = − + η(xµ, τ). (20) ∂τ δB Thus, the B field will dynamically update in this time parameter as B → B + δB, that is: δS[B] B −→ B − + η(xµ, τ). (21) δB Returning to the action (17), we make a number of simplifications which strip away structure considered unimportant to the model. Namely we set any variables that are deemed irrelevant equal to 1, drop the gluon interaction term D(x−y) and set the variables of the B terms in the integral term as equivalent (i.e., assuming no significant interaction between the fields at this level of the description). The term /∂δ(x − y) represents an essentially localised momentum term and is ignored. Similarly the trace operation is discarded as it simplifies too much of the behaviour of a system that has already been dramatically simplified. This results in a simpler action: Z S[B] = Ln [B] + d4xd4yB(x, y)2. (22)

Now, making a transition to a matrix based lattice representation in place of the continuous representation above, we obtain 2 S[Bij] = Bij + Ln [Bij] . (23) Substituting (23) into (21), and changing the symbol η to ω, an equivalent noise term which operates over the matrix representation of the B ﬁeld results in an update equation: µ ¶ δ 2 δ Bij → Bij − Bij + Ln [Bij] + ωij . (24) δBij δBij Finally, we make use of the calculus identities δB2 → B and δlnB → B−1 to extract the structure of the desired equation(1). Process Physics: Quantum Theories as Models of Complexity 97

It is worth emphasising that the use of the stochastic quantization procedure in this process suggests that a system emergent from this equation should exhibit quantum behaviour in the limit of sufficient numerical experiments that run for a long enough time. Note also that this is more of an extraction than a derivation; there is almost a sense of observation in the derivation itself, where we choose which variables to consider relevant and which to ignore. It is remarkable how much richness of behavior is retained in the behavior of (1), this is despite the dramatic loss of standard physical structure that is characteristic in this derivation. In this case it appears that a set of key characteristics are driving this behavior. Some of these include the nonlinear aspect of the equation and its grossly ‘nonlocal’ and holistic form, as well as the noise term that pushes its behavior away from equilibrium. Most important perhaps is the structure of the general equation, it is close enough to the bilocal action that it incorporates its key physical properties in some way. Indeed, it has been argued2 that the key structure remaining after this extraction is related to the Nambu–Goldstone (NG) modes30 that are used in the action sequencing of the QCD action. Indeed, the NG-modes form the basis of the choice of physical variables in this process.15 The GCM is particularly effective in revealing the NG phenomena that follows from the dynamical breaking of chiral symmetry.30 Indeed, the GCM results in the complete derivation of the Chiral Perturbation Theory phenomenology, but with the added feature that the induced NG effective action is nonlocal, so that the usual non-renormalisability problems do not arise. It is highly likely that NG-modes are particularly important in the modeling of complex emergent behaviour,2 a point that we shall return to shortly. It is expected that there will be a class of equations all of which exhibit behaviour similar to that discussed in section 1.1, but at this point in time (1) is the only one known. There is some reason to suppose that such a class of models is already being discovered in the field of complex systems science, consider for example a very simple network model on the rise and fall of societies46 that behaves very similarly to (1). Is it possible that a class of equations such as (1) can be used in the modeling of complex emergent behaviour? What can be determined about such a general class of models?

5. Quantum Theories as Models of Complexity It is very likely that an essential characteristic of such a class of models will lie in in quantum theory. This section will brieﬂy discuss this link 98 Kirsty Kitto between quantum theories and models of complex behaviour, a more in- depth discussion is currently in preparation. All quantum theories have the same fundamental structure. Indeed the implementation of any quantum theory involves following the same general recipe:

(1) First, a map from a classical state space, S to a complex number C is found. This map is often written in the form of the symbol ψ. (2) Depending upon the system under examination some time evolution equation is chosen from a set of possibilities including the Schrödinger equation, the Klein-Gordon equation, the Dirac equation etc., each of which map to another set of complex numbers. (3) Steps 1 and 2 are mathematically well defined and understood. How- ever, complex numbers are never revealed when a measurement is performed on a quantum system. Instead, the system is found to be in some classical state, related to the configuration of the experiment per- formed47,48 . The dynamics of this process are not understood, and there are a number of competing theories of quantum measurement, each of which lead to a different interpretation of quantum mechan- ics47,49 . Mathematically, the process of measurement is implemented by mapping the inner product of ψ at the point of time in which we are interested to some sort of probability space from which a set of predictions about the system are obtained, in a basis determined by the act of measurement itself.

Even the implementation of QFT’s involves the same procedure, albeit in a far more complex form2 , a point that becomes obvious with a shift to the modern path integral formulation50 . Although traditionally the quantum formalism has only been applied to a very particular set of systems, a wide variety of more novel applications are starting to appear, where quantum theories of macroscopic systems are being created, often quite successfully51 . For example, different varieties of the quantum formalism have been applied to situations such as: stock market analysis52 ; quantum models of the brain53,54 ; models of cognitive function and concepts55–57 ; modeling of the process of decision making in situations of ambiguity58 etc. This general use (some might argue abuse) of the quantum formalism suggests that it is indeed far more generally applicable than is traditionally considered to be the case, and indeed the above consideration of the form of the quantum formalism suggests a reason for this; there is no mention of macroscopic detectors or microscopic particles Process Physics: Quantum Theories as Models of Complexity 99 in this formalism, and there is no reason to suppose a priori that they are necessary. This is merely an historical bias resulting from the discovery of the quantum formalism as a description of a specific class of systems (i.e. microscopic ones). A clue to this apparent generality of the quantum formalism lies in the theorems, generally attributed to Bell, of nonlocality and contextuality.48 Each of these theorems rely upon showing that when a quantum system is entangled∗∗ there exists a set of observables for which it is impossible to consistently assign an eigenvalue i.e. the outcomes of measurements of apparently independent experiments are incompatible. The resolution to this incompatibility lies in a proper consideration of the experimental arrangement; performing one experiment always results in a change of the quantum system and rules out the possibility of performing an alternative one. Thus, it is impossible to completely describe a quantum system without reference to its context. Entangled quantum systems exhibit a form of nonseparable behaviour and should not be considered independently of the set of measurements performed upon them. This situation shares much similarity with systems exhibiting high end complexity. Such systems should not be considered independently of their context, and may show incompatible results depending upon the measurements to which they are subjected. For example, as was discussed in Sec- tion 3 the social context in which schizophrenia occurs can have a dramatic effect upon the course of a patients illness. Indeed, different patients may be classified as schizophrenic or not depending upon the culture in which they are being diagnosed.37 This situation is analogous to the incompatible measurements occurring in the quantum formalism, and hence it is expected that the very well developed quantum formalism could be used to provide models of such contextual dependency during measurement. This suggests that the quantum theoretic formalism can be understood as modeling generic situations of contextuality where a system cannot be considered reductively as a set of separable subcomponents uninfluenced by their environment, even in cases where two subsystems are spread over a distance.2

∗∗An entangled state consists of at least two noninteracting systems A and B, represented by the states ψA and φB on the Hilbert spaces HA and HB , the composite state of which is written ψA ⊗φB which cannot itself be separated in a meaningful manner as a product ψAφB . 100 Kirsty Kitto

This contextual dependence often manifests itself as randomness arising from a lack of knowledge about the outcome of experiments, which can be used to explain the appearance of randomness in systems exhibiting contextual behavior, including quantum ones59 . Two specific examples of the way in which the quantum formalism could be used to model such systems can be found quite quickly. Firstly, it is possible to formulate Bell-type inequalities which can be used to test whether a system is exhibiting contextual responses.12 Another possibility would consist of incorporating the effect that an observer might have upon a system through finding a set of basis states to represent the system that are compatible with the basis states available to an observer and then performing ‘measurements’ as is done in standard quantum theory.60 If QT is capable of describing such a rich set of behaviour then we might ask what is to be gained by a QFT as opposed to the simpler QT. The answer to this question lies at the heart of complex emergent behaviour. Some clues are supplied by early work performed by pioneers such as Primas who noted that it is possible to perform a perturbation expansion over multiple time scales,61 by Froölich who proposed that coherent phase correlations will play “a decisive role in the description of biological materials and their activity”,62 and Davydov who proposed the concept of a biological soliton that resists thermal fluctuations.63 Taken together these results suggest that QFT has many applications in nontraditional fields, but despite this initial promise there has been a tendency for the larger physics community to ignore these possible extra applications. The full power of QFT becomes evident when symmetry breaking is incorporated into the formalism. If a system falls into a situation where its ground state does not have the same continuous symmetry as its dynamics then it is considered to be exhibiting spontaneous symmetry breaking. Ac- cording to Goldstone’s theorem, a system in such a state will dynamically generate massless bosons, termed Nambu–Goldstone modes (NG-modes), in response to this break in symmetry. The number of NG-modes generated will equal the number of broken symmetries in the system.30 Due to their massless nature, NG-modes are long-range, they can move through an entire system with no loss of energy, providing it with long range coherence.54 Because of their boson status many NG-modes may occupy the same ground state without changing the energy of the system. In such systems there can be a number of structurally different ground states, each in a lowest energy configuration. This is not possible in a classical system; there is only one lowest energy state in the classical realm, which means Process Physics: Quantum Theories as Models of Complexity 101 that there is only one way in which a system can exhibit stable behaviour. On the other hand systems described by a QFT can exist in many different stable configurations, which allows them to change state from one stable ground state to another and in the process generate new behaviour of a very rich form. Because of this phenomenon, QFT is the only quantum formalism capable of generating truly emergent behaviour rather than merely modeling a set of components and their interactions.2,54,64 This is to be expected since its basis could be argued to lie in the necessity of modeling the creation and annihilation of particles in modern high energy experiments; QFT was invented in order to model the emergence of new particles. Thus, the origin of Process Physics in QFT and, in particular its dependence upon NG phenomena via the extraction of (1) from the GCM suggests that such a model can, not only exhibit contextual responses, but can truly generate complex emergent behaviour.

6. Generating Complex Emergent Behaviour — a New Modeling Approach? We shall conclude with a brief summary of some of the more recent developments and applications that have followed from this more process driven approach to modeling complex emergent behavior. The historical basis of Process Physics in QFT suggests that such theories are capable of describing far more systems than has generally been considered the case. As described above, the process of action sequencing applied in the extraction of hadronic structure from the GCM relies strongly upon the use of symmetry breaking and the associated emergence of Nambu–Goldstone modes which are chosen as the dynamically extracted new variables. It is expected that this process will provide the key to the discovery of a general mechanism whereby complex emergent behaviour can be generated2 . This idea has been explored2,12 using a very simple model of sympatric speciation which is based upon quantum field theory. The model utilises dynamical symmetry breaking and the associated emergence of NG-modes to model the emergence of new sets of species within a system undergoing dynamical change. This model of speciation can be considered as a model of the differentiation of a species, and it is expected that such models can be generalised to a model of the process of differentiation as it occurs in many systems exhibiting this form of behavior. For example, a key future goal is to use these concepts to create a model of the process of biological development. This process can be understood as consisting of two sub-processes, differentiation (the specification of different cell fates) 102 Kirsty Kitto and morphogenesis (the development of shape), and leads to the formation of an organism. It is expected that far more applications are possible. This is because differentiation can be understood as the process responsible for forming new word meanings in language, new religions, new cultures and societies etc. Hence, a dynamical model of differentiation could be broadly used in modeling processes of generation, evolution and the creation of novelty. A general class of such models would be very important in the modeling of systems exhibiting high end complexity. Interestingly, some models along these lines already exist. For example, a quantum field theoretic model of the brain has been developed over a number of years, which makes use of the infinite number of unitarily inequivalent ground states required by QFT to model processes such as the formation of long term memories as situations where new stable states describing the brain are created by the exposure of a person to new ideas etc. Short term memories are described as excitations of the ground state, and the process of recall is modeled using the excitation of NG-modes from the current ground state of the brain. This model explains the serial nature of association as a situation where an excitation gradually decays via intermediate states back into a ground state. The book by Vitiello54 contains a good overview of this work, as well as an exhaustive list of references to the more detailed papers describing these results. Thus, there is every reason to suppose that such models could be very useful in the dynamical modeling of emergent stable structures in general. Indeed, in the pioneering paper on the QFT brain model,65 Ricciardi and Umezawa suggest that a similar model might possibly be used to explain how a stable DNA code is dynamically generated. Another promising avenue for the quantum modeling of complexity lies in the modeling of human interactions with social, computational and ecological systems providing services to those humans. With the generalised understanding of quantum theories summarised in Section 5, we begin to understand something that has been appreciated by the so-called ‘soft’ sciences almost since their inception; that contradictory results can be obtained from the assumption that different choices of experiment are independent from one another. However, with an acceptance that contextual outcomes are the norm of the quantum formalism we acquire a new, less arbitrary, theory with which we can incorporate choice into scientific discourse. For example, the concept of a “service ecosystem” has recently been developed66 to describe the connection of different web services which are deployed, published, discovered and delivered to different business channels Process Physics: Quantum Theories as Models of Complexity 103 through specialist intermediaries. In such cases a richer range of semantics in service descriptions and support of fuzzier search goals becomes necessary in order to create adequate descriptions and results for the end users. Such systems are highly contextual, and as such, their modeling by the standard reductionist methodology is inappropriate. However, with a formalism that incorporates contextual outcomes into its very foundation, quantum theory provides a number of possibilities for a set of more appropriate modeling techniques. Instead of hardwiring a plethora of different services into any attempt to describe such systems, a more malleable and hence long term solution presents itself along the lines discussed in Section 5.60 Namely the effect of the user upon a system can be modeled through finding a set of basis states to represent the system that are compatible with the basis states available to functions performed by a user and then performing ‘measurements’ as is done in standard quantum theory Thus, we see a general possibility that quantum theories can provide a new class of models capable of incorporating more fully the choices made by an observer who takes a more active role in the behaviour of a system than is traditionally considered to be the case, i.e., is interacting with the system in choosing which measurements will be performed upon it. Finally, the simple relational model discussed in section 1.1 has been particularly successful as a pregeometric theory. It has been used as the basis of a higher level theory of quantum gravity1 which is proving remarkably successful in the explanation of a number of so far unexplained phenomena. However this theory is remarkably complex when compared to the more traditional object based approaches. This leads to the final question of this paper and indeed to our conclusions.

7. What has been Gained by a Shift to Process Physics? (In Conclusion) The shift to a process oriented physics is remarkably difficult to implement. All of our standard modes of analysis are based upon reductive, object based techniques that assume a passive observer unable to interact with the system of interest in any way other than via controllable, identified channels. In attempting to construct a more complex understanding of the world (which is in fact more complex) we lose many of the long held, almost sacrosanct, assumptions of the scientific method. The accompanying difficulty of analysis is to be expected. However, many of the phenomena and problems currently unexplainable, and even impossible to correctly frame in more standard approaches become at least approachable from this new 104 Kirsty Kitto perspective. Many mysteries of physics become less mysterious, and as this paper has attempted to show, some of the problems outside of standard physics also appear to be amenable to this new methodology. It has been argued in this paper that the success of such models will lie with the general class of quantum theories, and in particular, in the class of QFT’s. In conclusion I would like to point out that I still teach the standard object-driven and static physics to undergraduates, in much the same way that I begin a discussion of mechanics with Newton’s laws rather than with General Relativity. There is a very good reason for this, and it is not merely historical. Taking account of complexity is a very difficult process, and in many scenarios it is not necessary. Even in situations where context must be incorporated it is not always necessary, or even desirable to go to a fully-fledged QFT, and it is highly likely that there will be a large number of complex systems that will not be describable by any form of quantum formalism. Systems exhibiting high end complexity cannot be fully modeled by one language of description alone;2,12 we require more tools and more approaches before we can ever hope to understand this very important class of system, and the approach outlined above is merely one proposal in this direction.

Acknowledgement The author would like to acknowledge the support of her PhD supervisor R.T. Cahill, without whose support she could have taken a lifetime to generate these concepts.

References 1. R. T. Cahill (2005) Process Physics: From Information Theory to Quantum Space and Matter. Nova Science Publishers. 2. Kirsty Kitto (2006) modeling and Generating Complex Emergent Behaviour. PhD thesis, School of Chemistry Physics and Earth Sciences, The Flinders University of South Australia. 3. W. Ross Ashby (1956) An Introduction to Cybernetics. Chapman and Hall, London. 4. Robert Rosen (2000) Essays on Life Itself. Complexity in Ecological Systems Series. Columbia University Press, New York. 5. J. Horgan (1995) From complexity to perplexity. Scientiﬁc American, 272: pp 74–79. 6. Kirsty Kitto. Dynamical Hierarchies in Fundamental Physics. In Bilotta et al.67 , pp 55–62. 7. Stanley N. Salthe (1985) Evolving Hierarchical Systems: Their Structure and Representation. Columbia University Press, New York. Process Physics: Quantum Theories as Models of Complexity 105

8. R. V. O’Neill, D. L. DeAngelis, J. B. Waide, and T. F. H. Allen (1986) A Hierarchical Concept of Ecosystems, volume 23 of Monographs in Population Biology. Princeton University Press, Princeton, New Jersey. 9. D.T. Campbell (1974) ‘Downward causation’ in hierarchically organized biological systems. In F.J. Ayala and T. Dobzhansky, editors, Studies in the Philosophy of Biology, pp 179–186. Macmillan Press, London. 10. G. Nicholis and I. Prigogine (1997) Self-Organization in Non-Equilibrium Systems: From Dissipative Structures to Order through Fluctuations. J. Wi- ley and Sons. 11. A.S. Davydov (1982) Biology and quantum mechanics. Pergamon, Oxford. 12. Kirsty Kitto (2007) High End Complexity. International Journal of General Systems. In Press. 13. Per Bak (1996) How Nature Works: The Science of Self-Organized Critical- ity. Springer-Verlag, New York. 14. P. Bak, C. Tang, and K. Wiesenfeld. Self-organized criticality. Physical Re- view A, 38, pp 364–374, 1988. 15. Reginald T. Cahill and Susan M. Gunner (1998) The global colour model of QCD for Hadronic processes — a review. Fizika,B7, pp 171–202. 16. C. M Klinger (2005) Process Physics: Bootstrapping Reality fom the Limita- tions to Logic. PhD thesis, School of Chemistry Physics and Earth Sciences, The Flinders University of South Australia. 17. G. Nagels (1985) Space as a “bucket of dust”. General Relativity and Grav- itation, 17(6), pp 545–557. 18. R.T. Cahill and C.M. Klinger (2000) Self-Referential Noise as a Fundamen- tal Aspect of Reality. In D. Abbott and L. Kish, editors, Proceedings of the 2nd International Conference on Unsolved Problems of Noise and Fluctu- ations (UPoN’99), volume 511, page 43, New York, American Institute of Physics. 19. S. N. Salthe (2002) Summary of the principles of hierarchy theory. General Systems Bulletin, 31, pp 13–17. 20. Nils A. Baas (1994) Emergence, hierarchies, and hyperstructures. In C. Langton, editor, Artiﬁcial life III, pp 515–537. Addison–Wesley, Ad- vanced Book Program, Santa Fe, New Mexico. 21. Howard H. Pattee (1973) The Physical Basis and Origin of Hierarchical Control. In Hierarchy Theory: The challenge of complex systems, pp 71– 156. George Braziller Inc. 22. Herbert A. Simon (1996) The Sciences of the Artiﬁcial. MIT Press, Cam- bridge, Massachusetts, 3 edition. 23. Tom Lenaerts, Dominique Groß, and Richard Watson (2002) On the modeling of dynamical hierarchies: Introduction to the workshop WDH. In Bilotta et al.,67 pp 37–44. 106 Kirsty Kitto

24. George Kampis (1995) The Inside and Outside Views of Life. In F. Moran, A. Moreno, J.J. Merelo, and P. Chacon, editors, Advances in Artificial Life, Third European Conference on Artificial Life Granada, Spain, June 4–6, 1995, pp 95–102. Springer. 25. George Kampis (1991) Self-Modifying systems in biology and cognitive science. Pergamon Press Inc, Oxford. 26. Francisco Varela and Jonathan Shear (1999) editors. The View from Within: First-person approaches to the study of consciousness, 6 of Journal of Con- sciousness Studies. Imprint Academic, Thorverton, UK. 27. O.E. Rössler(1987) Endophysics. In J. Casti and A. Karlquist, editors, Real Brains — Artificial Minds. North-Holland, New York. 28. D. Finkelstein and S.R. Finkelstein (1983) Computer Interactivity Simulates Quantum Complementarity. International Journal of Theoretical Physics, 22, pp 753–779. 29. R. T. Cahill (2003) Process Physics: From Information Theory to Quantum Space and Matter. Process Studies Supplement, 5, pp 1–131. 30. Robert E. Marshak (1993) Conceptual Foundations of Modern Particle Physics. World Scientific, Singapore. 31. J.-J. Dugne, S. Fredriksson, and J. Hansson (2002) Preon trinity —a schematic model of leptons, quarks and heavy vector bosons. Europhysics Letters, 60(2), pp 188–194. 32. Mark A. Bedau, John S. McCaskill, Norman H. Packard, Steen Rasmussen, Chris Adami, David G. Green, Takashi Ikegami, Kinihiko Kaneko, and Thomas S. Ray (2000) Open problems in Artificial Life. Artificial Life, 6, pp 363–376. 33. Bruce Edmonds (1999) Syntactic Measures of Complexity. PhD thesis, Uni- versity of Manchester. 34. Mary Jane West-Eberhard (1989) Phenotypic Plasticity and the Origins of Diversity. Annual Review of Ecology and Systematics, 20, pp 249–278. 35. John Maynard Smith and EörsSzathmáry(1997) The Major Transitions in Evolution. Oxford University Press, Oxford. 36. J. Derrida (1982) Différance.In Margins Of Philosophy, pp 3–27. University of Chicago Press, Chicago. 37. Janis Hunter Jenkins and Robert John Barrett (2004) Schizophrenia, Cul- ture, and Subjectivity: The Edge of Experience. Cambridge University Press, New York. 38. Steven Pinker (2002) The Blank Slate: The Modern Denial of Human Na- ture. Viking Penguin, London. 39. R. T. Cahill and C. M. Klinger (1998) Pregeometric Model of the Spacetime phenomenology. Physics Letters A, 223, pp 313–319. 40. Susan M. Gunner (2004) modeling Low Energy Processes in Quantum Chro- modynamics using the Global Colour Model. PhD thesis, School of Chemistry Physics and Earth Sciences, The Flinders University of South Australia. 41. R. T. Cahill and C. D. Roberts (1985) Soliton bag models of hadrons from QCD. Physical Review D, 32(8), pp 2419–2428. Process Physics: Quantum Theories as Models of Complexity 107

42. R. T. Cahill (1989) Hadronisation of QCD. Australian Journal of Physics, 42, pp 171–186. 43. R. T. Cahill, J. Praschifka, and C. J. Burden (1989) Diquarks and the bosonisation of QCD. Australian Journal of Physics, 42, pp 161–169. 44. G. Parisi and Yongshi Wu (1981) Perturbation theory without gauge fixing. Scientia Sinica, 24, pp 483–490. 45. E. Gozzi (1983) Functional-integral approach to Parisi-Wu stochastic quantization: scalar theory. Physical Review D, 28(8), pp 1922–1930. 46. Markus Brede and John Finnigan (2006) On the Rise and Fall of Complex Societies. http://www.dar.csiro.au/css/August Workshop 06.shtm. Talk presented at Complex Systems in Focus CSIRO CSS Annual Workshop Aug 8–10 Coogee Bay Hotel, Sydney. 47. F. Laloë(2001) Do we really understand quantum mechanics? Strange correlations, paradoxes, and theorems. American Journal of Physics,69(6), pp 655–701. 48. N. David Mermin (1993) Hidden variables and the two theorems of John Bell. Reviews of Modern Physics, 65(3), pp 803–815. 49. John S. Bell (1987) Six possible worlds of quantum mechanics. pp 181–195, Cambridge University Press, Cambridge. 50. Michio Kaku (1993) Quantum Field Theory: a modern introduction. Oxford University Press, Oxford. 51. P. D. Bruza, W. Lawless, C.J. van Rijsbergen, and D. Sofge (2007) editors. Proceedings of the AAAI Spring Symposium on Quantum Interaction. March 27–29, Stanford University. AAAI Press. 52. Belal E. Baaquie (2004) Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates. Cambridge University Press. 53. Roger Penrose (1991) The emperor’s new mind. Penguin, New York. 54. Giuseppe Vitiello (2001) My Double Unveiled. John Benjamins Publishing Company, Amsterdam. 55. D. Aerts and L. Gabora (2005) A theory of concepts and their combinations I: the structure of the sets of contexts and properties. Kybernetes, 34, pp 151–175. 56. L. Gabora and D. Aerts (2002) Contextualizing Concepts using a Mathe- matical Generalization of the Quantum Formalism. Journal of Experimental and Theoretical Artificial Intelligence, 14, pp 327–358. 57. P.D. Bruza and R.J. Cole (2005) Quantum Logic of Semantic Space: An Ex- ploratory Investigation of Context Effects in Practical Reasoning. In S. Arte- mov, H. Barringer, A.S. d’Avila Garcez, L.C. Lamb, and J. Woods, editors, We Will Show Them: Essays in Honour of Dov Gabbay, volume 1, pp 339– 361. College Publications. 58. D. Aerts (2005) Towards a new democracy: Consensus through quantum parliament. In D. Aerts, B. D’Hooghe, and N. Note, editors, Worldviews, Science and Us, Redemarcating Knowledge and its Social and Ethical Im- plications. World Scientific, Singapore. 108 Kirsty Kitto

59. D. Aerts (1998) The hidden measurement formalism: what can be explained and where paradoxes remain. International Journal of Theoretical Physics, 37, pp 291–304. 60. P.D. Bruza, K. Kitto, and A. Barros (2007) Exploration of non-reductionist models of service ecosystems. In preparation. Abstract accepted at Com- plex’07 - The 8th Asia-Pacific Complex Systems Conference. 61. Hans Primas (1983) Chemistry, Quantum Mechanics and Reductionism. Perspectives in Theoretical Chemistry. Springer–Verlag, Berlin, 2 edition. 62. H. Fröhlich (1988) Theoretical physics and biology. In H. Fröhlich, editor, Biological coherence and response to external stimuli, pp 1–24. Springer– Verlag, Berlin. 63. A.S. Davydov (1991) Solitons in molecular systems. Kluwer, Dordrecht. 64. Hiroomi Umezawa (1993) Advanced Field Theory: Micro, macro, and thermal physics. American Institute of Physics, New York. 65. L.M. Ricciardi and H. Umezawa (1967) Brain and Physics of Many-Body Problems. Kibernetik, 4, pp 44–48. 66. Alistair Barros, Marlon Dumas, and Peter Bruza (2005) The Move to Web Service Ecosystems. BP Trends, 3(12). 67. Eleonora Bilotta, Dominique Groß, Tom Smith, Tom Lenaerts, Seth Bul- lock, Henrik Hautop Lund, Jon Bird, Richard Watson, Pietro Pantano, Luigi Pagliarini, Hussein Abbass, Russell Standish, and Mark Bedau (2002) editors. ALife VIII — workshops. University of New South Wales. 109

A Cross-disciplinary Framework for the Description of Contextually Mediated Change

a b Liane Gabora and Diederik Aerts a Department of Psychology, University of British Columbia Okanagan Campus, 3333 University Way, Kelowna BC, V1V 1V7, Canada b Leo Apostel Centre for Interdisciplinary Studies and Department of Mathematics, Brussels Free University, Brussel, Belgium a [email protected]

We present a mathematical framework (referred to as Context-driven Actual- ization of Potential, or CAP) for describing how entities change over time under the influence of a context. The approach facilitates comparison of change of state of entities studied in different disciplines. Processes are seen to differ according to the degree of nondeterminism, and the degree to which they are sensitive to, internalize, and depend upon a particular context. Our analysis suggests that the dynamical evolution of a quantum entity described by the Schrödinger equation is not fundamentally different from change provoked by a measurement often referred to as collapse but a limiting case, with only one way to collapse. The biological transition to coded replication is seen as a means of preserving structure in the face of context, and sexual replication as a means of increasing potentiality thus enhancing diversity through interaction with context. The framework sheds light on concepts like selection and fitness, reveals how exceptional Darwinian evolution is as a means of ‘change of state’, and clarifies in what sense culture (and the creative process underlying it) are Darwinian.

Keywords: Change of State, Collapse; Context; Dynamical Evolution; Evolu- tion; Natural Selection; Nondeterminism; Unifying Theory PACS(2006): 01.70.+w; 01.90.+g; 87.10.+e; 87.15.He; 89.75.Fb

1. Introduction This paper elaborates a general theory of change of state (a nascent, non- 1 mathematical draft of which is presented elsewhere ) with the goal of unit- ing physical, biological, and cultural evolution, not reductively, but through 2 a process that may be referred to as inter-level theorizing . Other unifying 3 theories have been put forward, such as that of Treur which takes temporal factorization as its unifying principle. Our scheme focuses on the role of 110 Liane Gabora and Diederik Aerts context ; i.e. on the fact that what entities of different kinds have in common is that they change through a reiterated process of interaction with a context. We refer to this fundamental process of change of state under the influence of a context as context-driven actualization of potential, or CAP. By “potential” we do not mean determined or preordained; indeed, different forms of evolution differ according to the degree of nondeterminism, as well as degree of contextuality and retention of context-driven change. We conclude with several examples of how the scheme has already born fruit: it suggests a unifying scheme for the two kinds of change in quantum 4 mechanics , illustrates why the earliest forms of life could not have evolved 5 through natural selection , and helps clarify how the concept of evolution 6,7 8,9 applies to culture and creative thought .

2. Deterministic and Nondeterministic Evolution under the Influence of a Context In this section we discuss the kinds of change that must be incoroporated in a general scheme for the description of the evolution of an entity under the influence of a changing or unchanging context. We use the term evolution to mean simply ‘change of state’. Thus our neither implies natural selection nor change in the absence of a measurement; the term is thus used as it was prior to both Darwin and Schrödinger. Since we do not always have perfect knowledge of the state of the entity, the context, and the interaction between them, a general description of an evolutionary process must be able to cope with nondeterminism. Processes differ with respect to the degree of determinism involved in the changes of state that the entity undergoes. Consider an entity—whether it be physical, biological, mental, or some other sort—in a state p(ti)ataninstantof time ti. If it is under the influence of a context e(ti), and we know with certainty that p(ti)changes to state p(ti+1)attimeti+1, we refer to the change of state as deterministic. Newtonian physics provides the classic example of deterministic change of state. Knowing the speed and position of a material object, one can predict its speed and position at any time in the future. In many situations, however, an entity in a state p(ti)attime tiunder the influence of a context e(ti)may change to any state in the set {p1(ti+1),p2(ti+1), ..., pn(ti+1), ...}. When more than one change of state is possible, the process is nondeterministic. Nondeterministic change can be divided into two kinds. In the first, the nondeterminism originates from a lack of knowledge concerning the state of the entity p(ti) itself. This means that deep down the change is Cross-disciplinary Framework for Contextually Mediated Change 111 deterministic, but since we lack knowledge about what happens at this deeper level, and since we want to make a model of what we know, the model we make is nondeterministic. This kind of nondeterminism is modeled by a stochastic theory that makes use of a probability structure that satisfies Kolmogorov’s axioms. Another possibility is that the nondeterminism arises through lack of knowledge concerning the context e(ti), or how that context interacts with the entity. Yet another possibility is that the nondeterminism is ontological i.e. the universe is at bottom intrinsically nondeterministic. It has been proven that in these cases, the stochastic model to describe this situation necessitates a non-Kolmogorovian probability model. A Kolmogorovian prob- 10−15 ability model (such as is used in population genetics) cannot be used . It is only possible to ignore the problem of incomplete knowledge of context if all contexts are equally likely, or if context has a temporary or limited effect. Because the entity has the potential to change to many different states (given the various possible states the context could be in, since we lack precise knowledge of it), we say that it is in a potentiality state with respect to context. This is schematically depicted in Figure 1. We stress that a potentiality state is not predetermined, just waiting for its time to come along, at least not insofar as our models can discern, possibly because we cannot precisely specify the context that will come along and actualize it. Note also that a state is only a potentiality state in relation to a certain (incompletely specified) context. It is possible for a state to be a potentiality state with respect to one context, and a deterministic state with respect to another. More precisely, a state that is deterministic with respect to a context can be considered a limit case of a potentiality state, with zero potentiality. In reality the universe is so complex we can never describe with complete certainty and accuracy the context to which an entity is exposed, and how it interacts with the entity. There is always some possibility of even very unlikely outcomes. However, there are situations in which we can predict the values of relevant variables with sufficient accuracy that we may consider the entity to be in a particular state, and other situations in which there is enough uncertainty to necessitate the concept of potentiality. Thus a formalism for describing the evolution of these entities must take into account the degree of knowledge we as observers have about the context. 112 Liane Gabora and Diederik Aerts

Fig. 1. Graphical representation of a general evolution process. Contexts e(t0), e(t1), e(t2)ande(t3)attimest0,t1,t2,andt3, are represented by vertical lines. States of the entity are represented by circles on vertical lines. At time t0 the entity is in state p(t0). Under the influence of context e(t0), its state can change to one of the states in the set {p1(t2),p2(t2),p3(t2), ..., pn(t2), ...}. These potential changes are represented by thin lines. Only one change actually takes place, the one represented by a thick line, i.e. p(t0) changes to p4(t1). At time t1 the entity in state p4(t1) is under the influence of another context e(t1), and can change to one of {p1(t2),p2(t2),p3(t2), ..., pn(t2), ...}. Again only one change occurs, i.e. p4(t1) changes to p3(t2). The process then starts all over again. Under the influence of a new context e(t2), the entity can change to one of {p1(t3),p2(t3),p3(t3),p4(t3), ..., pn(t3), ...}. Again only one change happens: p3(t2) changes to p5(t3). The dashed lines from states that have not been actualized at a certain instant indicate that much more potentiality is present at time t0 than explicitly shown. For example, if p(t0) had changed to p2(t1) instead of p4(t2) , which was possible at time t0,thencontexte(t1) would have exerted a different effect on the entity at time t1, such that a new vertical line at time t1 would have to be drawn, representing another pattern of change.

3. SCOP: A Mathematics that Incorporates Nondeterministic Change We have seen that a description of the evolutionary trajectory of an entity may involve nondeterminism with respect to the state of the entity, the Cross-disciplinary Framework for Contextually Mediated Change 113 context, or how they interact. An important step toward the development of a general framework for how entities evolve is to find a mathematical structure that can incorporate all these possibilities. There exists an elaborate mathematical framework for describing the change and actualization of potentiality through contextual interaction that was developed for quantum mechanics. However it has several limitations, including the linearity of the Hilbert space, and the fact that one can only describe the extreme case where change of state is maximally contextual. Other mathematical theories, such as State COntext Property (SCOP) systems, lift the quantum formalism out of its specific structural limitations, making it possible 16−29 to describe nondeterministic effects of context in other domains .The original motivation for these generalized formalisms was theoretical (as opposed to the need to describe the reality revealed by experiments). In the SCOP formalism, for instance, an entity is described by three sets Σ, M and L and two functions µ and ν. Σ represents that set of states of the entity, M the set of contexts, and L the set of properties of the entity. The function µ is a probability function that describes how state p ∈ Σ under the influence of context e ∈Mchanges to state q ∈ Σ. Mathemati- cally, this means that µ is a function defined as follows µ :Σ×M×Σ → [0, 1] (1) (q, e, p) → µ(q, e, p) where µ(q, e, p) is the probability that the entity in state p changes to state q under the influence of context e. Hence µ describes the structure of the contextual interaction of the entity under study.

3.1. Sensitivity to Context A parameter that diﬀerentiates evolving entities is the degree of sensitivity to context, or more precisely, the degree to which a change of state of context evokes a change of state of the entity. Degree of sensitivity to context depends on both the state of the entity and the state of the context. The degree of sensitivity to context is expressed by the probability of collapse. If the probability is close to 1, it means that this context almost with certainty (deterministically) causes the state of the entity to change to the collapsed state. 114 Liane Gabora and Diederik Aerts

If the probability is close to zero, it means that this context is unlikely to causes the state of the entity to change to the collapsed state. Specifically, if µ(q, e, p)=0,thene has no influence on the entity in state p,andif µ(q, e, p)=1,thene has a strong, deterministic influence on the entity in state p.Avalueofµ(q, e, p) between 0 and 1, which is the general situation, quantifies the influence of context e on the entity in state p.

3.2. Degree of Nondeterminism Suppose we consider the set {µ(q, e, p)|e ∈ M}⊂[0, 1] (2) which is a set of points between 0 and 1. Suppose this set equals the singleton {1}. This would mean that for all context e ∈ M the entity in state p changed to the entity in state q. Hence this would indicate a situation of ‘deterministic change independent of context’ for the entity, between state p and state q. On the contrary, if this set equals the singleton {0},thiswould mean that the entity in state p never changes to state q, again independent of context. The general situation of the set being a subset of the interval [0, 1] hence models in a detailed way the overal contextual way two states p and q are dynamically connected.

3.3. Weights of Properties We need a means of expressing that certain properties are more applicable to some entities than others, or more applicable to entities when they are in one state than when they are in another state. The function ν describes the weight (which is the renormalization of the applicability) of a certain property given a specific state. This means that ν is a function from the set Σ ×L to the interval [0, 1], where ν(p, a) is the weight of property a for the entity in state p.Wewrite ν :Σ×L→[0, 1] (3) (p, a) → ν(p, a). The function ν, contrary to µ, describes the internal structure of the entity under investigation. Again we can look to some special situations to clarify how ν models the internal structure. Suppose that we have ν(a, p)= 1. This means that for the entity in state p the property ahas weight equal to 1, which means that it is actual with certainty (probability equal to 1). Hence, this represents a situation where the entity, in state p,‘has’the Cross-disciplinary Framework for Contextually Mediated Change 115 property ‘in acto’. On the contrary, if ν(a, p) = 0, it means that for the entity in state p is not at all actual, but completely potential. A value of ν(a, p)between 0 and 1 describes in a refined way ‘how property a is’ when the entity is in state p. Hence the set {ν(a, p)|a ∈ L}⊂[0, 1] (4) gives a good description of the internal structure of the entity, i.e. how properties are depending on the different states the entity can be in.

4. Expressing Dynamics: Context-driven Actualization of Potential (CAP) Context-driven Actualization of Potential, or CAP, is the dynamics of entities modeled by SCOP. This which means that the mathematical model for CAP will be as follows: at moment ti we have a SCOP

(Σ,M,L,µ,ν)(ti)=(Σ(ti),M(ti),L,µ,ν)(5) where Σ(ti) is the set of states of the entity at time ti and M(ti)isthe setofrelevantcontextsattimeti. L is independent of time, since it is the collection of properties of the entity under consideration. Properties themselves do not change over time, but their status of ‘actual’ or ‘potential’ can change with the change of the state of the entity. One should in fact look to it the other way around: a property is an element of the entity idependent of time, or, it is exactly the elements independent of time that give rise to properties. That is also the reason that L and ν describe the internal context independent structure of the entity under consideration. In Figure 1 a typical dynamical pattern of CAP is presented. Four consecutive moments of time t0,t1,t2 and t3 are considered. It can be seen in the figure how, for example context e(t0) has an influence on the entity in state p(t0) which is such that the entity can change into one of the states of the set {p1(t1),p2(t1),p3(t1), ..., pn(t1)}.Thisisanexampleofageneral nondeterministic type of change. Similar types of changes are represented in the figure, under influence of contexts e(t1), e(t2)ande(t3). What is important to remark is that the actual change taking place is a path through the graph of the figure, but the states not touched by this path remain of influence for the overall pattern of change, since they are potentiality states.

The states p0(t0),pi1 (t1),pi2 (t2), ..., pin (tn), ... constitute the trajectory of the entity through state space, and describe its evolution in time. Thus, the general evolution process is broadly construed as the incremental change 116 Liane Gabora and Diederik Aerts that results from recursive, context-driven actualization of potential, or CAP. A model of an evolutionary process may consist of both deterministic segments, where the entity changes state in a way that follows predictably given its previous state and/or the context to which it is exposed, and/or nondeterministic segments, where this is not the case. With a generalized quantum formalism such as SCOP it is possible to describe situations with any degree of contextuality. In fact, classical and quantum come out as special cases: quantum at one extreme of complete contextuality, and classical 30,31,32 at the other extreme, complete lack of contextuality .Thisiswhyit lends itself to the description of context-driven evolution.

4.1. Degree to which Context-driven Change is Retained We saw that if µ(q, e, p)=1,thene has a strong, deterministic inﬂuence on the entity in state p. However, an entity may be sensitive—readily undergo change of state due to context—but through regulatory mechanisms or self- replication have a tendency to return to a previous state. Thus although it is susceptible to undergoing a change of state from p to q, its internal dymaics are such that it goes back to state p. A simple example of a situation where this is not the case, i.e. context-driven change is retained is a rock breaking in two. An example where it is the case, i.e. context-driven change is not retained is the healing of an injury or the birth of oﬀspring with none of the characteristics their parents acquired over their lifetimes. In the case of biological organisms, we are considering two interrelated entities, one embedded in the other: the organism itself, and its lineage.

4.2. Context Independence The extent to which a change of context threatens the survival of the entity can be referred to as context dependence. The degree to which an entity is able to withstand, not just its particular environment, but any environment, can be referred to as context independence. Sensitivity to and retention of context can lead, in the long run, to either context dependence or context independence. This can depend on the variability of the contexts to which an entity is exposed. A static, impoverished environment may provide contexts that foster specializations tailored to that particular environment, whereas a dynamic, rich, diverse environment may foster general coping mechanisms. Cross-disciplinary Framework for Contextually Mediated Change 117

Thus for example, a species that develops an intestine specialized for the absorption of nutrients from a certain plant that is abundant in its environment exhibits context dependence, whereas a species that becomes increasingly more able to consume any sort of vegetation exhibits context independence. Whether an entity exhibits context dependence or independence may simply reflect what one chooses to define as the entity of interest. If an entity splits into multiple ‘versions’ of itself (as through reproduction) each of which adapts to a different context and thus becomes more context dependent, when all versions are considered different lineages of one joint entity, that joint entity is becoming more context-independent.Thusfor example, while different mammalian species are becoming more context dependent, the kindom as a whole is becoming more context independent.

5. Ways Found by Universe to Actualize Potential We now look at how different kinds of evolution fit into the above framework, and how their trajectories differ with respect to the parameters introduced in the previous section. They are all means of actualizing potential that existed due to the state of the entity, the context, and the nature of their interaction, but differ widely with respect to these parameters.

5.1. The Evolution of Physical Objects and Particles We begin by examining three kinds of change undergone by physical entities. The first is the collapse of quantum particles under the influence of a measurement. The second is the evolution of quantum particles when they are not measured. The third is the change of state of macroscopic physical objects. Nondeterministic Collapse of a Quantum Particle Let us begin with change of state in the most micro of all domains, quantum mechanics. The central mathematical object in quantum mechanics is a complex Hilbert space H, which is a vector space over the field C of complex numbers. Vectors |p(ti) of this Hilbert space of unit length represent states p(ti) of a quantum particle. This is expressed in Hilbert space by the bra-ket of the vector with itself being equal to 1, hence p(ti) | p(ti) =1. (6) Ameasuremente(ti) on a quantum particle is described by a self-adjoint operator E(ti), which is a linear function on the Hilbert space, hence E(ti):H → H (7) 118 Liane Gabora and Diederik Aerts such that

E(ti)(a |p1(ti) + b |p2(ti))=aE(ti) |p1(ti) + bE(ti) |p2(ti) (8) A self-adjoint operator always has a set of special states associated with it, the eigenstates.Astatep(ti) described in the Hilbert space Hby means of the vector |p(ti) is an eigenstate of the measurement e(ti) represented by the self-adjoint operator E(ti) if and only if we have

E(ti) |p(ti) = a |p(ti) (9) where a ∈ C is a real number, and a is called the eigenvalue of the measurement e(ti) the quantum particle being in state p(ti). Let us denote the set of eigenstates corresponding to the measurement e(ti)by {p1(ti),p2(ti), ..., pn(ti), ...}, and the set of their corresponding eigenvalues by {a1,a2, ..., an, ...}. Let us mention for mathematical completeness, that only when the self- adjoint operator has a point spectrum do we encounter the above situation in quantum mechanics. Measurements described by operators with a continuous spectrum must be treated in a more sophisticated way. However, this is of no relevance to the points made here. An eigenstate pj(ti) does not change under the influence of the measurement e(ti) described by the self-adjoint operator E(ti), and the corresponding eigenvalue aj is the value obtained deterministically by the measurement. However the set of eigenstates corresponding to a measurement e(ti) is only a subset of the total set of states of the quantum particle. States that are not eigenstates of the measurement e(ti) are called superposition states with respect to the measurement e(ti). Suppose that q(ti) is such a superposition state of the quantum particle. The vector |q(ti) of the Hilbert space representing this superposition state can then always be written as a linear combination, i.e. a superposition, of the vectors {|p1(ti) , |p2(ti) , ..., |pn(ti) , ...}representing the eigenstates of the quantum particle. More specifically |q(ti) = λj |pj(ti) (10) j where λj ∈ C are complex numbers such that 2 |λj | =1. (11) j 2 The physical meaning of the coefficients λj is the following: |λj| is the probability that the measurement e(ti)will yield the value aj and as a consequence of performing the measurement e(ti), the superposition state |q(ti) Cross-disciplinary Framework for Contextually Mediated Change 119 will change (collapse) to the eigenstate |pj(ti)with probability equal to 2 |λj| . This change of state from a superposition state to an eigenstate is 2 referred to as collapse. In general (depending on how many |λj| are nonzero), many eigenstates are possible states to collapse to under the influence of this measurement. In other words, the collapse is non-deterministic. This means that a genuine superposition state is a state of potentiality with respect to the measurement. This suggests that what we refer to as a context is the same thing as what in the standard quantum case is referred to as a measurement. Thus a quantum entity exists in general in a superposition state, and a measurement causes it to collapse nondeterministically to an eigenstate of that measurement. The specifics of the measurement constitute the context that elicit one of the states that were previously potential. Its evolution cannot be examined without performing measurements—that is, introducing contexts—but the contexts unavoidably affect its evolution. The evolution of a quantum particle is an extreme case of nondeterministic change, as well as of context sensitivity, because its state at any point in time reflects the context to which it is exposed.

Evolution of Quantum Particles The other mode of change in standard quantum mechanics is the dynamical change of state when no measurement is executed, referred to as ‘evolution’. This evolution is described by the Schrödinger equation, and it is considered a fundamentally different kind of change from ‘collapse’ under the influence of a measurement. The Schrödinger evolution is described by the Hamiltonian H which is the self-adjoint operator correspond- ingtothemeasurementh(ti) of the energy of the quantum particle, and the Schrödinger equation −iHt |p(t + ti) = e |p(ti) (12) where |p(t + ti) is the vector representing the state p(t+ti)ofthequantum particle at time t + ti and |p(ti) is the vector representing the state p(ti) −iHt of the quantum particle at time ti, while e is the unitary operator describing the time translation of state p(ti) to state p(t + ti), and is Planck’s constant. It is possible to interpret the change of the state of a quantum entity as described by the Schrödinger equation as an effect of context, namely a context that is the rest of the universe. All the fields and matter present in the rest of the universe contribute to the effect. The change is deterministic. Specifically, if the quantum entity at a certain time 120 Liane Gabora and Diederik Aerts ti is in state p(ti), and the only change that takes place is this dynamical change governed by the Schrödinger equation, then state p(t + ti)atany time t + ti later than ti is determined. For historical reasons, physicists think of a measurement not as a context, but as a process that gives rise to outcomes that are read off a measurement apparatus. In this scheme of thought, the simplest measurements are assumed to be those with two possible outcomes. A measurement with one outcome is rightly not thought of as a measurement, because if the same outcome always occurs, nothing has been compared and/or measured. However, when measurements are construed as contexts, we see that the measurement with two possible outcomes is not the simplest change possible. It is the deterministic evolution process—which can be conceived as a measurement with one outcome—that is the simplest kind of change. This means that in quantum mechanics the effect of context on change is as follows:

• When the context is the rest of the universe, its influence on the state of a quantum entity is deterministic, as described by the Schrödinger equation. • When the context is a measurement, its influence on a genuine superposition state is nondeterministic, described as a process of collapse. • When the context is a measurement, its influence on an eigenstate is deterministic, the eigenstate is not changed by the measurement.

Thus, under the CAP framework, the two basic processes of change in quantum mechanics are united; what has been referred to as dynamical evolution is not fundamentally different from collapse. They are both processes of actualization of potentiality under the influence of a context. In dynamical evolution there is only one possible outcome, thus it is deterministic. In collapse, until the state of the entity becomes an eigenstate, there is more than one possible outcome, thus it is nondeterministic. It was mentioned that the standard quantum formalism contains some fundamental structural restrictions. For example, the state space, i.e. the complex Hilbert space, is a linear space. It was also mentioned that more general axiomatic approaches to quantum mechanica have been developed where these structural restrictions are not present. More specifically, the state space in such a quantum axiomatic approach does not need to be a linear space. The SCOP approach which is the underlying mathematical framework for the ‘context driven actualization of potential’ change Cross-disciplinary Framework for Contextually Mediated Change 121 envisaged here, is such an axiomatic quantum ‘and’ classical approach, i.e. both standard quantum mechanics and classical mechanics are special cases 33−41 of this SCOP formalism . There is an even more specific structural restriction of standard quantum mechanics, namely the fact that the Schrödinger equation is a linear equation, and can only represent changes of states described by a unitary transformation. For this reason, even staying within the standard quantum mechanical Hilbert space formalism for the state space, non-linear and hence non-unitary evolution more general than the Schrödinger one 42−45 have been considered and studied on many occasions . All these variations on standard quantum mechanics are special cases of the SCOP formalism, and hence CAP incorporates the types of changes modeled by them.

Evolution of Classical Physical Entities Classical physical entities are the paradigmatic example of lack of sensitivity to and internalization of context, and deterministic change of state. However, theorists are continually expanding their models to include more of the context surrounding an entity in order to better predict its behavior, which suggests that things are not so tidy in the world of classical physical objects as Newtonian physics suggests. Macro-level physical entities may 46 exhibit structure similar to the entanglement of quantum mechanics . This is the case when change of state of the entity cannot be predicted due 47,48 to lack of knowledge of how it interacts with its context .

5.2. Biological Evolution Some theorists seeking to develop an integrative framework for the physical and life sciences (Ref. 49) focus on tools or phenomena that are applicable to or appear in both such as power laws and adaptive landscapes. The real challenge, however, is to develop an adaptive framework that en- compasses what is unique about biological life, what makes some matter alive, and what give rise to lineages that undergo adaptive modification. This is closely tied with the origins of structure with the capacity to self- replicate, and the refinement of this capacity through the emergence of the 50,51 genetic code . Thus, this section is divided into four parts. The first concerns autocatalytic sets of polymers, which replicated themselves (slop- pily) without genetic information, and are therefore are widely believed to be the first forms that can be considered ‘alive’. The second concerns organisms after genetically coded replication was established but prior to sexual 122 Liane Gabora and Diederik Aerts reproduction. The third concerns sexually reproducing organisms. The fourth part is a general comment on the structure of biochemical change.

The Earliest Life Forms Early life forms were more sensitive to context and prone to internalize context than present-day life because their replication took place not according to instructions (such as a genetic code), but through happen- 52 stance interactions. In Kauffman’s model of the origin of life ,polymers catalyze reactions that generate other polymers, increasing their joint complexity, until together as a whole they form something that can more or 53 less replicate itself .Thesetisautocatalytically closed because although no polymer catalyzes its own replication, each catalyzes the replication of another member of the set. So long as each polymer is getting duplicated somewhere in the set, eventually multiple copies of all polymers exist. Thus it can self-replicate, and continue to do so indefinitely, or at least until it changes so drastically that its self-replicating structure breaks down. (No- tice that ‘death’ of such life forms is not a particularly noticeable event; the only difference between a dead organism and an alive one is that the alive one continues to spawn new replicants.) Replication is far from perfect, so ‘offspring’ are unlikely to be identical to their ‘parent’. Different chance encounters of polymers, or differences in their relative concentrations, or the appearance of new polymers, could all result in different polymers cat- alyzing a given reaction, which in turn altered the set of reactions to be catalyzed. Context was readily internalized by incorporating elements of the environment, thus there was plenty of room for heritable variation. Recent work has been shown that the dynamical structure of biochemical reactions is quantum-like, above and beyond their microscopic (and 54,55,56 obvious) quantum structure . This stems from the fact that not only in the micro-domain where standard quantum structures are known to exist, but also in other domains where change-of-state depends on how an entity interacts with its context, the resulting probabilities can be non- Kolmogorovian, and the appropriate formalisms for describing them are either the quantum formalisms or mathematical generalizations of them. Kolmogorovian probability models consider only actualized entities, and their actualized interactions, and assumes that all evolutionary change is steered by these actualized entities, and their actualized interactions.Within CAP, potential states of entities, and potential interactions between them, can be described. Cross-disciplinary Framework for Contextually Mediated Change 123

Genetic Code Impedes Retention of Context in Lineage

A significant transition in the history of life was the transition from un- coded, self-organized replication to replication as per instructions given by a genetic code. This has been beautifully described and modelled by Vet- 57 sigian, Goldenfeld, and Woese They refer to the transition from sloppy self-replication of autocatalytic form to precise replication using a genetic code as the Darwinian transition, since it is at this point that traits acquired at the level of individuals were no longer inherited and natural selection, a population-level mechanism of change, becomes applicable. We saw that prior to coded replication, a change to one polymer would still be present in offspring after budding occurred, and this could cause other changes that have a significant effect on the lineage further downstream. There was nothing to prohibit inheritance of acquired characteristics. But with the advent of explicit self-assembly instructions, acquired characteristics were no longer passed on to the next generation. The reason for this stems from 58 the fact that, as first noted by von Neumann , they are self-replicating automata, meaning that they contain a coded instruction set that is used in two distinct ways: (1) as a self-assembly code that is interpreted during development, and (2) as self-instructions that are passively copied during reproduction. It is this separation of how the code is used to generate offspring, and how the code is used during development, that is at the foundation of their lack of inheritance of acquired traits. Since acquired characteristics were no longer being passed on to the next generation, the process became more constrained, robust, and shielded from external influence. (Thus for example, if one cuts off the tail of a mouse, its offspring will have tails of a normal length.) A context-driven change of state of an organism only affects its lineage if it impacts the generation and survival of progeny (such as by affecting the capacity to attract mates, or engage in parental care). Clearly, the transition from un- coded to coded replication, while ensuring fidelity of replication, decreased long-term sensitivity to and internalization of context, and thus capacity for context independence. Since one generation was almost certainly identical to the next, the evolution became more deterministic. As a result, in comparison with entities of other sorts, biological entities are resistant to internalization and retention of context-driven change. Though the term ‘adaptation’ is most closely associated with biology, biological form is resistant to adaptation. This explains why it has been possible to develop a theory of biological evolution that all but ignores the problem of 124 Liane Gabora and Diederik Aerts incomplete knowledge of context. As we saw earlier, it is possible to ignore this problem if all contexts are equally likely, or if context has a limited effect on heritability. In biology, since acquired traits are not heritable, the only contextual interactions that exert much of an effect are those that affect the generation of offspring. Thus it is because classical stochastic models work fine when lack of knowledge concerns the state of the entity and not the context that natural selection has for so long been viewed as adequate for the description of biological evolution. In Aerts, Czachor, and 59 60 D’Hooghe and Aerts et al. , Darwinian evolution is analyzed with respect to potentiality using a specific example, and various possible (and speculative) consequences of this difference are put forward.

Sexual Reproduction With the advent of sexual reproduction, the contextuality of biological evolution increased. Consider an organism that is heterozygous for trait X with two alleles A and a.ThepotentialofthisAa organism gets actualized differently depending on the context provided by the genotype of the organism’s mate. In the context of an AA mate, the Aa organism’s potential is constrained to include only AA or Aa offspring. In the context of an aa mate, it has the potential for Aa or aa offspring, and once again some of this potential might get actualized. And so forth. But while the mate constrains the organism’s potential, the mate is necessary to actualize some of this potential in the form of offspring. In other words, the genome of the mate simultaneously makes some aspects of the Aa organism’s potentiality possible, and others impossible. An organism exists in a state of potentiality with respect to the different offspring (variants of itself) it could produce with a particular mate. In other words, a mate constitutes a context for which an organism is a state of potentiality. One can get away with ignoring this to the extent that one can assume mating is random. Note that since a species is delineated according to the capacity of individuals to mate with one another, speciation can be viewed as the situation wherein one variant no longer has the potential to create a context for the other for which its state is a potentiality state with respect to offspring. A species can be said to be adapted to the extent that its previous states could have collapsed to different outcomes in different contexts, and thus to the extent its form reflects the particular contexts to which it was exposed. Note also that although over time species becomes increasingly context dependent, collectively they are becoming more context independent. (For virtually any ecological niche there exists some branch of life that can cope with it.) Cross-disciplinary Framework for Contextually Mediated Change 125

Some (Ref. 61)argue for expansion of the concept of selection to other hi- 62 erarchical levels, e.g. group selection. We agree with Kitcher that ‘despite the vast amount of ink lavished upon the idea of “higher-order” processes’, once we have the causal story, it’s a matter of convention whether we say that selection is operating at the level of the species, the organism, the genotype, or the gene. It is not the concept of selection that needs expansion, but the embedding of selection in a framework for how change can occur. The actual is but the realized fragment of the potential, and selection works only on this fragment, what is already actual. We can now return to our question about what natural selection has to say about the fitness of the offspring you might have with one mate as opposed to another. The answer is of course, nothing, but why? Because the situation involves actualization of potential and nondeterminism with respect to context, and as we have seen, a nonclassical formalism is necessary to describe the change of state involved. The CAP perspective also clarifies why fitness has been so hard 63 to nail down. We agree with Krimbas that fitness is a property of neither organism nor environment, but emerges at the interface between them, and changes from case to case.

5.3. Change of State in Cognitive Processes 64−67 Campbell argues that a stream of creative thought is a Darwinian process. The basic idea is that we generate new ideas through variation and selection: ‘mutate’ the current thought in a multitude of different ways, select the variant that looks best, mutate it invariouswaysandselectthe best, and so forth, until a satisfactory idea results. Thus a stream of thought is treated as a series of tiny selections. This 68−71 view has been extended, most notably by Simonton . The problems 72,73 with this thesis are outlined in detail elsewhere , but one that is evident following our preceding discussion is that thoughts simply are not von Neumann self-replicating automata. Another is that selection theory requires multiple, distinct, simultaneously-actualized states. But in cognition, each thought or cognitive state changes the ‘selection pressure’ against which the next is evaluated; they are not simultaneously selected amongst. The mind can exist in a state of potentiality, and change through interaction with the context to a state that is genuinely new, not just an element of a pre-existing set of states. Creative thought is a matter of honing in on an idea by redescribing successive iterations of it from different real or 74 imagined perspectives ; i.e. actualizing potential through exposure to different contexts. Once again, the description of contextual change of state 126 Liane Gabora and Diederik Aerts introduces a non-Kolmogorovian probability distribution, and a classical formalism such as selection theory cannot be used. Thus an idea certainly changes as it gets mulled over in a stream of thought, and indeed it appears to evolve, but the process is not Darwinian. Campbell’s error is to treat a set of potential, contextually elicited states of one entity as if they were actual states of a collection of entities, or possible states with no effect of context, even though the mathematical structure of the two situations is completely different. In a stream of thought, neither are all contexts equally likely, nor does context have a limited effect on future iterations. So the assumptions that make classical stochastic models useful approximations do not hold.

5.4. Cultural Evolution Culture, even more often than creative thought, is interpretted in evolutionary terms. Some scientists view culture merely as a contributing to the biological evolution of our species. Increasingly, however, culture is viewd as an evolutionary process in and of its own (though one that still is intertwined with, and inﬂuences, biological evolution). In this section we look at how cultural evolution (as a second evolutionary process) ﬁts into the CAP framework.

Culture Evolves Without a Self-Assembly Code The basic unit of culture has been assumed to be the behavior or artifact, or the mental representations or ideas that give rise to concrete cultural forms. The meme notion further implies that these cultural units 75 are replicators , misleadingly, because it does not consist of self-assembly 76 instructions .(Anideamayretain structure as it passes from one individual to another, but does not replicate it.) Looking at cultural evolution from the CAP framework we ask: what is really changing through cultural processes? Because of the distributed nature of human memory, it is never is just one discrete ‘meme’affected by a cultural experience; it is ones view of how the world hangs together, ones’ model of reality, or worldview. A worldview is not merely a collection of discrete ideas or memes (nor do ideas or memes form an interlocking set like puzzle pieces) because each context impacts it differently; concepts and ideas are always 77−79 colored by the situation in which they are evoked . Indeed it has been 80 argued that a worldview is a replicator. We saw that living organisms prior to the genetic code—a pre-RNA set of autocatalytic polymers—were Cross-disciplinary Framework for Contextually Mediated Change 127 primitive replicators because they generate self-similar structure, but in a self-organized, emergent, piecemeal manner, eventually, for each polymer, there existed another that catalyzed its formation. Since there was no self-assembly instructions to copy from, there was no explicit copying going on. The presence of a given catalytic polymer, say X, simply speeded up the rate at which certain reactions took place, while another polymer, say Y, influenced the reaction that generated X. Just as polymers catalyze reactions that generate other polymers, retrieval of an item from memory can trigger another, which triggers yet another and so forth, thereby cross-linking memories, ideas, and so forth into a conceptual web. Like the autocatalytic sets 81 of poymers considered earlier, the result can be described as a connected 82,83 closure structure . Elements of a worldview are regenerated through social learning. Since as with Kauffman’s origin of life scenario the process occurrs in a self-organized, piecemeal autocatalytic manner, through bottom-up interactions rather than a top-down code, worldviews like the earliest life forms replicate with low fidelity, and their evolution is highly nondeterministic.

Inheritance of Acquired Traits in Culture As with the earliest pre-DNA forms of life, characteristics of a worldview acquired over a lifetime are heritable. We hear a joke and, in sharing it with others, give it our own slant. We create a disco version of Beethoven’s Fifth Symphony and a rap version of that. The evolutionary trajectory of a worldview makes itself known indirectly via the behavior and artifacts it manifests under the influence of the contexts it encounters. (For example, when you explain to a child how to brush ones’ teeth, certain facets of your worldview are revealed, while your writing of a poem reveals other facets.) Because acquired traits are heritable in culture, the probability of splitting into multiple variants is high. These variants can range from virtually identical to virtually impossible to trace back to the same ‘parent’ idea. They affect, and are affected by, the minds that encounter them. For example, books can affect all the individuals who read them, and these individuals subsequently provide new contexts for the possible further evolution of the ideas they described and stories they told. 128 Liane Gabora and Diederik Aerts

6. Conclusions This paper introduced a general framework for characterizing how entities evolve through context-driven actualization of potential (CAP). By this we mean an entity has the potential to change in different ways under different contexts. Some aspects of this potentiality are actualized when the entity undergoes a change of state through interaction with the particular context it encounters. The interaction between entity and context may also change the context, and the constraints and affordances it offers the entity. Thus the entity undergoes another change of state, and so forth, recursively. When evolution is construed as the incremental change that results from recursive, context-driven actualization of potential, the domains through which we have carved up reality can be united under one umbrella. Quan- tum, classical, biological, cognitive, and cultural evolution appear as different ways in which potential that is present due to the state of an entity, its context, and the nature of their interaction. They differ according to the degree of sensitivity to context, internalization of context, dependence upon a particular context. nondeterminism due to lack of knowledge concerning the state of the entity, and nondeterminism due to lack of knowledge concerning the state of the context. The reason potentiality and contextuality are so important stems from the fact that we inevitably have incomplete knowledge of the universe in which an entity is operating. When the state of the entity of interest and/or context are in constant flux, or undergoing change at a resolution below that which we can detect but nevertheless affect what emerges at the entity-context interface, this gives rise in a natural way to nondeterministic change. Nondeterminism that arises through lack of knowledge concerning the state of the entity can be described by classical stochastic models (Markov processes) because the probability structure is Kol- mogorovian. However, nondeterminism that arises through lack of knowledge concerning the interaction between entity and context introduces a 84 non-Kolmogorovian probability model on the state space, necessitating a nonclassical formalism. Historically, the first nonclassical formalism was the quantum formalism. This formalism has since been generalized to describe situations involving nonlinearity, and varying degrees of contextuality. Let us sum up a few of the more interesting results to come out of this framework. It has been thought that the two modes of change in quantum mechanics—dynamical evolution of the quantum entity as per the Schrödinger equation, and the collapse that takes place when the quantum entity is measured—were fundamentally different. However, when the Cross-disciplinary Framework for Contextually Mediated Change 129 measurement is seen to be a context, we notice that it is always a context that could actualize the potential of the entity in different ways. Indeed, if one knows the outcome with certainty one does not perform a measurement; it is only when there is more than one possible value that a measurement is performed. Thus the two modes of change in quantum mechanics are united; the dynamical evolution of a quantum entity as per the Schrödinger equation reduces to a collapse for which there was only one way to collapse (i.e. only one possible outcome), hence deterministic collapse. This also holds for the deterministic evolution of classical entities. This constitutes a true paradigm shift, for evolution and collapse have been thought to be two fundamentally different processes. Looking at biological evolution from the CAP perspective, self- replication appears as a means of testing the integrity of an entity—or rather different versions of an entity—against different contexts. While individuals and even species become increasingly context-dependent, the joint entity of living organisms becomes increasingly context-independent.The genetic code afforded primitive life protection against contextually-induced disintegration of self-replication capacity, at the cost of decreased diversity. The onset of sexual reproduction increased potentiality, and thus possible trajectories for biological form. The CAP framework supports the notion that fitness is a property of neither organism nor environment, but emerges at the interface between them. The concept of potential fitness includes all possible evolutionary trajectories under all possible contexts. Since it involves nondeterminism with respect to context, unless context has a limited effect or all possible contexts are equally likely, a nonclassical formalism is necessary to describe the novel form that results when an organism interacts with its environment in a way that makes some of its potential became actual (where actual fitness refers only to the realized segment of its potentiality). It now becomes clear why natural selection has been able to tell us much about changes in frequencies of existing forms, but little about how new forms emerge in the first place! The CAP framework provides a perspective from which we can see why the neo-Darwinian view of evolution has been satisfactory for so long, and it wasn’t until after other processes become prominently viewed in evolutionary terms that the time was ripe for potentiality and contextuality to be taken seriously. We also see how unique the genetic code is, and the consequent lack of retention of context-driven change. The effects of contextual interaction in biology are in the long-run largely invisible; context affects lineages only by influencing the number and nature of offspring. Natural selection is such an exceptional means of 130 Liane Gabora and Diederik Aerts change, it is no wonder it does not transfer readily to other domains. Note that it is often said that because acquired traits are inherited in culture, culture should not be viewed in evolutionary terms. It is ironic that this critique also applies to the earliest stage of biological evolution itself. What was true of early life is also true of the replication of worldviews: acquired characteristics can be inherited. Modern life is unique in this sense. The same argument holds for what happens in a stream of creative thought. The mathematical formulation of the theory of natural selection requires that in any given iteration there be multiple distinct, actualized states. In cognition however, each successive mental state changes the context in which the next is evaluated; they are not simultaneously selected amongst. Creative thought is a matter of honing in on an idea by redescribing successive iterations of it from different real or imagined perspectives; actualizing potential through exposure to different contexts. Thus selection theory is not applicable to the formal description of a stream of thought, and to the extent that creative thought powers cultural change, it is of limited applicability there as well. Once again, a nonclassical formalism is necessary. The notion of culture as a Darwinian process probably derives from the fact that the means through which a creative mind manifests itself in the world—language, art, and so forth—exist as discrete entities such as stories and paintings. This can lead to the assumption that discrete creative artifacts in the world spring forth from corresponding discrete, pre-formed entities in the brain. This in turn leads to the assumption that novelty gets generated through that most celebrated of all change-generating mechanisms, Darwinian selection, and that ideas and artifacts must therefore be replicators. However, an idea or artifact is not a replicator because it does not consist of coded self-assembly instructions, and thus does not make copies of itself. Moreover, ideas and artifacts do not arise out of separate, distinct compartments in the brain, but emerge from a dynamically and contextually modifiable, web-like memory structure, a melting pot in which different components continually merge and blend, get experienced in new ways as they arise in new contexts and combinations. The CAP perspective suggests instead that the basic unit and the replicator of culture is an integrated network of knowledge, attitudes, ideas, and so forth; that is, an internal model of the world, or worldview, and that ideas and artifacts are how a worldview reveals itself under a particular context. Cross-disciplinary Framework for Contextually Mediated Change 131

Acknowledgments We would like to thank Ping Ao for comments on the manuscript. This research was supported by Grant G.0339.02 of the Flemish Fund for Scientiﬁc Research and a research grant from Foundation for the Future.

References 1. Gabora, L. & Aerts, D. (2005) Interdisciplinary Science Reviews, 30(1), 69-88. 2. Maull N. (1977) Stud. Hist. & Phil. Sci. 8, 143–162. 3. Treur J. (2007) Cog. Syst. Res. 8, in press. 4. Aerts D. 2002) ‘Being and change foundations of a realistic operational formalism’. In D.Aerts, M. Czachor & T. Durt (eds.), Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics, 71-110 World Scientiﬁc, Singapore. 5. Gabora L. (2006) J. Theor. Bio. 241(3), 443–450. 6. Gabora L. 2005) & Aerts D. Distilling the essence of an evolutionary process, and implications for a formal description of culture. In W. Kistler (ed.), Proceedings of Center for Human Evolution Workshop #5: Cultural Evolution, May 2000, 1–19 (Foundation for the Future, Seattle WA. 7. Gabora, L. (2007) Cog. Syst. Res. special issue. 8. Gabora L. (2005) J. Creat. Behav. 39(4), 65–87. 9. Gabora, L. (2007) Creat. Res. J. , 19, (4), pp 361–369. 10. Accardi L. (1982) & Fedullo A. Nuovo Cimento 34(7), 161–172. 11. Aerts D. (1986) J. Math. Phys. 27, 202–210. 12. Aerts D. (1994) Found. Phys. 24, 1227–1259. 13. Aerts D. (1994) & Aerts S. Found. Sci. 1, 85–97. 14. Pitowsky I. ‘Quantum Probability, Quantum Logic’, Lecture Notes in Physics 321, Springer, Berlin. 15. Randall C. & Foulis D. (1976) ‘A Mathematical Setting for Inductive Rea- soning’, in C. Hooker (ed.), Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science III, 169 (Reidel, Dordrecht, The Netherlands). 16. Pitowsky I. ( 1976) ‘Quantum Probability, Quantum Logic’, Lecture Notes in Physics 321 (Springer, Berlin, 1989). 17. Randall C. & Foulis D. ‘A Mathematical Setting for Inductive Reasoning’, in C. Hooker (ed.), Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science III, 169 (Reidel, Dordrecht, The Nether- lands). 18. Aerts D. (1993) Int’l J. Theor. Phys. 32, 2207–2220. 19. Aerts D. (1994) & Durt T. Found. Phys. 24, 1353–1368. 20. Aerts D. & Gabora L. (2005) Kybernetes 34, 167–191. 21. Aerts D. & Gabora L. (2005) Kybernetes 34, 192–221. 132 Liane Gabora and Diederik Aerts

22. Foulis D. & Randall C. (1981) ‘What are Quantum Logics and what Ought they to Be?’, in E. Beltrametti & B. van Fraassen (eds.), Current Issues in Quantum Logic 35 (Plenum, New York). 23. Foulis D., Piron C., & Randall C. (1983) Found. Phys. 13(813), 813–841. 24. Jauch J. (1968) Found. Quant. Mech. (Addison-Wesley, Reading, MA.) 25. Mackey G. (1963) Math. Found. Quant. Mech. (John Benjamin, Reading, MA). 26. Piron C. (1976) Found. Quant. Phys. (W.A. Benjamin, Reading, MA). 27. Piron C. (1989) Helvetica Physica Acta 62(82), 439–68. 28. Piron C. (1990) Mécanique Quantique: Bases et Applications (Press Poly- technique de Lausanne, Lausanne, Switzerland) 29. Randall C. & Foulis D. (1978) ‘The Operational Approach to Quantum Me- chanics’,inC.Hooker(ed.),Physical Theories as Logico-Operational Struc- tures 167 (Reidel, Dordrecht, The Netherlands). 30. Aerts D. (2002) ‘Being and change foundations of a realistic operational formalism’. In D. Aerts, M. Czachor & T. Durt (eds.), Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics 71–110. (World Scientific, Singapore). 31. Piron C. (1976) Found. Quant. Phys. (W.A. Benjamin, Reading, MA). 32. Aerts D. (1983) J. Math. Phys. 24, 2441–2454. 33. Aerts D. (2002) ‘Being and change foundations of a realistic operational formalism’. In D. Aerts, M. Czachor & T. Durt (eds.), Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics, pp 71–110 (World Scientific, Singapore). 34. Randall C. & Foulis D. (1976) ‘A Mathematical Setting for Inductive Rea- soning’, in C. Hooker (ed.), Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science III, 169 (Reidel, Dordrecht, The Netherlands). 35. Aerts D. & Durt T. (1994) Found. Phys. 24, 1353–1368. 36. Foulis D. & Randall C. (1981) ‘What are Quantum Logics and what Ought they to Be?’, in E. Beltrametti & B. van Fraassen (eds.), Current Issues in Quantum Logic 35 (Plenum, New York). 37. Piron C. (1976) Found. Quant. Phys. (W.A. Benjamin, Reading, MA). 38. Piron C. (1989) Helvetica Physica Acta 62(82), 439–468. 39. Piron C. (1990) Mécanique Quantique: Bases et Applications (Press Poly- technique de Lausanne, Lausanne, Switzerland). 40. Randall C. & Foulis D. ‘The Operational Approach to Quantum Mechanics’, in C. Hooker (ed.), Physical Theories as Logico-Operational Structures 167 (Reidel, Dordrecht, The Netherlands, 1978). 41. Aerts D. (1983) J. Math. Phys. 24, 2441–2454. 42. Gisin N. (1990) Phys. Lett. A 143,1. 43. Mielnik B. (2001) Phys. Lett. A 289,1-8. 44. Polchinski J. (1991) Phys. Rev. Lett. 66, 379–400. 45. Weinberg S. (1989) Ann. Phys. 194, 336-386. 46. Aerts D., Aerts S., Broekaert J. & Gabora L. (2000) Found. Phys. 30, 1387– 1414. Cross-disciplinary Framework for Contextually Mediated Change 133

47. Aerts D. (2002) ‘Being and change foundations of a realistic operational formalism’. In D. Aerts, M. Czachor & T. Durt (eds.), Probing the Structure of Quantum Mechanics: Nonlinearity, Nonlocality, Probability and Axiomatics, pp 71–110 (World Scientific, Singapore). 48. Aerts D. (1983) J. Math. Phys. 24, 2441–2454. 49. Ao P. (2005) Laws of Darwinian evolutionary theory. Phys. Life Rev. 2, 117–156. 50. Langton C. G. In C. Langton, C. Taylor, J. D. Farmer & S. Rasmussen (eds.) (1992) Artificial Life II (Addison-Wesley, Redwood City, CA). 51. Schrdinger E. (1944) What Is Life? The Physical Aspect of the Living Cell (Cambridge University Press, Cambridge). 52. Kauffman S. (1993) Origins of Order (Oxford University Press, New York). 53. The reason this works is because when polymers interact, the number of different polymers increases exponentially, but the number of reactions by which they can interconvert increases faster than their total number. Thus, as their diversity increases, so does the probability that some subset of the total reaches a critical point where there is a catalytic pathway to every member. 54. Aerts D., Czachor M., Gabora L. & Polk P. (2006) AIP Conference Pro- ceedings 810, 19-33. 55. Aerts D., Czachor M., Gabora L., Kuna M., Posiewnik A., Pykacz J. & Syty M. (2003) Phys. Rev. E 67, 051926. 56. Aerts D. & Czachor M. (2006) Nonlinearity 19, 575-589. 57. Vetsigian, K., Woese, C., and Goldenfeld, N. (2006) Proceedings of the Na- tional Academy Science USA 103: 10696–10701. 58. Von Neumann J. (1966) The Theory of Self-Reproducing Automata (Uni- versity of Illinois Press, Chicago). 59. Aerts D., Czachor M. & D’Hooghe B. (2006) ‘Towards a quantum evolutionary scheme: violating Bell’s inequalities in language’. In N. Gontier, J. P. Van Bendegem & D. Aerts (eds.), Evolutionary Epistemology, Language and Culture - A non adaptationist systems theoretical approach [Theory and De- cision Library Series A: Philosophy and Methodology of the Social Sciences. Series editor: Julian Nida-Ruemelin] (Dordrecht, Springer). 60. Aerts D., Bundervoet S., Czachor M., D’Hooghe B., Gabora L. & Polk P. (2007) ‘On the foundations of the theory of evolution’. In M. Locker (ed.), Systems Theory in Philosophy and Religion. Vols. I&II (IIAS, Windsor, Ont). 61. Gould S. J. (1991) Bully for Brontosaurus: Reflections in Natural History 63-66 (W.W. Norton & Company, New York). 62. Kitcher P. (2004) Bio. & Phil. 19(1), 1–15. 63. Krimbas C.B. (2004) Bio. & Phil. 19(2), 185–204. 64. Campbell D.T. (1960) Psych. Rev. 67, 380-400. 65. Campbell D.T. (1965) ‘Variation and selective retention in socio-cultural evolution’. In H.R. Barringer, G I. Blanksten & R.W. Mack (eds), Social change in developing areas: A reinterpretation of evolutionary theory (Cam- bridge, Schenkman). 134 Liane Gabora and Diederik Aerts

66. Campbell D.T. (1974) ‘Evolutionary epistemology’. In P.A. Schilp (ed.), The Philosophy of Karl Popper (Open Court, LaSalle, IL). 67. Campbell D.T. (1987) ‘Evolutionary epistemology’. In G. Radnitzky & W.W. Bartley III (eds.) Evolutionary epistomology: Rationality, and the sociology of knowledge (Open Court, LaSalle, IL). 68. Simonton D.K. (1998) J. Creat. Behav. 32(3), 153–58. 69. Simonton D.K. (1999) Origins of Genius: Darwinian Perspectives on Cre- ativity (Oxford, New York). 70. Simonton D.K. (1999b) ‘Creativity as Blind Variation and Selective Reten- tion: Is the Creative Process Darwinian?’, Psychological Inquiry 10, 309–28. 71. Simonton D.K. (2007) Creat. Res. J. in press. 72. Gabora L. (2005) J. Creat. Behav. 39(4), 65–87. 73. Gabora, L. (2007a) Creativity Research J. . 74. Gabora, L. (2007b) Creativity Research J. . 75. Dawkins R. (1976) The Selfish Gene (Oxford University Press, Oxford). 76. Gabora L. (2004) Bio. & Phil. 19(1), 127–143. 77. Barsalou L. (1982) Mem. & Cog. 10, 82–93. 78. Gabora L. & Aerts D. (2002) ‘Contextualizing concepts’. In Proceedings of the 15th International FLAIRS Conference, Pensacola Florida, May 14-17, 2002, AAAI. 79. Gabora L. & Aerts D. (2002) J. Experi. & Theor. AI 14(4), 327-358. 80. Gabora L. (2004) Bio. & Phil. 19(1), 127–143. 81. Kauffman S. (1993) Origins of order (Oxford University Press, New York). 82. Gabora L. (1998) Psycoloquy, 9(67), pp 1–26. 83. Gabora L. (2000) ‘Conceptual closure: Weaving memories into an interconnected worldview’. In G. Van de Vijver & J. Chandler (eds.), Closure: Emergent Organizations and their Dynamics. Ann. NY Acad. Sci. 901, 42- 53. 84. For example Bayes’ formula for conditional probability is not satisfied. 135

Quantum-like Probabilistic Models Outside Physics

Andrei Khrennikov International Center for Mathematical Modeling in Physics and Cognitive Sciences University of V¨axj¨o,S-35195, Sweden [email protected]

We present a quantum-like (QL) model in that contexts (complexes of e.g. mental, social, biological, economic or even political conditions) are represented by complex probability amplitudes. This approach gives the possibility to apply the mathematical quantum formalism to probabilities induced in any domain of science. In our model quantum randomness appears not as irreducible randomness (as it is commonly accepted in conventional quantum mechanics, e.g. by von Neumann and Dirac), but as a consequence of obtaining incomplete information about a system. We pay main attention to the QL description of processing of incomplete information. Our QL model can be useful in cognitive, social and political sciences as well as economics and artiﬁcial intelligence. In this paper we consider in a more detail one special application — QL modeling of brain’s functioning. The brain is modeled as a QL-computer.

Keywords: Incompleteness of quantum mechanics; Quantum-like Representa- tion of Information; Quantum-like Models in Biology; Psychology; Cognitive and Social Sciences and Economy; Context; Complex Probabilistic Amplitude PACS(2006): 03.65.Fd; 03.67.a; 03.65.w; 05.30.d; 03.65.Ta; 03.65.Ca; 87.10.+e; 89.75.k; 89.75.Fb; 89.65.s; 89.65.Gh

1. Introduction: Quantum Mechanics as Operation with Incomplete Information Let us assume that, in spite of a rather common opinion, quantum mechanics is not a complete theory. Thus the wave function does not provide a complete description of the state of a physical system. Hence we assume that the viewpoint of Einstein, De Broglie, Schrödinger,Bohm, Bell, Lamb, Lande, ‘t Hooft and other believers in the possibility to provide a more detailed description of quantum phenomena is correct, but the viewpoint of Bohr, Heisenberg, Pauli, Fock, Landau and other believers in the completeness of quantum mechanics (and impossibility to go beyond it) is wrong. We remark that the discussion on completeness/incompleteness of 136 Andrei Khrennikov quantum mechanics is also known as the discussion about hidden variables — additional parameters which are not encoded in the wave function. In this paper we would not like to be involved into this great discussion, see Ref. 12 for recent debates. We proceed in a very pragmatic way by taking advantages of the incompleteness viewpoint on quantum mechanics. Thus we would not like to be waiting for until the end of the Einstein–Bohr debate. This debate may take a few hundred years more.∗ What are advantages of the Einstein’s interpretation of the QM- formalism? The essence of this formalism is not the description of a special class of physical systems, so called quantum systems, having rather exotic and even mystical properties, but the possibility to operate with incomplete information about systems. Thus according to Einstein one may apply the formalism of quantum mechanics in any situation which can be characterized by incomplete description of systems. This (mathematical) formalism could be used in any domain of science, see Refs. 4, 7–9 : in cognitive and social sciences, economy, information theory. Systems under consideration need not be exotic. However, the complete description of them should be not available or ignored (by some reasons). We repeat again that in this paper it is not claimed that quantum mechanics is really incomplete as a physical theory. It is not the problem under consideration. We shall be totally satisfied by presenting an information QL model such that it will be possible to interpret the wave function as an incomplete description of a system. It might occur that our model could not be applied to quantum mechanics as a physical theory. However, we shall see that our model can be used at least outside physics. Therefore we shall speak about a quantum-like (QL) and not quantum model. We shall use Einstein’s interpretation of the formalism of quantum mechanics. This is a special mathematical formalism for statistical decision making in the absence of complete information about systems. By using this formalism one permanently ignore huge amount of information. How- ever, such an information processing does not induce chaos. It is extremely consistent. Thus the QL information cut off is done in a clever way. This is the main advantage of the QL processing of information. We also remark that (may be unfortunately) the mathematical formalism for operating with probabilistic data represented by complex

∗Even if quantum mechanics is really incomplete it may be that hidden parameters would be found only on scales of time and space that would be approached not very soon, cf.3 Quantum-like Probabilistic Models Outside Physics 137 amplitudes was originally discovered in a rather special domain of physics, so called quantum physics. This historical fact is the main barrier on the way of applications of QL methods in other domains of science, e.g. biology, psychology, sociology, economics. If one wants to apply somewhere the mathematical methods of quantum mechanics (as e.g. Hameroff10,11 and Penrose12,13 did in study of brain’s functioning) he would be typically constrained by the conventional interpretation of this formalism, namely, the orthodox Copenhagen interpretation. By this interpretation quantum mechanics is complete. There are no additional (hidden or ignored) parameters completing the wave function description. Quantum randomness is not re- ducible to classical ensemble randomness, see von Neumann14Birkhoff and von Neumann15 .† It is not easy to proceed with this interpretation to macroscopic applications. In this paper we present a contextualist statistical realistic model for QL representation of incomplete information for any kind of systems: biological, cognitive, social, political, economical. Then we concentrate our considerations to cognitive science and psychology7,16 . In particular, we shall describe cognitive experiments to check the QL structure of mental processes. The crucial role is played by the interference of probabilities for mental observables. Recently one such experiment based on recognition of images was performed, see17,7 . This experiment confirmed our prediction on the QL behavior of mind. In our approach “quantumness of mind” has no direct relation to the fact that the brain (as any physical body) is composed of quantum particles, see Refs. 10−13. We invented a new terminology “quantum-like (QL) mind.” Cognitive QL-behavior is characterized by a nonzero coe–cient of interference λ (“coefficient of supplementarity”). This coefficient can be found on the basis of statistical data. There are predicted not only cos θ-interference of probabilities, but also hyperbolic cosh θ-interference. The latter interference was never observed for physical systems, but we could not exclude this possibility for cognitive systems. We propose a model of brain functioning as a QL-computer. We shall discuss the difference between quantum and QL-computers. From the very beginning we emphasize that our approach has nothing

†The founders of the orthodox Copenhagen interpretation, Bohr and Heisenberg as well as e.g. Pauli and Landau, did not discuss so much probabilistic aspects of this interpretation. This is an astonishing historical fact, since quantum mechanics is a fundamentally probabilistic theory. 138 Andrei Khrennikov to do with quantum reductionism. Of course, we do not claim that our approach implies that quantum physical reduction of mind is totally impossible. But our approach could explain the main QL-feature of mind — interference of minds — without reduction of mental processes to quantum physical processes. Regarding the quantum logic approach we can say that our contextual statistical model is close mathematically to some models of quantum logic18 , but interpretations of mathematical formalisms are quite diﬀerent. The crucial point is that in our probabilistic model it is possible to combine realism with the main distinguishing features of quantum probabilistic formalism such as interference of probabilities, Born’s rule, complex probabilistic amplitudes, Hilbert state space, and representation of (realistic) observables by operators.

2. Levels of Organization of Matter and Information and their Formal Representations This issue is devoted to the principles and mechanisms which let matter to build structures at different levels, and their formal representations. We would like to extend this issue by considering information as a more general structure than matter, see Ref. 19. Thus material structures are just information structures of a special type. In our approach it is more natural to speak about principles and mechanisms which let information to build structures at difierent levels, and their formal representations. From this point of view quantum mechanics is simply a special formalism for operation in a consistent way with incomplete information about a system. Here a system need not be a material one. It can be a social or political system, a cognitive system, a system of economic or financial relations. The presence of OBSERVER collecting information about systems is always assumed in our QL model. Such an observer can be of any kind: cognitive or not, biological or mechanical. The essence of our approach is that such an observer is able to obtain some information about a system under observation. As was emphasized, in general this information is not complete. An observer may collect incomplete information not only because it is really impossible to obtain complete information. It may occur that it would be convenient for an observer or a class of observers to ignore a part of information, e.g. about social or political processes.‡

‡We mention that according to Freud’s psychoanalysis human brain can even repress some ideas, so called hidden forbidden wishes and desires, and send them into the unconsciousness. Quantum-like Probabilistic Models Outside Physics 139

Any system of observers with internal or external structure of self- organization should operate even with incomplete information in a consistent way. The QL formalism provides such a possibility. We speculate that observers might evolutionary develop the ability to operate with incomplete information in the QL way — for example, brains. In the latter case we should consider even self-observations: each brain performs measurements on itself. We formulate the hypothesis on the QL structure of processing of mental information. Consequently, the ability of QL operation might be developed by social systems and hence in economics and finances. If we consider Universe as an evolving information system, then we shall evidently see a possibility that Universe might evolve in such a way that different levels of its organization are coupled in the QL setting. In particular, in this model quantum physical reality is simply a level of organization (of information structures, in particular, particles and fields) which is based on a consistent ignorance of information (signals, interactions) from a prequantum level. In its turn the latter may also play the role of the quantum level for a pre-prequantum one and so on. We obtain the model of Universe having many (infinitely many) levels of organization which are based on QL representations of information coming from the previous levels. In same way the brain might have a few levels of transitions from the classical-like (CL) to the QL description. At each level of representation of information so called physical, biological, mental, social or economic laws are obtained only through partial ignorance of information coming from previous levels. In particular, in our model the quantum dynamical equation, the Schrödinger’sequation, can be obtained as a special (incomplete) representation of dynamics of classical probabilities, see Section 12.

3. VäxjöModel: Probabilistic Model for Results of Observations A general statistical realistic model for observables based on the contextual viewpoint to probability will be presented. It will be shown that classical as well as quantum probabilistic models can be obtained as particular cases of our general contextual model, the Vãxjõmodel. This model is not reduced to the conventional, classical and quantum models. In particular, it contains a new statistical model: a model with hyperbolic cosh-interference that induces ”hyperbolic quantum mechanics”.7 A physical, biological, social, mental, genetic, economic, or financial 140 Andrei Khrennikov context C is a complex of corresponding conditions. Contexts are fundamental elements of any contextual statistical model. Thus construction of any model M should be started with fixing the collection of contexts of this model. Denote the collection of contexts by the symbol C (so the family of contexts C is determined by the model M under consideration). In the mathematical formalism C is an abstract set (of “labels” of contexts). We remark that in some models it is possible to construct a set-theoretic representation of contexts — as some family of subsets of a set Ω. For example, Ω can be the set of all possible parameters (e.g. physical, or mental, or economic) of the model. However, in general we do not assume the possibility to construct a set-theoretic representation of contexts. Another fundamental element of any contextual statistical model M is a set of observables O : each observable a ∈ O can be measured under each complex of conditions C ∈ C. For an observable a ∈ O, we denote the set of its possible values (“spectrum”) by the symbol Xa. We do not assume that all these observables can be measured simultaneously. To simplify considerations, we shall consider only discrete observables and, moreover, all concrete investigations will be performed for dichotomous observables. Axiom 1: For any observable a ∈ O and its value x ∈ Xa, there are deflned contexts, say Cx, corresponding to x-selections: if we perform a measurement of the observable a under the complex of physical conditions Cx, then we obtain the value a = x with probability 1. We assume that the set of contexts C contains Cx-selection contexts for all observables a ∈ O and x ∈ Xa. For example, let a be the observable corresponding to some question: a = + (the answer “yes”) and a = − (the answer “no”). Then the C+- selection context is the selection of those participants of the experiment who answering “yes” to this question; in the same way we define the C−- selection context. By Axiom 1 these contexts are well defined. We point out that in principle a participant of this experiment might not want to reply at all to this question. By Axiom 1 such a possibility is excluded. By the same axiom both C+ and C−-contexts belong to the system of contexts under consideration. Axiom 2: There are deflned contextual (conditional) probabilities P(a = x|C) for any context C ∈ C and any observable a ∈ O. Thus, for any context C ∈ C and any observable a ∈ O, there is defined the probability to observe the fixed value a = x under the complex of conditions C. Quantum-like Probabilistic Models Outside Physics 141

Especially important role will be played by probabilities:

a|b p (x|y) ≡ P(a = x|Cy), a, b ∈ O, x ∈ Xa, y ∈ Xb, where Cy is the [b = y]-selection context. By axiom 2 for any context C ∈ C, there is deﬁned the set of probabilities:

{P(a = x|C): a ∈ O}.

We complete this probabilistic data for the context C by contextual probabilities with respect to the contexts Cy corresponding to the selections [b = y] for all observables b ∈ O. The corresponding collection of data D(O,C) consists of contextual probabilities:

P(a = x|C), P(b = y|C), P(a = x|Cy), P(b = y|Cx), ..., where a, b, ... ∈ O. Finally, we denote the family of probabilistic data D(O,C) for all contexts C ∈ C by the symbol D(O, C)(≡ ∪C∈CD(O,C)). Definition 1. (VäxjöModel) An observational contextual statistical model of reality is a triple

M = (C, O, D(O, C)) (1) where C is a set of contexts and O is a set of observables which satisfy to axioms 1,2, and D(O, C) is probabilistic data about contexts C obtained with the aid of observables belonging O. We call observables belonging the set O ≡ O(M) reference of observables. Inside of a model M observables belonging to the set O give the only possible references about a context C ∈ C.

4. Representation of Incomplete Information Probabilities P(b = y|C) are interpreted as contextual (conditional) probabilities. We emphasize that we consider conditioning not with respect to events as it is typically done in classical probability,20 but conditioning with respect to contexts — complexes of (e.g. physical, biological, social, mental, genetic, economic, or ﬁnancial) conditions. This is the crucial point. On the set of all events one can always introduce the structure of the Boolean algebra (or more general σ-algebra). In particular, for any two events A and B their set-theoretic intersection A ∩ B is well deﬁned and it determines a new event: the simultaneous occurrence of the events A and B. 142 Andrei Khrennikov

In contract to such an event-conditioning picture, if one have two contexts, e.g. complexes of physical conditions C1 and C2 and if even it is possible to create the set-theoretic representation of contexts (as some collections of physical parameters), then, nevertheless, their set-theoretic intersection C1 ∩ C2 (although it is well defined mathematically) need not correspond to any physically meaningful context. Physical contexts were taken just as examples. The same is valid for social, mental, economic, genetic and any other type of contexts. Therefore even if for some model M we can describe contexts in the set-theoretic framework, there are no reasons to assume that the collection of all contexts C should form a σ-algebra (Boolean algebra). This is the main difference from the classical (noncontextual) probability theory.20 One can consider the same problem from another perspective. Suppose that we have some set of parameters Ω (e.g. physical, or social, or mental). We also assume that contexts are represented by some subsets of Ω. We consider two levels of description. At the first level a lot of information is available. There is a large set of contexts, we can even assume that they form a σ-algebra of subsets F. We call them the first level contexts. There is a large number of observables at the first level, say the set of all possible random variables ξ :Ω → R (here R is the real line). By introducing on F a probability measure P we obtain the classical Kolmogorov probability model (Ω, F, P), see Ref. 20. This is the end of the classical story about the probabilistic description of reality. Such a model is used e.g. in classical statistical physics. We point our that any Kolmogorov probability model induces a Växjö model in such a way: a) contexts are given by all sets C ∈ F such that P(C) 6= 0; b) the set of observables coincides with the set of all possible random variables; c) contextual probabilities are defined as Kolmogorovian conditional probabilities, i.e. by the Bayes formula: P(a = x|C) = P(ω ∈ C : a(ω) = x)/P(C). This is the Växjömodel for the first level of description. Consider now the second level of description. Here we can obtain a non- Kolmogorovian Vãxjõmodel. At this level only a part of information about the first level Kolmogorovian model (Ω, F, P) can be obtained through a special family of observables O which correspond to a special subset of the set of all random variables of the Kolmogorov model (Ω, F, P) at the first level of description. Roughly speaking not all contexts of the first level, F can be “visible” at the second level. There is no sufficiently many observables “to see” all contexts of the first level — elements of the Kolmogorov Quantum-like Probabilistic Models Outside Physics 143

σ-algebra F. Thus we should cut off this σ-algebra F and obtain a smaller family, say C, of visible contexts. Thus some Växjömodels (those permitting a set-theoretic representation) can appear starting with the purely classical Kolmogorov probabilistic framework, as a consequence of ignorance of information. If not all information is available, so we cannot use the first level (classical) description, then we, nevertheless, can proceed with the second level contextual description. We shall see that starting with some Växjömodels we can obtain the quantum-like calculus of probabilities in the complex Hilbert space. Thus in the opposition to a rather common opinion, we can derive a quantum- like description for ordinary macroscopic systems as the results of using of an incomplete representation. This opens great possibilities in application of quantum-like models outside the micro-world. In particular, in cognitive science we need not consider composing of the brain from quantum particles to come to the quantum-like model. Example 1. (Firefly in the box) Let us consider a box which is divided into four sub-boxes. These small boxes which are denoted by ω1, ω2, ω3, ω4 provides the the first level of description. We consider a Kolmogorov probability space: Ω = {ω1, ω2, ω3, ω4}, the algebra of all finite subsets F of Ω and a probability measure determined by probabilities P(ωj) = pj, where 0 < pj < 1, p1 + ... + p4 = 1. We remark that in our interpretation it is more natural to consider elements of Ω as elementary parameters, and not as elementary events (as it was done by Kolmogorov).

We now consider two diﬀerent disjoint partitions of the set Ω :

A1 = {ω1, ω3},A2 = {ω2, ω4},

B1 = {ω1, ω2},B2 = {ω3, ω4}. We can obtain such partitions by dividing the box: a) into two equal parts by the vertical line: the left-hand part gives A1 and the right-hand part A2; b) into two equal parts by the horizontal line: the top part gives B1 and the bottom part B2. We introduce two random variables corresponding to these partitions: ξa(ω) = xi, if ω ∈ Ai and ξb(ω) = yi ∈ if ω ∈ Bi. Suppose now that we are able to measure only these two variables, denote the corresponding observables by the symbols a and b. We project the Kolmogorov model under consideration to a non-Kolmogorovian V¨axj¨omodel by using the observables a and b — the second level of description. At this 144 Andrei Khrennikov

Fig. 1. The ﬁrst level (complete) description. level the set of observables O = {a, b} and the natural set of contexts C :Ω,A1,A2,B1,B2,C1 = {ω1, ω4},C2 = {ω2, ω3} and all unions of these sets. Here “natural” has the meaning permitting a quantum-like representation (see further considerations). Roughly speaking contexts of the second level of description should be large enough to “be visible” with the aid of observables a and b.

Intersections of these sets need not belong to the system of contexts (nor complements of these sets). Thus the Boolean structure of the original ﬁrst level description disappeared, but, nevertheless, it is present in the latent form. Point-sets {ωj} are not “visible” at this level of description. For example, the random variable

η(ωj) = γj, j = 1, ..., 4, γi 6= γj, i 6= j, is not an observable at the second level. Such a model was discussed from positions of quantum logic, see Ref. 6. There can be provided a nice interpretation of these two levels of description. Let us consider a firefly in the box. It can fly everywhere in this box. Its locations are described by the uniform probability distribution P (on the σ-algebra of Borel subsets of the box). This is the first level of description. Such a description can be realized if the box were done from glass or Quantum-like Probabilistic Models Outside Physics 145

Fig. 2. The second level description: the a-observable.

Fig. 3. The second level description: the b-observable if at every point of the box there were a light detector. All Kolmogorov random variables can be considered as observables. Now we consider the situation when there are only two possibilities to observe the firefly in the box: 1) to open a small window at a point a which is located in such a way (the bold dot in the middle of the bottom side of the box) that it is possible 146 Andrei Khrennikov to determine only either the firefly is in the section A1 or in the section A2 of the box; 2) to open a small window at a point b which is located in such a way (the bold dot in the middle of the right-hand side of the box) that it is possible to determine only either the firefly is in the section B1 or in the section B2 of the box.

In the first case I can determine in which part, A1 or A2, the firefly is located. In the second case I also can only determine in which part, B1 or B2, the firefly is located. But I am not able to look into both windows simultaneously. In such a situation the observables a and b are the only source of information about the firefly (reference observables). The Kolmogorov description is meaningless (although it is incorporated in the model in the latent form). Can one apply a quantum-like description, namely, represent contexts by complex probability amplitudes? The answer is to be positive. The set of contexts that permit the quantum-like representation consists of all subsets C such that P(Ai|C) > 0 and P(Bi|C) > 0, i = 1, 2 (i.e. for sufficiently large contexts). We have seen that the Boolean structure disappeared as a consequence of ignorance of information. Finally, we emphasize again that the Växjömodel is essentially more general. The set-theoretic representation need not exist at all.

5. Quantum Projection of Boolean Logic Typically the absence of the Boolean structure on the set of quantum propositions is considered as the violation of laws of classical logic, e.g. in quantum mechanics.15 In our approach classical logic is not violated, it is present in the latent form. However, we are not able to use it, because we do not have complete information. Thus quantum-like logic is a kind of projection of classical logic. The impossibility of operation with complete information about a system is not always a disadvantages. Processing of incomplete set of information has the evident advantage comparing with “classical Boolean” complete information processing — the great saving of computing resources and increasing of the speed of computation. However, the Boolean structure cannot be violated in an arbitrary way, because in such a case we shall get a chaotic computational process. There should be developed some calculus of consistent ignorance by information. Quantum formalism provides one of such calculi. Of course, there are no reasons to assume that processing of information through ignoring of its essential part should be rigidly coupled to a special Quantum-like Probabilistic Models Outside Physics 147 class of physical systems, so called quantum systems. Therefore we prefer to speak about quantum-like processing of information that may be performed by various kinds of physical and biological systems. In our approach quantum computer has advantages not because it is based on a special class of physical systems (e.g. electrons or ions), but because there is realized the consistent processing of incomplete information. We prefer to use the terminology QL-computer by reserving the “quantum computer” for a special class of QL-computers which are based on quantum physical systems. One may speculate that some biological systems could develop in the process of evolution the possibility to operate in a consistent way with incomplete information. Such a QL-processing of information implies evident advantages. Hence, it might play an important role in the process of the natural selection. It might be that consciousness is a form of the QL- presentation of information. In such a way we really came back to White- head’s analogy between quantum and conscious systems.22

6. Contextual Interpretation of ‘Incompatible’ Observable Nowadays the notion of incompatible (complementary) observables is rigidly coupled to noncommutativity. In the conventional quantum formalism observables are incompatible iff they are represented by noncommuting self-adjoint operatorsa ˆ and ˆb : [â, ˆb] 6= 0. As we see, the Växjömodel is not from the very beginning coupled to a representation of information in a Hilbert space. Our aim is to generate an analogue (may be not direct) of the notion of incompatible (complementary) observables starting not from the mathematical formalism of quantum mechanics, but on the basis of the Växjömodel, i.e. directly from statistical data. Why do I dislike the conventional identification of incompatibility with noncommutativity? The main reason is that typically the mathematical formalism of quantum mechanics is identified with it as a physical theory. Therefore the quantum incompatibility represented through noncommutativity is rigidly coupled to the micro-world. (The only possibility to transfer quantum behavior to the macro-world is to consider physical states of the Bose–Einstein condensate type.) We shall see that some Växjömodels can be represented as the conventional quantum model in the complex Hilbert space. However, the Växjömodel is essentially more general than the quantum model. In particular, some Växjömodels can be represented not in the complex, but in hyperbolic Hilbert space (the Hilbert module over the two dimensional Clifford algebra with the generator j : j2 = +1). 148 Andrei Khrennikov

Another point is that the terminology — incompatibility — is misleading in our approach. The quantum mechanical meaning of compatibility is the possibility to measure two observables, a and b simultaneously. In such a case they are represented by commuting operators. Consequently incompatibility implies the impossibility of simultaneous measurement of a and b. In the Växjömodel there is no such a thing as fundamental impossibility of simultaneous measurement. We present the viewpoint that quantum incompatibility is just a consequence of information supplementarity of observables a and b. The information which is obtained via a measurement of, e.g. b can be non trivially updated by additional information which is contained in the result of a measurement of a. Roughly speaking if one knows a value of b, say b = y, this does not imply knowing the fixed value of a and vice versa, see Ref. 16 for details. We remark that it might be better to use the notion “complementary,” instead of “supplementary.” However, the first one was already reserved by Nils Bohr for the notion which very close to “incompatibility.” In any event Bohr’s complementarity implies mutual exclusivity that was not the point of our considerations. Supplementary processes take place not only in physical micro-systems. For example, in the brain there are present supplementary mental processes. Therefore the brain is a (macroscopic) QL-system. Similar supplementary processes take place in economy and in particular at financial market. There one could also use quantum-like descriptions.8 But the essence of the quantum-like descriptions is not the representation of information in the complex Hilbert space, but incomplete (projection-type) representations of information. It seems that the Växjömodel provides a rather general description of such representations. We introduce a notion of supplementary which will produce in some cases the quantum-like representation of observables by noncommuting operators, but which is not identical to incompatibility (in the sense of impossibility of simultaneous observations) nor complementarity (in the sense of mutual exclusivity).

Deﬁnition 2. Let a, b ∈ O. The observable a is said to be supplementary to the observable b if

pa|b(x|y) 6= 0, (2) for all x ∈ Xa, y ∈ Xb.

Let a = x1, x2 and b = y1, y2 be two dichotomous observables. In this Quantum-like Probabilistic Models Outside Physics 149 case (2) is equivalent to the condition: pa|b(x|y) 6= 1, (3) for all x ∈ Xa, y ∈ Xb. Thus by knowing the result b = y of the b-observation we are not able to make the definite prediction about the result of the a- observation. Suppose now that (3) is violated (i.e., a is not supplementary to b), for example: a|b p (x1|y1) = 1, (4) a|b and, hence, p (x2|y1) = 0. Here the result b = y1 determines the result a = x1. In future we shall consider a special class of Växjömodels in that the a|b a|b 2 matrix of transition probabilities P = (p (xi|yj))i,j=1 is double stochas- a|b a|b a|b a|b tic: p (x1|y1)+p (x1|y2) = 1; p (x2|y1)+p (x2|y2) = 1. In such a case the condition (4) implies that a|b p (x2|y2) = 1, (5) a|b and, hence, p (x1|y2) = 0. Thus also the result b = y2 determines the result a = x2. We point out that for models with double stochastic matrix a|b a|b 2 P = (p (xi|yj))i,j=1 the relation of supplementary is symmetric! In general it is not the case. It can happen that a is supplementary to b : each a-measurement gives us additional information updating information obtained in a preceding measurement of b (for any result b = y). But b can be non-supplementary to a. Let us now come back to Example 1. The observables a and b are supplementary in our meaning. Consider now the classical Kolmogorov model and suppose that we are able to measure not only the random variables ξa and ξb — observables a and b, but also the random variable η. We denote the corresponding observable by d. The pairs of observables (d, a) and (d, b) are non-supplementary: a|d a|d p (x1|γi) = 0, i = 3, 4; p (x2|γi) = 0, i = 1, 2, and, hence, a|d a|d p (x1|γi) = 1, i = 1, 2; p (x2|γi) = 1, i = 3, 4.

Thus if one knows , e.g. that d = γ1 then it is deﬁnitely that a = x1 and so on. 150 Andrei Khrennikov

7. A Statistical Test to Find Quantum-like Structure We consider examples of cognitive contexts: 1) C can be some selection procedure that is used to select a special group SC of people or animals. Such a context is represented by this group SC (so this is an ensemble of cognitive systems). For example, we select a group Sprof.math. of professors of mathematics (and then ask questions a or (and) b or give corresponding tasks). We can select a group of people of some age. We can select a group of people having a “special mental state”: for example, people in love or hungry people (and then ask questions or give tasks). 2) C can be a learning procedure that is used to create some special group of people or animals. For example, rats can be trained to react to special stimulus. We can also consider social contexts. For example, social classes: proletariat-context, bourgeois-context; or war-context, revolution-context, context of economic depression, poverty-context, and so on. Thus our model can be used in social and political sciences (and even in history). We can try to find quantum-like statistical data in these sciences. We describe a mental interference experiment. Let a = x1, x2 and b = y1, y2 be two dichotomous mental observables: x1=yes, x2=no, y1=yes, y2=no. We set X ≡ Xa = {x1, x2},Y ≡ Xb = {y1, y2} (“spectra” of observables a and b). Observables can be two different questions or two different types of cognitive tasks. We use these two fixed reference observables for probabilistic representation of cognitive contextual reality given by C. We perform observations of a under the complex of cognitive conditions C :

the number of results a = x pa(x) = . the total number of observations

So pa(x) is the probability to get the result x for observation of the a under the complex of cognitive conditions C. In the same way we find probabilities pb(y) for the b-observation under the same cognitive context C. As was supposed in axiom 1, cognitive contexts Cy can be created corresponding to selections with respect to fixed values of the b-observable. The context Cy (for fixed y ∈ Y ) can be characterized in the following way. By measuring the b-observable under the cognitive context Cy we shall obtain the answer b = y with probability one. We perform now the a-measurements Quantum-like Probabilistic Models Outside Physics 151 under cognitive contexts Cy for y = y1, y2, and find the probabilities: number of the result a = x for context C pa|b(x|y) = y number of all observations for context Cy where x ∈ X, y ∈ Y. For example, by using the ensemble approach to prob- a|b ability we have that the probability p (x1|y2) is obtained as the frequency of the answer a = x1 = yes in the ensemble of cognitive system that have already answered b = y2 = no. Thus we first select a sub-ensemble of cognitive systems who replies no to the b-question, Cb=no. Then we ask systems belonging to Cb=no the a-question. It is assumed (and this is a very natural assumption) that a cognitive system is “responsible for her (his) answers.” Suppose that a system τ has answered b = y2 = no. If we ask τ again the same question b we shall get the same answer b = y2 = no. This is nothing else than the mental form of the von Neumann projection postulate: the second measurement of the same observable, performed immediately after the first one, will yield the same value of the observable). Classical probability theory tells us that all these probabilities have to be connected by the so called formula of total probability:

a b a|b b a|b p (x) = p (y1)p (x|y1) + p (y2)p (x|y2), x ∈ X.

However, if the theory is quantum-like, then we should obtain7 the formula of total probability with an interference term:

a b a|b b a|b p (x) = p (y1)p (x|y1) + p (y2)p (x|y2) (6)

p¯a(x) − p¯b(y )¯pa|b(x|y ) − p¯b(y )¯pa|b(x|y ) λ¯(a = x|b, C) = p 1 1 2 2 , b a|b b a|b 2 p¯ (y1)¯p (x|y1)¯p (y2)¯p (x|y2) 152 Andrei Khrennikov where the symbolp ¯ is used for empirical probabilities – frequencies. An empirical situation with λ(a = x|b, C) 6= 0 would yield evidence for quantum- like behaviour of cognitive systems. In this case, starting with (experimentally calculated) coefficient of interference λ(a = x|b, C) we can proceed either to the conventional Hilbert space formalism (if this coefficient is bounded by 1) or to so called hyperbolic Hilbert space formalism (if this coefficient is larger than 1). In the first case the coefficient of interference can be represented in the trigonometric form λ(a = x|b, C) = cos θ(x), Here θ(x) ≡ θ(a = x|b, C) is the phase of the a-interference between cognitive contexts C and Cy, y ∈ Y. In this case we have the conventional formula of total probability with the interference term:

a b a|b b a|b p (x) = p (y1)p (x|y1) + p (y2)p (x|y2) (8) q b a|b b a|b +2 cos θ(x) p (y1)p (x|y1)p (y2)p (x|y2). In principle, it could be derived in the conventional Hilbert space formalism. But we chosen the inverse way. Starting with (8) we could introduce a “mental wave function” ψ ≡ ψC (or pure quantum-like mental state) belonging to this Hilbert space. We recall that in our approach a mental wave function ψ is just a representation of a cognitive context C by a complex probability amplitude. The latter provides a Hilbert representation of statistical data about context which can be obtained with the help of two ﬁxed observables (reference observables).

8. The Wave Function Representation of Contexts In this section we shall present an algorithm for representation of a context (in fact, probabilistic data on this context) by a complex probability amplitude. This QL representation algorithm (QLRA) was created by the author.7 It can be interpreted as a consistent projection of classical probabilistic description (the complete one) onto QL probabilistic description (the incomplete one). Let C be a context. We consider only contexts with trigonometric interference for supplementary observables a and b — the reference observables for coming QL representation of contexts. The collection of all trigonometric contexts, i.e., contexts having the coeﬃcients of supplementarity (with respect to two ﬁxed observables a and b) bounded by one, is denoted by the tr tr symbol C ≡ Ca|b. We again point out to the dependence of the notion of a trigonometric context on the choice of reference observables. Quantum-like Probabilistic Models Outside Physics 153

We now point directly to dependence of probabilities on contexts by using the context lower index C. The interference formula of total probability (6) can be written in the following form: X q a b a|b b a|b pC (x) = pC (y)p (x|y) + 2 cos θC (x) Πy∈Y pC (y)p (x|y) . (9) y∈Xb By using the elementary formula: √ √ √ D = A + B + 2 AB cos θ = | A + eiθ B|2,

a for A, B > 0, θ ∈ [0, 2π], we can represent the probability pC (x) as the square of the complex amplitude (Born’s rule):

a 2 pC (x) = |ϕC (x)| , (10) where a complex probability amplitude is deﬁned by

q q b a|b iθC (x) b a|b ϕ(x) ≡ ϕC (x) = pC (y1)p (x|y1) + e pC (y2)p (x|y2) . (11)

We denote the space of functions: ϕ : Xa → C, where C is the ﬁeld of complex numbers, by the symbol Φ = Φ(Xa, C). Since Xa = {x1, x2}, the Φ is the two dimensional complex linear space. By using the representation (11) we construct the map J a|b : C tr → Φ(X, C) which maps contexts (complexes of, e.g. physical or social conditions) into complex amplitudes. This map realizes QLRA. The representation (10) of probability is nothing other than the famous Born rule. The complex amplitude ϕC (x) can be called a wave function of the complex of physical conditions (context) C or a (pure) state. We a set ex(·) = δ(x − ·). The Born’s rule for complex amplitudes (10) can be rewritten in the following form:

a a 2 pC (x) = |(ϕC , ex)| , (12) where the scalar product in the space Φ(X,C) is deﬁned by the standard formula: X (ϕ, ψ) = ϕ(x)ψ¯(x). (13)

x∈Xa a The system of functions {ex}x∈Xa is an orthonormal basis in the Hilbert space H = (Φ, (·, ·)). 154 Andrei Khrennikov

Let Xa be a subset of the real line. By using the Hilbert space representation of the Born’s rule we obtain the Hilbert space representation of the expectation of the reference observable a: X X 2 a a E(a|C) = x|ϕC (x)| = x(ϕC , ex)(ϕC , ex) = (ˆaϕC , ϕC ),

x∈Xa x∈Xa where the (self-adjoint) operatora ˆ : H → H is determined by its eigenvec- a a tors:ae ˆ x = xex, x ∈ Xa. This is the multiplication operator in the space of complex functions Φ(X, C): aϕˆ (x) = xϕ(x). It is natural to represent this reference observable (in the Hilbert space model) by the operatora. ˆ We would like to have Born’s rule not only for the a-observable, but also for the b-observable. b b 2 pC (y) = |(ϕ, ey)| , y ∈ Xb. Thus both reference observables would be represented by self-adjoint oper- a b ators determined by bases {ex}, {ey}, respectively. b How can we deﬁne the basis {ey} corresponding to the b-observable? Such a basisq can be found starting with interference of probabilities. We b b a|b √ set uj = pC (yj), pij = p (xj|yi), uij = pij , θj = θC (xj). We have: b b b b ϕ = u1e1 + u2e2, (14) where

b b iθ1 iθ2 e1 = (u11, u12), e2 = (e u21, e u22). (15) a|b We consider the matrix of transition probabilities P = (pij ). It is always a stochastic matrix: pi1 + pi2 = 1, i = 1, 2). We remind that a matrix is called double stochastic if it is stochastic and, moreover, p1j +p2j = 1, j = 1, 2. The b a|b system {ei } is an orthonormal basis iﬀ the matrix P is double stochastic and probabilistic phases satisfy the constraint: θ2 −θ1 = π mod 2π, see Ref. 7 for details. It will be always supposed that the matrix of transition probabilities Pa|b is double stochastic. In this case the b-observable is represented by the ˆ b operator b which is diagonal (with eigenvalues yi) in the basis {ei }. The Kolmogorovian conditional average of the random variable b coincides with the quantum Hilbert space average: X b ˆ tr E(b|C) = ypC (y) = (bφC , φC ),C ∈ C .

y∈Xb Quantum-like Probabilistic Models Outside Physics 155

9. Brain as a System Performing a Quantum-like Processing of Information The brain is a huge information system that contains millions of elementary mental states . It could not “recognize” (or “feel”) all those states at each instant of time t. Our fundamental hypothesis is that the brain is able to create the QL-representations of mind. At each instant of time t the brain creates the QL-representation of its mental context C based on two supplementary mental (self-)observables a and b. Here a = (a1, ..., an) and b = (b1, ..., bn) can be very long vectors of compatible (non-supplementary) dichotomous observables. The reference observables a and b can be chosen (by the brain) in different ways at different instances of time. Such a change of the reference observables is known in cognitive sciences as a change of representation. A mental context C in the a|b− representation is described by the mental wave function ψC . We can speculate that the brain has the ability to feel this mental field as a distribution on the space X. This distribution is given 2 by the norm-squared of the mental wave function: |ψC (x)| . In such a model it might be supposed that the state of our consciousness is represented by the mental wave function ψC . The crucial point is that in this model consciousness is created through neglecting an essential volume of information contained in subconsciousness. Of course, this is not just a random loss of information. Information is selected through QLRA, see (11): a mental context C is projected onto the complex probability amplitude ψC . The (classical) mental state of sub-consciousness evolves with time C → C(t). This dynamics induces dynamics of the mental wave function ψ(t) =

ψC(t) in the complex Hilbert space. Further development of our approach (which we are not able to present here) induces the following model of brain’s functioning9 : The brain is able to create the QL-representation of mental contexts, C → ψC (by using the algorithm based on the formula of total probability with interference).

10. Brain as Quantum-like Computer The ability of the brain to create the QL-representation of mental contexts induces functioning of the brain as a quantum-like computer. The brain performs computation-thinking by using algorithms of quantum computing in the complex Hilbert space of mental QL-states. We emphasize that in our approach the brain is not quantum computer, 156 Andrei Khrennikov but a QL-computer. On one hand, a QL-computer works totally in accordance with the mathematical theory of quantum computations (so by using quantum algorithms). On the other hand, it is not based on superposition of individual mental states. The complex amplitude ψC representing a mental context C is a special probabilistic representation of information states of the huge neuronal ensemble. In particular, the brain is a macroscopic QL-computer. Thus the QL-parallelism (in the opposite to conventional quantum parallelism) has a natural realistic base. This is real parallelism in the working of millions of neurons. The crucial point is the way in which this classical parallelism is projected onto dynamics of QL-states. The QL- brain is able to solve NP -problems. But there is nothing mysterious in this ability: an exponentially increasing number of operations is performed through involving of an exponentially increasing number of neurons. We point out that by coupling QL-parallelism to working of neurons we started to present a particular ontic model for QL-computations. We shall discuss it in more detail. Observables a and b are self-observations of the brain. They can be represented as functions of the internal state of brain ω. Here ω is a parameter of huge dimension describing states of all neurons in the brain: ω = (ω1, ω2, ..., ωN ):

a = a(ω), b = b(ω).

The brain is not interested in concrete values of the reference observables at ﬁxed instances of time. The brain ﬁnds the contextual probability dis- a b tributions pC (x) and pC (y) and creates the mental QL-state ψC (x), see QLRA – (11). Then it works with the mental wave function ψC (x) by using algorithms of quantum computing.

11. Two Time Scales as the Basis of the QL-representation of Information The crucial problem is to ﬁnd a mechanism for producing contextual probabilities. We think that they are frequency probabilities that are created in the brain in the following way.There are two scales of time: a) internal scale, τ-time; b) QL-scale, t-time. The internal scale is ﬂner than the QL- scale. Each instant of QL-time t corresponds to an interval ∆ of internal time τ. We might identify the QL-time with mental (psychological) time and the internal time with physical time. We shall also use the terminology: pre-cognitive time-scale - τ and cognitive time-scale - t. During the interval ∆ of internal time the brain collects statistical data Quantum-like Probabilistic Models Outside Physics 157 for self-observations of a and b. The internal state ω of the brain evolves as

ω = ω(τ, ω0).

This is a classical dynamics (which can be described by a stochastic differential equation). At each instance of internal time τ there are performed nondisturbative self-measurements of a and b. These are realistic measurements: the brain gets values a(ω(τ, ω0)), b(ω(τ, ω0)). By finding frequencies of realization of fixed values for a(ω(τ, ω0)) and b(ω(τ, ω0)) during the interval ∆ of internal a b time, the brain obtains the frequency probabilities pC (x) and pC (y). These probabilities are related to the instant of QL-time time t corresponding a b to the interval of internal time ∆ : pC (t, x) and pC (t, y). We remark that in these probabilities the brain encodes huge amount of information — millions of mental “micro-events” which happen during the interval ∆. But the brain is not interested in all those individual events. (It would be too disturbing and too irrational to take into account all those fluctuations of mind.) It takes into account only the integral result of such a pre-cognitive activity (which was performed at the pre-cognitive time scale). For example, the mental observables a and b can be measurements over different domains of brain. It is supposed that the brain can “feel” probabil- a b ities(frequencies) pC (x) and pC (y), but not able to “feel” the simultaneous probability distribution pC (x, y) = P (a = x, b = y|C). This is not the problem of mathematical existence of such a distribution. This is the problem of integration of statistics of observations from different domains of the brain. By using the QL-representation based only on prob- a b abilities pC (x) and pC (y) the brain could be able to escape integration of information about individual self-observations of variables a and b related to spatially separated domains of brain. The brain need not couple these domains at each instant of internal (pre-cognitive time) time τ. It couples a them only once in the interval ∆ through the contextual probabilities pC (x) b and pC (y). This induces the huge saving of time and increasing of speed of processing of mental information. One of fundamental consequences for cognitive science is that our mental images have the probabilistic structure. They are products of transition from an extremely fine pre-cognitive time scale to a rather rough cognitive time scale. Finally, we remark that a similar time scaling approach was developed in Ref. 3 for ordinary quantum mechanics. In Ref. 3 quantum expectations appear as results of averaging with respect to a prequantum time scale. 158 Andrei Khrennikov

There was presented an extended discussion of possible choices of quantum and prequantum time scales. We can discuss the same problem in the cognitive framework. We may try to estimate the time scale parameter ∆ of the neural QL-coding. There are strong experimental evidences, see, Ref 21 that a moment in psychological time correlates with ≈ 100 ms of physical time for neural activity. In such a model the basic assumption is that the physical time required for the transmission of information over synapses is somehow neglected in the psychological time. The time (≈ 100 ms) required for the transmission of information from retina to the inferiotemporal cortex (IT) through the primary visual cortex (V1) is mapped to a moment of psychological time. It might be that by using

∆ ≈ 100ms we shall get the right scale of the QL-coding. However, it seems that the situation is essentially more complicated. There are experimental evidences that the temporial structure of neural functioning is not homogeneous. The time required for completion of color information in V4 (≈ 60 ms) is shorter that the time for the completion of shape analysis in IT (≈ 100 ms). In particular it is predicted that there will be under certain conditions a rivalry between color and form perception. This rivalry in time is one of manifestations of complex level temporial structure of brain. There may exist various pairs of scales inducing the QL-representations of information.

12. The Hilbert Space Projection of Contextual Probabilistic Dynamics Let us assume that the reference observables a and b evolve with time: x = x(t), y = y(t), where x(t0) = a and y(t0) = b. To simplify considerations, we consider evolutions which do not change ranges of values of the reference observables: Xa = {x1, x2} and Xb = {y1, y2} do not depend on time. Thus, for any t, x(t) ∈ Xa and y = y(t) ∈ Xb. In particular, we can consider the very special case when the dynamical reference observables correspond to classical stochastic processes: x(t, ω)), y(t, ω)), where x(t0, ω) = a(ω) and y(t0, ω) = b(ω). Under the previous assumption these are random walks with two-points state spaces Xa and Xb. However, we recall that in general we do not assume the existence of Kolmogorov measure-theoretic representation. Quantum-like Probabilistic Models Outside Physics 159

Since our main aim is the contextual probabilistic realistic reconstruction of QM, we should restrict our considerations to evolutions with the trigonometric interference. We proceed under the following assumption: (CTRB) (Conservation of trigonometric behavior) The set of trigonometric contexts does not depend on time: C tr = C tr = C tr . x(t)|y(t) x(t0)|y(t0) a|b By (CTRB) if a context C ∈ C tr , i.e. at the initial instant of x(t0)|y(t0) time the coefficients of statistical disturbance |λ(x(t0) = x|y(t0),C)| ≤ 1, then the coefficients λ(x(t) = x|y(t),C) will always fluctuate in the segment [0, 1]. § For each instant of time t, we can use QLRA, see (11): a context C can be represented by a complex probability amplitude: q x(t)|y(t) y(t) x(t)|y(t) ϕ(t, x) ≡ ϕC (x) = pC (y1)p (x|y1) q x(t)|y(t) y(t) iθC (x) x(t)|y(t) +e pC (y2)p (x|y2). We remark that the observable y(t) is represented by the self-adjoint oper- atory ˆ(t) defined by its with eigenvectors: µ p ¶ µ p ¶ p (x |y ) p (x |y ) b p t 1 1 b iθC (t) pt 1 2 e1t = e2t = e pt(x2|y1) − pt(x2|y1)

x(t)|y(t) x(t)|y(t) b where pt(x|y) = p (x|y), θC (t) = θC (x1) and where we set ejt ≡ y(t) x(t)|y(t) x(t)|y(t) ej . We recall that θC (x2) = θC (x1) + π, since the matrix of transition probabilities is assumed to be double stochastic for all instances of time. We shall describe dynamics of the wave function ϕ(t, x) starting with following assumptions (CP) and (CTP). Then these assumptions will be completed by the set (a)-(b) of mathematical assumptions which will imply the conventional Schr¨odingerevolution. (CP)(Conservation of b-probabilities) The probability distribution of the y(t) b(t0) b-observable is preserved in process of evolution: pC (y) = pC (y), y ∈ X , for any context C ∈ C tr . This statistical conservation of the b- b a(t0)|b(t0) quantity will have very important dynamical consequences. We also assume that the law of conservation of transition probabilities holds: (CTP) (Conservation of transition probabilities) Probabilities pt(x|y) are conserved in the process of evolution: pt(x|y) = pt0 (x|y) ≡ p(x|y).

§Of course, there can be considered more general dynamics in which the trigonometric probabilistic behaviour can be transformed into the hyperbolic one and vice versa. 160 Andrei Khrennikov

Under the latter assumption we have:

b b b i[θC (t)−θC (t0)] b e1t ≡ e1t0 , e2t = e e2t0 . ˆ ˆ ˆ For such an evolution of the y(t)-basis b(t) = b(t0) = b. Hence the whole stochastic process y(t, ω) is represented by one ﬁxed self-adjoint operator ˆb. This is a good illustration of incomplete QL representation of information. Thus under assumptions (CTRB), (CP) and (CTP) we have:

b b b b b b iξC (t,t0) b b ϕ(t) = u1e1t + u2e2t = u1e1t0 + e u2e2t0 , q b b(t0) where uj = pC (yj), j = 1, 2, and ξC (t, t0) = θC (t) − θC (t0). Let us consider the unitary operator Uˆ(t, t0): H → H deﬁned by this transfor- b b b b b ˆ mation of basis: et0 → et . In the basis et0 = {e1t0 , e2t0 } the U(t, t0) can be represented by the matrix: µ ¶ 1 0 Uˆ(t, t0) = . 0 eiξC (t,t0) We obtained the following dynamics in the Hilbert space H:

ϕ(t) = Uˆ(t, t0)ϕ(t0). (16) This dynamics looks very similar to the Schrödinger dynamics in the Hilbert space. However, the dynamics (16) is essentially more general than Schrödinger’sdynamics. In fact, the unitary operator Uˆ(t, t0) = Uˆ(t, t0,C) depends on the context C, i.e., on the initial state ϕ(t0): Uˆ(t, t0) ≡ Uˆ(t, t0, ϕ(t0)). So, in fact, we derived the following dynamical equation: ϕ(t) = Uˆ(t, t0, ϕ0)ϕ0, where, for any ϕ0, Uˆ(t, t0, ϕ0) is a family of unitary operators. The conditions (CTRB), (CP) and (CTP) are natural from the physical viewpoint (if the b-observable is considered as an analog of energy, see further considerations). But these conditions do not imply that the Hilbert space image of the contextual realistic dynamics is a linear unitary dynamics. In general the Hilbert space projection of the realistic prequantum dynamics is nonlinear. To obtain a linear dynamics, we should make the following assumption: (CI) (Context independence of the increment of the probabilistic phase) The ξC (t, t0) = θC (t) − θC (t0) does not depend on C. iξ(t,t0) Under this assumption the unitary operator Uˆ(t, t0) = diag(1, e ) does not depend on C. Thus the equation (16) is the equation of the linear unitary evolution. The linear unitary evolution (16) is still essentially more general than the conventional Schrödingerdynamics. To obtain the Schrödingerevolution, we need a few standard mathematical assumptions: Quantum-like Probabilistic Models Outside Physics 161

¶ (a) Dynamics is continuous: the map (t, t0) → Uˆ(t, t0) is continuous ; (b) Dynamics is deterministic; (c) Dynamics is invariant with respect to time- shifts; Uˆ(t, t0) depends only on t − t0 : Uˆ(t, t0) ≡ Uˆ(t − t0). The assumption of determinism can be described by the following relation: φ(t; t0, φ0) = φ(t; t1, φ(t1; t0, φ0)), t0 ≤ t1 ≤ t, where φ(t; t0, φ0) = Uˆ(t, t0)φ0. It is well known that under the assumptions (a), (b), (c) the family of (linear) unitary operators Uˆ(t, t0) corresponds to the one parametric group − i Htˆ of unitary operators: Uˆ(t) = e h , where Hˆ : H → H is a self-adjoint operator. Here h > 0 is a scaling factor (e.g. the Planck constant). We have: Hˆ = diag(0, E), where hθ (t) − θ (t )i E = −h C C 0 . t − t0 Hence the Schrödingerevolution in the complex Hilbert space corresponds to the contextual probabilistic dynamics with the linear evolution of the probabilistic phase: E θ (t) = θ (t ) − (t − t ). C C 0 h 0 We, finally, study the very special case when the dynamical reference observables correspond to classical stochastic processes: x(t, ω)), y(t, ω)). This is a special case of the Växjömodel: there exist the Kolmogorov representation of contexts and the reference observables. Let us consid- xer a stochastic process (rescaling of the process y(t, ω)) : H(t, ω) = 0 if y(t, ω) = y1 and H(t, ω) = E if y(t, ω) = y2. Since the probability distributions of the processes y(t, ω)) and H(t, ω) coincide, we have H(t) H(t0) H(t) H(t0) pC (0) = pC (0); pC (E) = pC (E). If E > 0 we can interpret H(t, ω) as the energy observable and the operator Hˆ as its Hilbert space image. We emphasize that the whole “energy process” H(t, ω) is represented by a single self-adjoint nonnegative operator Hˆ in the Hilbert space. This is again a good illustration of incomplete QL representation of information. This operator, “quantum Hamiltonian”, is the Hilbert space projection of the energy process which is defined on the “prespace” Ω. In principle, random variables H(t1, ω),H(t2, ω), t1 6= t2, can be very different (as functions of ω). We have only the law of statistical H(t) H(t0) conservation of energy: pC (z) ≡ pC (z), z = 0,E.

¶We recall that there is considered the ﬁnite dimensional case. Thus there is no problem of the choice of topology. 162 Andrei Khrennikov

13. Concluding Remarks We created an algorithm – QLRA – for representation of a context (in fact, probabilitic data on this context) by a complex probability amplitude. QLRA can be applied consciously by e.g. scientists to provide a consistent theory which is based on incomplete statistical data. By our interpretation quantum physics works in this way. However, we can guess that a complex system could perform such a self-organization that QLRA would work automatically creating in this system the two levels of organization: CL-level and QL-level. Our hypothesis is that the brain is one of such systems with QLRA-functioning. We emphasize that in the brain QLRA-representation is performed on the unconscious level (by using Freud’s terminology: in the unconsciousness). But the final result of application of QLRA is presented in the conscious domain in the form of feelings, associations and ideas, see Ref. 23. and Ref. 24. We guess that some complex social systems are able to work in the QLRA-regime. As well as for the brain for such a social system, the main advantage of working in the QLRA-regime is neglecting by huge amount of information (which is not considered as important). Since QLRA is based on choosing a fixed class of observables (for a brain or a social system they are self-observables), importance of information is determined by those observables. The same brain or social system can use parallely a number of different QL representations based on applications of QLRA for different reference observables. We can even speculate that physical systems can be (self-) organized through application of QLRA. As was already pointed out, Universe might be the greates user of QLRA.

Conclusion The mathematical formalism of quantum mechanics can be applied outside of physics, e.g. in cognitive, social, and political sciences, psychology, economics and ﬁnances.

References 1. A. Yu. Khrennikov (2002) editor, Quantum Theory: Reconsideration of Foundations. VäxjöUniv. Press, Växjö. 2. G. Adenier and A.Yu. Khrennikov (2005) editors, Foundations of Probability and Physics-3. American Institute of Physics, Melville, NY, 750. 3. A. Yu. Khrennikov (2006) To quantum mechanics through random fluctuations at the planck time scale. http://www.arxiv.org/abs/hep-th/0604011. 4. D. Aerts and S. Aerts (1994) Foundations of Science 1, (1) pp 85–97. Quantum-like Probabilistic Models Outside Physics 163

5. Accardi L (1997) Urne e Camaleoni: Dialogo sulla realta, le leggi del caso e la teoria quantistica, Il Saggiatore, Rome. 6. K. Svozil (1998) Quantum Logic. Springer, Berlin. 7. A. Yu. Khrennikov (2004) Information Dynamics in Cognitive, Psychologi- cal and Anomalous Phenomena. Kluwer Academic, Dordreht. 8. O. Choustova (2006) Quantum bohmian model for financial market. Physica A 374, 304–314 . 9. A. Yu. Khrennikov (2006) Quantum-like brain: Interference of minds. BioSystems 84, 225–241. 10. S. Hameroff (1994) Quantum coherence in microtubules. A neural basis for emergent consciousness? J. of Consciousness Studies 1, pp 91–118. 11. S. Hameroff (1994) Quantum computing in brain microtubules? The Penrose-Hameroff Orch Or model of consciousness. Phil. Tr. Royal Sc., Lon- don A pp 1–28. 12. R. Penrose (1989)The emperor’s new mind. Oxford Univ. Press, New-York. 13. R. Penrose (1994) Shadows of the mind. Oxford Univ. Press, Oxford. 14. J. von Neumann (1955) Mathematical foundations of quantum mechanics, Princeton Univ. Press, Princeton, N.J. 15. G. Birkhoff and J. von Neumann (1936) The logic of quantum mechanics, Ann. Math. 37, pp 823–643. 16. A. Yu. Khrennikov (2005) The principle of supplementarity: A contextual probabilistic viewpoint to complementarity, the interference of probabilities, and the incompatibility of variables in quantum mechanics, Foundations of Physics 35(10), pp 1655–1693. 17. E. Conte, O. Todarello, A. Federici, F. Vitiello, M. Lopane, A. Khrennikov and J. P. Zbilut (2006) Some remarks on an experiment suggesting quantum- like behavior of cognitive entities and formulation of an abstract quantum mechanical formalism to describe cognitive entity and its dynamics, Chaos, Solitons and Fractals 31, pp 1076–1088. 18. G. W. Mackey (1963) Mathematical Foundations of Quantum Mechanics, W. A. Benjamin Inc., New York. 19. A. Yu. Khrennikov (1999) Classical and quantum mechanics on information spaces with applications to cognitive, psychological, social and anomalous phenomena, Foundations of Physics 29, pp 1065–1098. 20. A. N. Kolmogoroff (1933) Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer Verlag, Berlin. 21. K. Mori (2002) On the relation between physical and psychological time. In Toward a Science of Consciousness, Tucson University Press, Tucson, Arizona, p. 102. 22. A. N. Whitehead, (1929) Process and Reality: An Essay in Cosmology, Macmillan Publishing Company, New York. 23. A. Yu. Khrennikov (1998) Human subconscious as the p-adic dynamical system. J. of Theor. Biology 193, pp 179–196. 24. A. Yu. Khrennikov (2002) Classical and quantum mental models and Freud’s theory of unconscious mind, VäxjöUniv. Press, Växjö. This page intentionally left blank 165

Phase Transitions in Biological Matter

Eliano Pessa Centro Interdipartimentale di Scienze Cognitive, Universit`adi Pavia and Dipartimento di Psicologia, Universit`adi Pavia Piazza Botta, 6 , 27100 Pavia, Italy [email protected]

In this paper we will deal with usefulness of physical theory of phase transitions in order to describe phenomena of change occurring in the biological world. In particular, we will assess the role of quantum theory in accounting for the emergence of diﬀerent forms of coherence seemingly characterizing a number of biological behaviours. In this regard we will introduce some arguments that, while supporting the convenience (as well as the unavoidability) of resorting to a quantum-theoretical framework to describe biological emergence, point to the need for a suitable generalization of actual quantum theory. Some possible ways to achieve such a generalization will be shortly discussed.

Keywords: Biophysics; Phase Transition; Biological Emergence PACS(2006): 87.16.Ac; 87.10.+e; 05.70.a; 05.70.Fh; 64.60.i; 89.75.k; 89.75.Fb

1. Introduction The world of biological phenomena (here the attribute ‘biological’ will be used in the widest sense, including even psychological, social, and economic phenomena) is characterized by a number of phenomena of change, whose understanding, forecasting, and (when possible) control is the main goal of many scientific disciplines, such as Biology, Neuroscience, Psychology, So- ciology, Economics. Most phenomena of this kind seem to consist in some form of emergence (this word is here used in an intuitive sense) of a new, coherent, form of organization. Surely the appearance of life itself, under the form of a new individual or of a new species, belongs to this category. But the latter includes even the formation of a flock of birds flying together, the synchronization between electric activities of spatially distant brain neurons, evidenced by the electroencephalographic recordings, the deep understanding of a new concept by a student (the so-called insight), 166 Eliano Pessa the establishment of a new social habit. It is obvious that scientists dealing with phenomena of this kind have been attracted by the conceptual structure of physical theory of phase transitions. The latter, after all, has been dealing with similar problems since more than one century, achieving a remarkable success in explaining and controlling a number of phenomena, even if in a restricted domain. This circumstance has triggered the explosive growth of models trying to use the methods of physical theory of phase transitions (hereafter shortly denoted as TPT) to describe biological changes. Quantum Biology, Quantum Brain Theory, Dynamical Approach in Psychology, Econophysics are all examples of this new way of thinking. Faced with this situation, the basic question is: can TPT be directly applied, as it stands (that is, without modifications or generalizations), to describe biological changes? Would the answer to the basic question be yes, then a further question would arise: what general methodology (if any) should be used to apply TPT to biological problems? Would the answer be no, then the next question would be: how should we modify or generalize TPT? Within this paper we will introduce a number of arguments supporting a partially negative answer to the basic question. The expression ‘partially negative’ means that our conclusion is, not in favor of a reject of TPT, but rather of a suitable (and unavoidable) generalization of it when applied to biological realm. As a consequence, in the final part of this paper we will introduce a number of proposals in order to generalize TPT so as to allow its application to a wider range of phenomena. We feel, however, that these generalizations will turn out to be useful, sooner or later, even in applications concerning strictly physical domains. The organization of the paper can be described as follows. In section two we will summarize the features of TPT which are relevant for our discussion. In section three there will be a synthetic description of main mechanisms underlying biological emergence phenomena so far hypothesized within the actual context of TPT. As these mechanisms are grounded on quantum theory, section four will be devoted to a discussion of the possible role of this latter within biological phenomena, with particular reference to the problem of decoherence. As most arguments based on Quantum Field Theory are heavily dependent on infinite volume limits, section five will contain a discussion of finite volume effects (domains, defects, vortices, and like) arising in phase transitions, as these effects can be of utmost importance when trying to apply TPT to biological realm. Section six will contain a discussion about a number of proposals of generalization of TPT, in order to make easier its application to biological phenomena, while section Phase Transitions in Biological Matter 167 seven will be devoted to a synthetic conclusion ensuing from the arguments presented within the paper.

2. The Main Features of TPT Within this exposition we will assume that the reader is familiar with standard formalism of Thermodynamics, as well as with the one of Statistical Mechanics, and of usual TPT. In this way we can focus only on main conceptual aspects, neglecting the explanation of concepts and formulae which can be easily found on standard textbooks. We start by remarking that, roughly speaking, TPT adopts a framework based on a distinction between microscopic and macroscopic descriptions. The former deal with a world of elementary constituents whose interactions are described through (hopefully) simple laws, while the latter refer to the appearance of phenomena on our normal observational scale. The science of macroscopic descriptions coincides with Thermodynamics and phase transitions belong to phenomena dealt with by this discipline. Namely phase transitions are considered as existing and observable only on a macroscopic scale, even if produced by phenomena occurring at the microscopic level. The connection between macroscopic and microscopic level should be assured by Statistical Mechan- ics, allowing, in principle, a deduction of features of macroscopic phenomena starting only from a knowledge of laws ruling the behaviour of microscopic constituents. Within this context it is possible to introduce a suitable definition of the concept of phase, and characterize in a precise way the conditions associated with a phase transition. We will not enter into details about the complicated history of this subject, and we will limit ourselves to mention that, according to actual views, phase transitions are associated to the presence of discontinuities of derivatives of thermodynamical Gibbs potential of the system. When these discontinuities affect first-order derivatives, we speak of first-order phase transitions, while, in presence of continuous first-order derivatives and of discontinuities in derivatives whose order is greater than the first, we speak of second-order phase transitions. We follow here the classification scheme of phase transitions first introduced by Landau, and then adopted by most modern authors, neglecting the existence of other classification schemata (like the older one proposed by Ehrenfest) which don’t matter for our purposes. Leaving aside technical details (for a summary of which we refer to textbooks on this subjects; see, for instance, Sewell 1986, Chap. 4), we stress that the connection mentioned above between phase transitions and 168 Eliano Pessa discontinuities of derivatives of Gibbs potential is nothing but a consequence of a number of hypotheses, the most important of which can be thus summarized: A) the microscopic dynamics can be described in terms of a suitable set of extensive conserved quantities, including the energy (whence we are in presence of a Hamiltonian dynamics), which are bounded and transform covariantly with respect to space translations; B) it is possible to define in an unambiguous way, starting from these microscopic quantities, their global macroscopic densities, which act as intensive variables at the macroscopic level; C) the macroscopic densities, defined in B), can be used to characterize in an unambiguous way the macroscopic states in the sense that, every time they are constrained to take well defined values, the entropy density of the system is maximized by precisely one translationally invariant state. Already at this level, we can easily realize that these hypotheses can be devoid of any meaning in most biological phenomena. In first place, most observations about these phenomena occur only on a macroscopic scale, in total absence of any hypothesis about the existence of some microscopic entities producing the observed effects. For instance, a psychologist looking at the behaviour of a person troubled by problems will usually try to account for what has been observed in terms of hypothetical (macroscopic) mental processes, without resorting to some unknown microscopic entities (of course, it would be possible to explain the behaviour in terms of brain neurons electrical activity, but the gap between the level of neuronal discharges and the one of human behaviours is still too great to allow this kind of explanation). In second place, even when we are in presence of a microscopic description dealing with interacting elementary components responsible for the behaviours observed on a macroscopic scale, almost never this description can be cast under a Hamiltonian form. Namely in the biological world almost all components are driven by a dissipative dynamics, owing to their interaction with external environment, so that it is virtually impossible to find conserved quantities. Biological cells or individuals belonging to a society are typical examples of open systems, for which there is no form of energy conservation (and perhaps the concept itself of ‘equilibrium’ cannot be defined). However, even when the hypotheses A), B), C) are satisfied, they are not enough for building a theory of phase transitions. The latter has been built by resorting to a suitable combination of phenomenological observations, of phenomenological theories describing them, of suitable theoretical Phase Transitions in Biological Matter 169 frameworks into which the latter have been embedded, and of statistical arguments supporting all this machinery. TPT can be thus seen as a sort of complicated device used to accounting for experimental data and, when possible, forecasting them. The operation of this device is not easy to understand and often unsatisfactory. We will shortly summarize the main aspects of this operation, by listing some essentials of actual TPT. Phenomenological observations There are two kinds of observations which played a major role in shap- ing the actual TPT. The first kind includes the behaviours observed when temperature approaches critical temperature, that is the divergence of (generalized) susceptibility, the divergence of amplitude of fluctuations of order parameter, the critical slowing down, and the discontinuity in the curve giving specific heat as a function of temperature. The second kind refers to existence of universality classes, evidenced by the fact that different phase transitions are associated to (almost) the same set of critical exponents appearing in the laws describing macroscopic behaviours near the critical point. It is to be stressed that generally we lack similar observations in biological contexts. In most domains, each phenomenon of change is studied in its own right, without any comparison with other change phenomena, in absence of any search for similarities between different contexts of change. On the other hand, it is to be acknowledged that many models of biological changes include an explicit computation of critical exponents, thus postulating the existence, close to the critical point, of yet unobserved phenomena which should parallel the ones observed in the physical domain. Besides, it is to be taken into account that, within biological realm, observations of the kinds quoted above would be very difficult to make. However we must remark that perhaps, in some cases, observations of this nature have been already made, but get unnoticed. This is particularly the case of electrical recordings of neural activity. The latter are associated to an apparently ‘noisy’ character which makes difficult to extract from the raw signal the typical behaviours which could occur close to a critical point, even because it is very difficult to identify the correct ‘order parameter’ to be used in these contexts. In recent times (see Beggs and Plentz 2003, 2004) it has been reported the observation of neuronal avalanches close to a critical network state, whose EEG was characterized by a characteristic activity pattern. However, other authors (see, for instance, Bédard et al., 2006) questioned the connection between EEG power spectra and the occurrence of critical states. 170 Eliano Pessa

Phenomenological theories In this regard, as it is well known, two main approaches constitute the skeleton of macroscopic TPT: the Ginzburg–Landau theory and the Renor- malization Group. While neglecting any technical description of these topics (which can be easily found on standard textbooks; see, for instance, Patashinskij and Pokrovskij 1979; Amit 1984; Huang 1987; Tolédanoand Tolédano1987; Goldenfeld 1992; Benfatto and Gallavotti 1995; Cardy 1996; Domb 1996; Minati and Pessa 2006, Chap. 5), we will limit ourselves to mention the outstanding importance of two main ideas: on one hand, the identification of phase transitions with symmetry breaking phenomena (and in practice with bifurcation phenomena in Ginzburg–Landau functional), and, on the other hand, the scaling hypothesis, allowing the elimination of all irrelevant parameters close to critical point. While both ideas are correct only in a limited number of cases, however they underlie a huge number of models of phase transitions, so that every concrete computation, done within this context, is, in a direct or indirect way, based on them. Theoretical frameworks In many cases the framework adopted in building models within TPT is the one of classical physics. This doesn’t mean that quantum theory is neglected. In most contexts the interactions allowing the occurrence of phase transitions have a quantum origin (the typical case being the one of ferromagnetism). However in all these cases quantum theory (most often quantum mechanics) is used only to deduce effective potentials which, in further computations, can be considered as entirely classical. The typical case is given by London dispersion forces whose quantum description ends in effective potentials like the celebrated one of Lennard–Jones. The latter are then used in subsequent computations as if they were classical entities, completely neglecting their quantum origin. However, even the approach based on classical physics has its pitfalls, the main one being that this approach entails the belief in classical thermodynamics and in classical statistical physics. The latter, as it is well known (cfr. Rumer and Ryvkin 1980; Akhiezer and Péletminski1980; De Boer and Uhlenbeck 1962), is based on the so-called correlation weakening principle stating that, during a relaxation process, all long-range correlations between the individual motions of single particles tend to vanish when the volume tends to infinity. This, in turn, implies that, while spontaneous symmetry breaking phenomena, like the ones described by Landau theory, are allowed by classical physics without any problem (see, for instance, Green- berger 1978; Sivardière1983; Drugowich de Fel´ıcioand Hipólito1985), they Phase Transitions in Biological Matter 171 give rise to new equilibrium states which are unstable with respect to thermal perturbations, even of small amplitude (one of the first proofs of this fact was given in Stein 1980). At first sight, such a circumstance could not appear as a problem. Af- ter all, nobody pretends to build models granting for structures absolutely insensitive to any perturbation. However, a deeper analysis shows that instability with respect to thermal perturbations is equivalent to instability with respect to long-wavelength disturbances, and whence entails the impossibility of taking infinite volume limits. The latter is a serious flaw, as one of the main pillars of TPT, that is the divergence of correlation length at the critical point, due to the occurrence of infinite-range correlations, has a meaning if and only if we go at the infinite volume limit. Therefore the aforementioned results imply that classical physics is a framework un- tenable if we are searching for a wholly coherent formulation of TPT, in which phenomenological models are in agreement with statistical physics. How to find an alternative framework? The most obvious candidate is given by quantum theories. However, in this regard, it is immediate to recognize that ordinary quantum mechanics must be ruled out, mostly because the celebrated Von Neumann theorem shows that all possible representations of the state vector of a given quantum system are unitarily equivalent. This means that different representations give rise to the same values of probabilities of occurrence of the results of all possible measurements related to the physical system under consideration, independently of the particular representation chosen (for a more recent discussion on this topic see Halvorson 2001; Rédeiand Stölzner2001; Halvorson 2004). In this way it is clearly impossible to describe the different phases of a same quantum system (obviously deeply differing as regards their physical properties), and, a fortiori, phase transitions. At this point the only remaining possibility is to make resort to Quan- tum Field Theory (QFT). The attractiveness of QFT stems from the fact that within QFT, and only within it, there is the possibility of having different, non-equivalent, representations of the same physical system (cfr. Haag 1961; Hepp 1972; a more recent discussion on the consequences arising from this result, often denoted as ‘Haag Theorem’, can be found in Bain 2000; Arageorgis et al. 2002; Ruetsche 2002). As each representation is associated with a particular class of macroscopic states of the system (via quantum statistical mechanics) and this class, in turn, can be identified with a particular thermodynamical phase of the system (for a proof of the correctness of such an identification, see Sewell, 1986), we are forced to conclude that 172 Eliano Pessa only QFT allows for the existence of different phases of the system itself. Within this approach all seems to work very well. Namely the occurrence of a phase transition through the mechanism of spontaneous symmetry breaking (SSB) implies the appearance of collective excitations, which can be viewed as zero-mass particles carrying long-range interactions. They are generally called Goldstone bosons (cfr. Goldstone et al. 1962; for a more general approach see Umezawa 1993). Such a circumstance endows these systems with a sort of generalized rigidity, in the sense that, acting upon one side of the system with an external perturbation, such a perturbation can be transmitted to a very distant location, essentially unaltered. The reason for the appearance of Goldstone bosons is that they act as order- preserving messengers, preventing the system from changing the particular ground state chosen at the moment of the SSB transition. Moreover, the interaction between Goldstone bosons explains the existence of macroscopic quantum objects (Umezawa 1993; Leutwyler 1997). Here, it must be stressed that the long-range correlations associated with a SSB arise as a consequence of a disentanglement between the condensate mode and the rest of system (Shi 2003). This finding, related to the fact that within QFT we have a stronger form of entanglement than within Quantum Me- chanics (Clifton and Halvorson 2001), explains how the structures arising from a SSB in QFT have a very different origin from those arising from a SSB in classical physics. Namely, while in the latter case we need an exact, and delicate, balance between short-range activation (due to non-linearity) and long-range inhibition (due, for instance, to diffusion), a balance which can be broken even by a small perturbation, in the former case we have systems which, already from the starting, lie in an entangled state, with strong correlations which cannot be altered as much by the introduction of perturbations. Despite these advantages, however, the situation within QFT-based approach is not so idyllic at is could appear at a first sight. Namely the mathematical machinery of QFT is essentially based on propagators (or Green functions) which allow to compute only transition probabilities between asymptotic states, without any possibility of describing the transient dynamics occurring during the transitions themselves. And, what is worse, it is easy to prove that, in most cases, such a dynamics must be described by classical physics. In this regard we recall that quantum effects are negligible when, at a given temperature T , the following relationship is satisfied:

kT/(~ωc) >> 1 (1) Phase Transitions in Biological Matter 173 where ωc denotes a suitable characteristic frequency and, as customary, k and ~ denote, respectively, the Boltzmann constant and the Planck constant divided by 2π. Now, close to transition point, we can identify the characteristic frequency with the inverse of spontaneous relaxation time τ which, owing to the phenomenon of critical slowing down, can be written, by resorting to scaling arguments, as: τ ≈ ξz (2) in which ξ is the correlation length and z denotes the so-called dynamical critical exponent. In turn, the correlation length in proximity of the critical point follows the scaling relationship:

ξ ≈ δν . (3) Here ν is another critical exponent and δ denotes a suitable measure of the distance with respect to critical point. For instance, if the ordered phase corresponds to T < Tc , where Tc is the critical temperature, we will chose δ = (Tc − T )/Tc. By substituting (2) and (3) into (1) and identifying kT with kTc , we will ﬁnd that (1) becomes:

−ν z kTc/(~δ ) >> 1. (4) Now, as experimental data evidenced that the value of ν is not very far from – 1 (close to – 0.7), we can simplify (4) under the form:

z kTc >> ~δ . (5)

It is immediate to see that (5) is always satisfied, provided Tc be greater than zero (when Tc = 0 the possibility is left open for quantum phase transitions; see in this regard, for a short review of some examples, Brandes 2005, pp. 449–455). Namely this occurs not only for the trivial case in which δ = 0, but even for nonzero values of δ. For instance, if we deal with a 3-dimensional system and we accept for z the value 3/2 (on which most physicists seem to agree), it is easy to see that, in correspondence to the o critical point of superfluid transition of liquid Helium, that is Tc = 2.18 K, if we choose δ = 10− 6 then the left hand member of the inequality (5) has a value approximately of 3.008 × 10− 2 3 J, while the right hand member has a value of 1.054 × 10− 4 3 J. In short: the framework for describing phase transitions is based on QFT, but the description of what occurs in correspondence to a phase 174 Eliano Pessa transition can be done only in classical terms. No doubt, a very strange situation! This raises a number of further problems, because the transition from the quantum regime, existing far from critical point, and the classical regime, holding close to critical point, is nothing but a phenomenon of decoherence, due to the interaction with the external environment. But, what are the physical features of the latter process (and of the symmetric process of recoherence, taking place after the phase transition and restating the quantum regime)? How to describe the external environment? How these phenomena can influence the formation of structures (like defects) surviving even after the completion of phase transition and signalling its past occurrence? All these questions cannot, unfortunately, be answered within the traditional framework of QFT. Namely these problems have been dealt with by resorting to suitable generalizations of it. We will postpone a discussion of these fundamental topics until the following sections four and five, being here enough to mention the main problems connected to the choice of a framework for TPT. Statistical arguments The main feature related to the use of statistical arguments (and of statistical models) within TPT is that often the latter rely on canonical ensembles and deal therefore with systems in thermal equilibrium with the environment. All that remains to be done, within such a context, is to compute the partition function (admittedly a very difficult task). However, processes far from equilibrium (which are of interest when dealing with biological phenomena) are out of reach by these methods.

3. Attempts to Use TPT to Model Biological Emergence Before dealing with the topic of this section, we must take into account that biological phenomena (and their models) need to be subdivided into diﬀerent classes as a function of their spatial and temporal scales. The latter can be arranged within a sort of hierarchy, which can roughly summarized as follows:

(1) phenomena occurring on atomic or molecular scales, on times of the order of nanoseconds, such as, for instance, the ones related to interactions between electrical dipoles in water solutions, or to conformational changes of macromolecules; (2) phenomena occurring on a cell scale, on times of the order of milliseconds, or dozens of milliseconds, such as the discharge of a neuron; (3) phenomena occurring on the macroscopic scale of a single organ, or Phase Transitions in Biological Matter 175

of a whole living being, on times of the order of seconds, or tenths of seconds, such as, for instance, the visual recognition of a perceived image; (4) phenomena occurring on the macroscopic scale of a whole organism, on times of the order of minutes, or hours, such as ﬁlling a questionnaire, solving a problem, understanding a concept; (5) phenomena occurring on the macroscopic scale of a whole organism, but on times of the order of weeks, months, years, such as the growth of a tree, the acquisition of mathematical competence, the reorganization of personality; (6) phenomena occurring on the macroscopic scale of a set of organisms, on times which often are very long, such as the evolution of a species, the change of economic relationships, the acquisition of new social habits.

At this point, two remarks are in order. The first is that it appears as highly implausible that changes occurring at different levels of this hierarchy can be dealt with by the same theory of phase transitions. The second is that it seems very difficult that phase transitions occurring at the lowest levels of this hierarchy can have a direct influence on phenomena taking place at the highest levels. As a consequence, it seems unlikely that actual TPT can account for all change phenomena occurring at any level, from a) to f). And it appears likewise unlikely that phase transitions occurring, for instance, at level a) can explain the occurrence of phenomena occurring at level e), such as the emergence of consciousness. These considerations let us understand why the applications of traditional TPT have been, so far, limited to level a), with little incursions at level b). However we could justify the need for applying a single TPT to all levels of the previous hierarchy by resorting to one of two different (but mutually compatible) strategies. The first consists in building a many-level TPT (obviously generalizing the actual TPT). This is a very difficult enterprise, as actual statistical physics is typically a two-level theory. However the improvement of our knowledge about the interactions between defects and, more in general, about mesoscopic physics, could be a promising starting point for continuing on this way. The second strategy is based on the assumption that changes occurring at level n be nothing but emergent phenomena related to interactions occurring at level n – 1, and that this emergence is based on mechanisms which are universal (or belonging to universality classes) and independent from n. We could name this statement as hypothesis of universality of interlevel transitions. It has been, more or less implicitly adopted, for instance, by the followers of the so-called connectionist approach in Psychology (the 176 Eliano Pessa approach dates back to the Eighties; for more recent reviews see Chris- tiansen and Chater 2001; Quinlan 2003; Houghton 2005). As a matter of fact, a look at experimental data about human behaviour would seem, at first sight, to prove the existence of phase transitions endowed with features very similar to the ones observed in physical world. Developmental curves for prehension in infants (Wimmer et al. 1998) or for analogical reasoning in children (Hosenfeld et al. 1997), and even for cognitive changes within a psychotherapy (Tang et al. 2005) look very similar to the curve characterizing the lambda transition for liquid helium. Therefore we could consider the hypothesis of universality of interlevel transitions as a working hypothesis which, even if not proved, so far doesn’t seem absurd or contradicting experimental findings. In any case, all these considerations point to a theory which should go beyond traditional TPT. The application of the latter to biological phenomena has been so far limited mostly to level a) and is based on two main mechanisms, which here we call Davydov effect and Fröhlicheffect. Before shortly discussing them, we must remark that at this level the use of quantum framework is widely diffused and accepted since long time, as most phenomena regarding biological macromolecules involve entities such as electrons or ions for which quantum effects can be important (see Helms 2002). Even most chemical reactions of biological relevance, such as enzyme-catalyzed reactions, are dominated by tunneling phenomena which are typically of quantum nature (see, for a review, Kohen and Klinman 1999, and, for a recent example, Masgrau et al. 2006). Even neurons could behave as quantum systems (Bershadskii et al. 2003), and odor recognition mechanism appears to be based on quantum effects (Brookes et al. 2007). Let us now take first into consideration the Davydov effect. It consists in the production of solitons moving on long biomolecular chains, when the metabolic energy inflow is able to produce a localized deformation on the chains themselves. The original context in which Davydov introduced his ideas was the one of protein α-helices in which different neighbouring monomers interact through hydrogen bonds (the typical case is given by muscle myosin). The energy liberated in the hydrolysis of ATP in a site corresponding to a particular monomer gives rise to a vibration excitation of the latter, and this excitation propagates from one monomer to the other, owing to the dipole-dipole interaction between them. However this excitation interacts even with the hydrogen bonds, producing a deformation of spatial lattice structure of the polymer. Such a deformation, in presence of suitable conditions, propagates along the protein under the form of a Phase Transitions in Biological Matter 177 soliton. The origin of Davydov effect, therefore, lies in the non-linear coupling between the vibrational excitation and lattice displacements. Since the appearance of first papers by Davydov on this subject (see Davydov 1973; Davydov and Kislukha 1973; Davydov 1979, 1982) the literature about it has grown in such a way that it is impossible even to quote only the most relevant papers (here we will limit ourselves to mention only papers that, in a way or in another, contain reviews of this topic, such as Scott 1992; Förner1997; Cruzeiro–Hansson and Takeno 1997; Brizhik et al. 2004; Georgiev 2006). Before shortly discussing the Davydov model, we must point out that, strictly speaking, it doesn’t describe a phase transition, but deals, rather, with the state of affairs which could occur after the phase transition has taken place. Namely the goal of this model is only to argument in favour of a possible collective effect whose occurrence is allowed by the fact that model parameter values are confined within suitable ranges. While being obvious that such a confinement results from a previous phase transition in which parameter values crossed some critical point, both the mechanism of this transition and the phase eventually preceding it are not investigated. As regards the explicit formulation of Davydov model, it can be done in many different ways. Davydov himself started from a general Hamiltonian, describing the dynamics of the excitations, of phonons, and their interactions within a 1-dimensional lattice with N sites and constant spacing a, in which each site contains a monomer of mass M. The Hamiltonian operator of this system is written, in second-quantized form, as a sum of three terms:

H = Hex + Hph + Hint (6)

where Hex is the exciton Hamiltonian: X + + + Hex = [(ε − D)An An − J(An An−1 − An An+1)] (7a) n Hph is the phonon Hamiltonian: X 2 2 Hph = (1/2) [κ(un − un−1) + (pn/M)] (7b) n while Hint is the interaction Hamiltonian: X + Hint = χ [(un+1 − un−1)An An]. (7c) n + In the previous formulae An (An) denotes the fermionic creation (annihilation) operator for a quantum excitation in the site n, while un is the displacement operator and pn the momentum operator in the same site (the latter, however, have a different character, as phonons are bosons). More- over, ε is the excitation energy released by a single external input (like ATP 178 Eliano Pessa hydrolysis), D the deformation excitation energy, J is the absolute value of dipole-dipole interaction energy between neighbouring sites, κ the elasticity constant of the lattice, and χ a parameter giving the strength of coupling between vibrational excitations and lattice displacements. At this point many different routes can be followed. For instance one could derive from the Hamiltonian the Heisenberg equations of motion for the operators and then try to derive from the latter suitable consequences. However Davydov adopted a different approach. First of all, he focussed his interest on the stationary states of the Hamiltonian (6), hypothesizing that the latter were containing all information needed to describe the phenomena he was searching for. In the second place, he assumed (the so-called Davydov ansatz) that these stationary states were of the form: X + ψs(t) = an(t) exp[σ(t)]An |0i (8) n where the complex function an(t) describes, through its squared modulus, the distribution of excitations within the molecular chain. This function is normalized in a suitable way (we omit the normalization rule, to save space). Besides, the operator σ(t) is given by: X σ(t) = −(i/~) [βn(t)pn − πn(t) un] (9) n

It can be shown that βn(t) and πn(t) describe, respectively, the average values of displacements and of momenta of molecules in the stationary state. When imposing the condition that, in correspondence to stationary states, the functional hψs| H |ψsi has a minimum value, it is possible to obtain, by exploiting the commutation rules of the different operators appearing in (6), a set of equations which, if we go to a continuum limit in which the site numbers n are replaced by a continuous spatial coordinate x, become a set of partial differential equations for the unknown functions a(x,t), β(x, t) (as regards the function π(x, t) it is possible to show that it can be expressed through β(x, t), as expected on intuitive grounds). A number of manipulations (for details we refer to the literature, chiefly to Davydov 1979, which, despite some mistakes in the formulae, is still one of the best papers on this subject) let to obtain an equation for the function a(x, t) having the form: 2 [(i~) ∂t − Λ + J ∂xx + G |a(x, t)| ] a(x, t) = 0 (10) where:

2 2 1/2 Λ = ε − D + κ − 2 J, G = 4 χ /[κ(1 − s )], s = ν/νac, νac = (κ/M) (11) Phase Transitions in Biological Matter 179

It is easy to recognize in (10) a particular version of Non-linear Schr¨odinger equation which, as it is well known, allows for the existence of solitonic solutions. We will limit ourselves, here, to show the explicit form of the solitonic solution for |a(x, t)|2, given by:

2 2 |a(x, t)| = 2 µ sec h [µ (x − vt − x0)]. (12) Here v denotes the velocity of propagation of soliton along the molecular chain, always lesser than the velocity of propagation of longitudinal sound waves νac. Moreover, the symbol µ is defined by: µ = χ2/[κ(1 − s2)]. (13) Leaving aside the question of biological plausibility of the mechanism proposed by Davydov, we remark that the approach described above leaves open even the question of the stability of soliton with respect to external perturbations, for instance of thermal nature. Without dealing with this difficult problem, here we will limit ourselves to stress that it could be better tackled by adopting, already from the starting, a QFT-based approach. Within the latter (for an extensive list of references we refer to Umezawa et al. 1982; Umezawa 1993; in the following we will partly follow the scheme adopted in Del Giudice et al. 1985; a useful reference is also Matsumoto et al. 1979) the Nonlinear Schrödingerequation (10) is no longer a c-number equation arising as a consequence of the choice of the Hamiltonian (6), but rather a field equation for an excitation quantum field described by a suitable field operator ψ(x, t). It is convenient, in order to save space, to rewrite this field equation in the shortened form: Q (∂) ψ = I(ψ) (14) where, in conformity with (10), the operator Q (∂) is given by:

Q (∂) = (i~) ∂t − Λ + J ∂xx (15) while the ‘current’ operator is: I(ψ) = − Gψ+ψψ (16) Now, in order to endow the theory of this field with a physical meaning, we must connect it with the so-called asymptotic ‘free’ field, the only one which, in principle, is accessible to observation, do not being perturbed by the interactions described by current operator. The asymptotic field ϕ is therefore described by the field equation: Q (∂) ϕ = 0. (17) 180 Eliano Pessa

The complex boson field ϕ is to be conceived as a quasi-particle field, describing the vacuum excitations. The vacuum itself, of course, is nothing but the degenerate vacuum state arising after a symmetry-breaking transition and, in our case of Davydov soliton, consists in the state of the molecular chain in presence of Goldstone bosons (the quasi-particle excitations) granting for the stability of the particular choice of a ground state, occurred after the transition. It is to be taken into account that the interacting field ψ must be expressed as a function of the asymptotic field ϕ, in order to have a physically meaningful theory. This implies that we must have a map of the form:

ψ = F [ϕ]. (18)

In practice such a map (in some contexts called dynamical map) can be found, for instance, by recasting (16) under the form of an integral equation (the so-called Yang–Feldman equation) which, by using (17), can be written synthetically as:

ψ = ϕ + Q−1(∂) ∗ I(ψ) (19) where the asterisk denotes the convolution operator and Q−1(∂) is nothing but the Green function for the ϕ field. At first sight the equation (19) could be solved only by resorting to suitable iteration procedures, which should let express ψ in terms of suitable powers of ϕ, thus specifying the concrete form of the map (18) up to the wanted degree of approximation. However, in order to implement such a computation, we need further information. Namely, as mentioned in the previous section, within a QFT-based framework, we have the possibility of different, not unitarily equivalent, representations of the same system. In our case these representations correspond to the different vacua, arising as a consequence of vacuum degeneracy occurring after the symmetry-breaking phase transition. In order to have physical predictions, we must choose a particular kind of vacuum among the multiplicity of the possible ones. Otherwise the dynamical map is devoid of any physical significance. Namely both (18) and (19) allow the freedom for a choice of the boundary conditions, essentially because the dynamical map holds only in a weak sense, that is relating expectation values and not operators. In order to eliminate the ambiguity in the choice of boundary conditions, let us now introduce a c-number function f(x, t) satisfying the equation:

Q (∂)f = 0 (20) Phase Transitions in Biological Matter 181 and a space-time dependent translation of the asymptotic quantum ﬁeld ϕ deﬁned by: ϕ → ϕ + f. (21) The (21) is also called the boson transformation. It is to be remarked that it is a canonical transformation, as it leaves unchanged the canonical form of commutation relations. Moreover, it can be associated to a generator ∆, allowing to write (21) under the form: ϕ0 = exp(−i∆) ϕ exp(i∆) (22) the explicit form of ∆ being given, in the case of Davydov soliton and within a reference frame comoving with it, by: Z

∆ = − dx [g(x, t) ∂tϕ − ϕ ∂tg(x, t)], g(x, t) = θ(x) f0(x, t) (23) where θ(x) is the Heaviside step function, while f0(x, t) denotes the solution of (20) when the velocity of soliton is set to zero. Let us now denote by |0i the vacuum state for the ﬁeld ϕ. It is, then, convenient to introduce the state |f0i deﬁned by:

|f0i = exp (i∆) |0i . (24) Easy considerations (here, as usually, omitted), based on the fact that the generator ∆ induces a space-time displacement even of the annihilation operator relative to the vacuum state |0i, show that |f0i is a coherent state, as it is an eigenstate of the latter operator. In order to exploit the consequences of the previous formalism, let us first denote by the symbol ψf the interacting quantum field, satisfying (14) or (19), when the asymptotic field, undergoing the boson transformation, has been defined by (21) or (22). It is then immediate to recognize from (19) that ψf fulfils the following form of Yang–Feldman equation: ψf = ϕ + f + Q−1(∂) ∗ I(ψf ) (25) Taking the vacuum expectation value of (25) we will obtain: Φf = h0| ψf |0i = f + h0| Q−1(∂) ∗ I(ψf ) |0i (26) Here we made use of the fact that the vacuum expectation value of the quasi- particle excitation field ϕ is vanishing. The macroscopic order parameter Φf , defined through (26), has clearly a quantum nature, implicitly contained in the developments in powers of ~ needed to obtain an explicit (though approximate) form of the right hand member of this equation. In this regard 182 Eliano Pessa the lowest level of approximation (the so-called Born approximation) is that in which ~ tends to zero and h0| I(ψf ) |0i tends to I(Φf ). In this case (26) becomes: Φf = f + Q−1(∂) ∗ I(Φf ). (27) When applying to both members of (27) the operator Q(∂) we will immediately obtain, by taking into account (20), the equation: Q(∂)Φf = I(Φf ) (28) that is the classical equation describing the Davydov soliton. This time, however, differently from the approach adopted by Davydov, we can more easily understand the origin of the order parameter Φf . Namely from (22) and (24) we can immediately recognize that:

hf0| ϕ |f0i = f (29) so that f appears as a coherent condensation of quasi-particle excitations described by the quantum field ϕ. In absence of the self-interaction term (27) therefore tells us that even Φf is nothing but a coherent condensate of quasi-particles. However, this condensate evolves dynamically as a soliton, owing to non-linear self-interaction contained in the right hand member of (28). We underline that the above picture is valid only within an approximation which is of zero order in ~. Nobody prevents, however, from going to higher-order approximations, where more complicated effects should emerge. All these computations, as well as the stability analysis of the soliton, are allowed by the adoption of a QFT-based framework, in which Gold- stone bosons, arising as a consequence of a symmetry breaking, undergo a coherent condensation, giving rise to observable effects, whose detailed nature depends essentially on the interactions connecting the quantum fields. Within this approach all effects are from the beginning endowed with a quantum nature and we don’t need additional assumptions about the coupling between phonons and vibrational excitations. This occurs because in this case phonons coincide just with the quasi-particle excitations acting as Goldstone bosons. As a proof for this interpretation we can add the fact that the expectation value of density probability of ϕ in the coherent state |f0i fulfils the equation of sound propagation (like phonons). Before going further we remark that, despite the high inner coherence and the conceptual economy allowed by this approach, it leaves unsolved a number of problems, mainly connected to the study of transient processes taking place during the phase transition, not after nor before. Unfortunately these processes have a fundamental role in determining the outcome of a 184 Eliano Pessa

From the latter relationships it would be possible to sketch an hypothetical, even if simplistic, path leading from the critical point of a symmetry- breaking phase transition to the stable propagation of solitons along the macromolecule. As it is possible to see, the critical point of the phase transition can be identified with the condition G = 0. Very close to this critical point we deal with a very small (even if positive) value of G, and therefore of b, owing to (35). As the behaviour is classical, the equation (32) has a solution which, while being a soliton, is very flattened, as it follows from (34). However, such a first solitonic ‘hint’ is just a cue for the subsequent quantum regime, occurring when, as a consequence of the growth of G, the classical description is no longer valid and Goldstone bosons enter into play. Namely the latter give rise to a coherent condensate favoured just by the initial soliton solution, in such a way as to induce the QFT-based dynamics previously described when discussing the boson transformation. In other words, the initial classical solitonic solution ‘forces’ the choice of a particular vacuum state, among the ones present after the symmetry breaking, a choice kept stable against perturbations by the essentially quantum mechanism of coherent boson condensation. Let us now shortly discuss the other mechanism quoted at the beginning of this section, that is the Fröhlich effect. The latter, in its essence, consists in the excitation of a single (collective) vibrational mode within a system of electric dipoles interacting in a nonlinear way with a suitable source of energy, or with a thermal bath. These dipoles are thought to be present both in water constituting the intracellular and extracellular liquid as well as in biological macromolecules, owing to ionization state connected to the existence of high-intensity electric fields close to cellular membranes. This effect can be seen as a sort of Bose condensation of phonons describing dipole vibrational modes, and therefore it appears as a truly quantum effect. The original model proposed by Fröhlich (see Fröhlich 1968; the heuristic arguments introduced in the following have been adapted from the exposition given in Sewell 1986, pp. 209–214) was described in term of kinetic equations describing the rate of changes of occupation numbers ni (i = 1, ..., N) of N phonon modes with frequencies ωi. The equations have the explicit form: dni/dt = si − ai[ni − (1 + ni) exp(−βωi)] X − bik[ni(1 + nk) exp(−βωi) − nk(1 + ni) exp(−βωk)]. (36) k

Here β = ~/kT while si is a constant pumping term. The term with coefficient ai describes the emission and absorption of quanta due to Phase Transitions in Biological Matter 185 interactions with thermal bath, and the term with coefficient bik describes the exchanges of quanta between different modes. These equations can be further simplified by assuming rates which do not depend on mode indices. Then straightforward mathematical transformations reduce them to the form:

dni/dt = s − ni − gθ ni + gν (1 + ni) exp(−βωi) (37) where s is the pump parameter, g a suitable coupling constant and ν, θ are deﬁned by: X X ν = nk/N, θ = (1 + ni) exp (−β ωi)/N. (38) k i When looking at the equilibrium solution of (37), that is the stationary distribution of occupation numbers, it is possible to cast it under the form:

ni = [1 + (1/g) exp(−βωi)]/ {exp[β(ωi − µ)] − 1} (39) where the quantity µ is defined by: µ = (1/β) ln [g s/(1 + g θ)]. (40) It is immediate to recognize that (39) defines a Bose–Einstein distribution, apart from the multiplicative factor [1 + (1/g) exp (−β ωi)]. Within it µ plays the role of a chemical potential. It is then possible to see that, when the pump parameter s tends to a suitable critical value, the chemical potential µ tends to the lowest frequency ω1 so that n1 diverges: a Bose–Einstein condensation of phonon quanta occurs (for a rigorous proof of this assertion we address the reader to the quoted literature, as well as to Del Giudice et al. 1982). It is to be remarked that the critical value of pump parameter defines a true phase transition. Despite the fact that original Fröhlich model was formulated without resorting to a microscopic Hamiltonian, it has been possible, however, to recast it within the language of QFT (see Del Giudice et al. 1985). The QFT-based interpretation of Fröhlich effect is, anyway, slightly more complicated than the one of Davydov effect, as dipoles need a (3+1)-dimensional model, while for solitons on macromolecular systems (1+1)-dimensional models were enough. In the case of Fröhlich effect the coherent state can still be interpreted as deriving from a condensation of quasi-particle Gold- stone modes, but the picture is made more complex because a 3-dimensional space forces to introduce a symmetry breaking of a SU(2)-invariant model, implying a richer mathematical structure and different kinds of Goldstone bosons. In any case, it is to be stressed that in the course of time a number of Hamiltonian descriptions of Fröhlich effect have been proposed, all 186 Eliano Pessa evidencing how Davydov and Fröhlich effects be strictly interrelated, so as to be nothing but different consequences of the same theoretical framework (see, in this regard, Bolterauer et al. 1991; Nestorovic et al. 1998; Mesquita et al. 1998, 2004). Before ending this short discussion of Fröhlich effect, we remark that the adoption of QFT-based perspective allows a natural coupling between the latter and Davydov effect, giving rise to a possible general theory about the origin of biological coherence (see Del Giudice et al. 1985, 1988). Namely, as biological macromolecules are embedded within water, it is possible to hypothesize the existence of a general effect, arising from the coupling of two successive phases. In the first the random release of energy produced by metabolic chemical reactions is channelled (and, in a sense, stored) in an ordered way through solitons travelling along 1-dimensional molecular chains, owing to Davydov effect. In the second phase the motion of solitons themselves induces an electric polarization in the water surrounding the molecular chains themselves. The fact that this polarization has been induced by a non-thermal phenomenon, such as the travelling of a soliton, breaks the rotational symmetry of water dipoles, and, as a consequence, triggers the occurrence of Goldstone bosons (the so-called dipole wave quanta). The condensation of these latter gives rise to a Fröhlich-like effect, resulting in an ordered (electret) state of the system of water dipoles, associated to coherence and appearance of long-range correlations. The net result of the two phases is that a disordered energetic input has produced a coherent state. It is important to remark that this effect is based on a single dynamical scheme, the one of QFT-based description of symmetry breaking. Without entering into further details on this topic, we recall that the approach based on QFT gave rise to a highly interesting theory of the operation of one of most important biological systems: the brain. This Quantum Brain Theory (the huge amount of literature on this topic is summarized in Jibu and Yasue 1995, 2004; Vitiello 2001; Del Giudice et al. 2005) had a number of important applications, such as the study of the role of cytoskeleton in neural cells (Tuszyński et al. 1997, 1998; Hagan et al. 2002; Tuszyński 2006) the operation of memory (Ricciardi and Umezawa 1967; Stuart et al. 1978, 1979; Vitiello 1995; Alfinito and Vitiello 2000; Pessa and Vitiello 2004; Freeman and Vitiello 2006), and the basis for consciousness (Penrose 1994; Hameroff and Penrose 1996; Hameroff et al. 2002; Tuszyński 2006). We can now summarize the discussion performed in this section by saying that, provided we adopt a QFT-based description of phase transitions Phase Transitions in Biological Matter 187 interpreted as symmetry breaking phenomena, traditional TPT is able to account for phase transitions in biological matter at the lowest, microscopic level. However, such a possibility exists only within a perspective in which Goldstone bosons are considered as entities able to produce physical effects through condensation phenomena, a view actually not shared by all physicists. In any case, it is to be stressed that the application itself of QFT to biological phenomena evidenced a number of failures of traditional TPT. These can be listed as follows: 1) within the biological realm the most important process is the interaction between a system and its environment; as it is well known the latter could destroy quantum coherence, so as to transform the picture presented in this section in nothing but a dream, a sort of illusion in disagreement with the realm of experimental observations; 2) the formalism of QFT is based on an Hamiltonian structure, in turn related to a physics based on energy conservation principle; however, most biological phenomena appear as typically dissipative and whence not re- ducible to an Hamiltonian treatment; 3) within the biological world one must take into account finite volume effects, which destroy the picture presented above, as the lack of infinite volume limits opens the way to the formation of defects, impurities, and sometimes of disorder; 4) the previous models are useful only to describe the emergence of biological coherence at the microscopic level; however they are unable to account for the plurality of levels existing in biological matter and to provide the basis for a multi-level approach in which we can describe the formation of a new level from a lower level, in a way which is partially independent from the level taken into consideration; 5) the solution of the problems described in the points 1) - 4) requires the introduction of a description of the environment and whence of a general theory of organism-environment interactions; such a theory, however, is out of reach of actual TPT. In the next sections we will try to discuss the problems 1)-5), by introducing some arguments which perhaps could be useful to the enterprise of generalizing traditional TPT so as to account for biological changes.

4. Decoherence and Dissipation The phenomenon of decoherence consists in the suppression of non-diagonal (interference) terms in the density operator of a quantum system, owing to a number of different possible influences, most of which include the action of 188 Eliano Pessa external environment. This results in the destruction of interference effects connected to entangled states which, in turn, are at the basis of quantum coherence (for reviews on this topic see, for instance, Giulini et al. 1996; Zurek 2003). The existence of this phenomenon induced a number of authors to make claims about the uselessness of quantum theories in the biological domain, as the decoherence time would be, in normal situations in which biological systems operate, so small as to make the classical description of these latter the only one feasible (see, for instance, Tegmark 2000; it is, however, to be taken into account the criticism by Alfinito et al. 2001). A short discussion of decoherence phenomenon should start by listing the possible mechanisms for decoherence so far taken into consideration. Following a tentative classification introduced by Anastopoulos (1999) we will distinguish between two different kinds of mechanisms: the extrinsic and the intrinsic ones. Within the former the decoherence is produced as a consequence of a suitable coupling between the system and the external environment, while the latter give rise to decoherence owing to the form itself of system dynamics, which selects in a very short time particular pointer states. Among the extrinsic mechanisms we can quote: d.1) the interaction with a thermal bath; d.2) the interaction with external noise; d.3) the interaction with a dissipating environment; d.4) the interaction with a chaotic environment; d.5) the introduction of finite volume or of special boundary conditions. We first stress that this list is rather rough, as there can be a significant overlap between different categories — such as, for instance, d.2) and d.3) — when the environment is described by detailed microscopic models. In the second place we must remark that in some cases these mechanisms fail to achieve decoherence (see, for instance, Ford and O’Connell, 2001, as well as the considerations contained in Pascazio 2004, concerning the role of noise in decoherence). In any case it is to be underlined that the assessment of the role of these mechanisms, as well as the estimates of decoherence time, are based mostly on a microscopic description of the environment itself. In this regard, the mechanisms listed above differ one from another in the possibility of giving an explicit description of this kind. While in the case d.1) the environment can be modelled through a system of oscillators in equilibrium at a given temperature, coupled to the system in a suitable way, and in d.2) we can resort to some kind of Brownian motion, in the case d.3) we have only a macroscopic description of a dissipation, whose microscopic description is not fixed in advance. Phase Transitions in Biological Matter 189

A simple example is given by the elementary damped harmonic oscillator:

2 2 2 d x/dt + γ dx/dt + ω0 x = 0 (41)

Here the friction coefficient γ could be related, via the traditional fluctuation-dissipation theorem (in the Callen–Welton interpretation), to the autocorrelation function of force fluctuations. However, this tells us nothing about the role of friction in inducing decoherence, for instance, in a system of coupled quantum oscillators. We should need a further information on a microscopic theory of forces acting between the undamped oscillator and the environment. And — what is worse — even this theory would be useless if, as showed by many authors (see, for instance, Cuglian- dolo et al. 1997; Pérez–Madrid et al. 2003), we were in a context in which fluctuation-dissipation theorem no longer holds. Unfortunately, it seems that such a form of description of dissipation is unavoidable, at least if we want to work within the framework of QFT. As shown by a number of authors (Calzetta and Hu 2000; Calzetta et al. 2001, 2002), the dynamical content of QFT is completely given by the so-called Haag map, which is nothing but an explicit version of Yang– Feldman equation (19), containing an infinite series of approximations in terms of asymptotic fields. It is to be remarked that each term of this series depends on a suitable correlation function of asymptotic fields, whose order depends on the term taken into consideration. As the series is infinite, it will contain correlation function of every order. Besides, we will have that, in general, correlations of different orders are reciprocally connected, so that, for instance, 2-point correlation functions will depend on 3-point correlation functions, and so on. In most cases of interest correlations of higher order can be practically neglected, so that the theory can stop the expansion at a given correlation order. This is the case of traditional QFT, which is limited to 2-point correlation functions. However, in presence of an external environment, the latter can influence higher-order correlations, so as to make important their contribution to lower-order correlations. What occurs, at this point, is that the traditional theory must account for these influences under the form of dissipative terms, generally having a macroscopic form, owing to the practical impossibility of monitoring in a detailed way the influence of changes in higher-order correlations produced by the external environment. This explains why, in principle, every QFT-based description is automatically dissipative in presence of an environment. These arguments support the claim that every application of QFT to biological phenomena should necessary include, within model equations, a 190 Eliano Pessa description of dissipative effects. At a first sight, such a claim seems to give rise to new problems. Namely QFT, like Quantum Mechanics, is based on an Hamiltonian description of dynamics, and, as it is well known, it is very difficult, if not impossible, to cast even the simplest models of dissipative dynamics in an Hamiltonian form. In this regard, it suffices to deal with the elementary example of damped harmonic oscillator described by (41). This case, as shown by a number of authors (see, for instance, Denman and Buch 1973; Nagem et al. 1991) can be described by a (time-dependent) Hamiltonian having the form:

2 2 2 H(x, p) = (1/2)[p exp(−γ t) + ω0 x exp(γ t)] (42) However this Hamiltonian does not solve the problems arising when we try to quantize the damped harmonic oscillator through the well known procedures. Namely from the Schr¨odingerequation associated to (41) it is possible to derive (see, for instance, Edwards 1979; Bolivar 1998) that the uncertainty principle assumes the form:

2 2 1/2 ∆x ∆p = (~ω0/2Ω) exp(−γ t), Ω = [ω0 − (γ /4)] (43) As it is easy to understand, this relationship leads to physically unaccept- able consequences when we let t → +∞. Many different strategies have been proposed to deal in a correct way with the problem of quantization of dissipative systems. Among them we quote the one based on suitable modifications of quantization rules (for an overview see Tarasov 2001), and the one, inspired by Caldeira–Leggett approach (see, among the others, Caldeira and Leggett 1983), which, after introducing a suitable microscopic model of the interaction with the dissipative environment, derives a generalized master equation for the quantum density matrix (the so-called Lindblad form; see, for instance, Lindblad 1976; for an overview see Rajagopal 1998). The latter strategy, while in agreement with general principles of Quantum Mechanics, entails a complex mathematical apparatus, and is heavily based on the choices made in describing the environment (examples of this strategy can be found in Rajagopal and Rendell 2001; Mensky and Stenholm 2003). A more convenient and simpler strategy has been introduced by Celegh- ini, Rasetti and Vitiello, who, developing earlier proposals by Bateman (1931) and Feshbach and Tikochinsky (1977), described the influence of dissipating environment by doubling the original dissipative system through the introduction of a time-reversed version of it, which acts as an absorber of the energy dissipated by the original system (see Celeghini et al. 1992; Phase Transitions in Biological Matter 191 for an overview of both the principles underlying this method and its applications it is highly recommended the reading of Vitiello 2001). In this regard, a dissipative system like the damped harmonic oscillator described by (41) is doubled by introducing its “mirror” (i.e. time-reversed) system given by:

2 2 2 d y/dt − γ dy/dt + ω0 y = 0. (44) It is possible to show that the dynamics of the whole system including both (41) and (44) can be described by the following (time-invariant) Hamilto- nian:

2 2 H(x, y, px, py) = px py + (γ/2)(y py − x px) + [ω0 − (γ /4)] x y. (45) It has to be remarked that in this way the reduction to an Hamiltonian form has been obtained at the price of doubling the number of degrees of freedom. However, this lets us introduce the usual quantization procedures without additional troubles. In particular, when resorting to an occupation number representation we must introduce two diﬀerent kinds of creation and annihilation operators:

+ + a = g (px − i Ω x), a = g (px + i Ω x), b = g (py − i Ω y), b = g (py + i Ω y) (46) where Ω has been defined by (43), while g denotes the quantity (1/2 ~ Ω)1/2. The commutation relations fulfilled by these operators have the form: [a, a+] = 1 = [b, b+], [a, b] = 0 = [a, b+]. (47) In terms of these operators the Hamiltonian (45) becomes: H = ~ Ω(a+b+b+a)+(i ~ Γ/2)[a+a+ −b+b+ −(a a−b b)], Γ = γ/2. (48) As the form (48) is very far from a diagonal one, we can perform a partial diagonalization through the canonical transformation to new operators A and B: √ √ A = (1/ 2)(a + b),B = (1/ 2)(a − b). (49) The latter allows to write the Hamiltonian (49) under the more tractable form: H = ~ Ω(A+A + B+B) + (i ~ Γ)(A+B+ − AB). (50) A straightforward extension of this Hamiltonian to QFT consists in writing (50) under the form of an infinite collection of damped harmonic oscillators: 192 Eliano Pessa

X X + + + + H = ~Ωk(A Ak + B Bk) + (i~Γk)(A B − AkBk) (51) k k k k k k where k denotes the spatial momentum. The simple arguments presented above put into evidence the existence of two diﬀerent possible strategies in using the doubling mechanism. The ﬁrst one starts from a macroscopic classical description of dissipation, introduced by hand, like in (41), and resorts to doubling mechanism in order to obtain a quantum or a QFT counterpart, shown in (50) or (51), so as to deal with a dissipative system without abandoning the traditional quantum formalism. The second one tries to take advantage of eventual (of course, not canonical) transformations leading from the traditional harmonic oscillator Hamiltonian to the damped oscillator Hamiltonian, in order to derive, from a whatsoever non-dissipative Hamiltonian, the corresponding dissipative version, expressed in terms of damped oscillators of the form (41). Obviously, both strategies assume that Γ or Γk be known in advance or can be derived by a suitable microscopic theory. Here we will show a simple example of implementation of the second strategy, starting from the well known harmonic oscillator Hamiltonian:

H = ~ Ω a+a. (52)

Here the zero point energy was discarded to speed up the computations, by redeﬁning the value of energy in the ground state. Now the problem is to introduce a formal procedure in order to transform (52) into the form (48). This can be done through the following steps:

• t.1) introduction of new operators α, β through the canonical transformation: √ √ a = (1/ 2)(α + β), a+ = (1/ 2)(α+ + β+) (53)

• t.2) discarding from the new Hamiltonian, written in terms of α, β, all contributions having the form α+α + β+β, as they cancel one with the other when applied to the vacuum state (they have opposite eigenvalues); this procedure leads to the Hamiltonian:

H = (1/2) ~ Ω(α+β + β+α) (54)

• t.3) introduction of new operators A, B through the (not canonical) formal transformation:

α+ = A+ + A + B+ + B (55a) Phase Transitions in Biological Matter 193

α = (iΓ/2Ω)A+ +[1+(iΓ/2Ω)] A−(iΓ/2Ω) B+ +[1−(iΓ/2Ω)] B (55b)

β+ = A+ − A + B+ − B (55c)

β = (iΓ/2Ω)A+ +[1−(iΓ/2Ω)] A−(iΓ/2Ω) B+ +[1+(iΓ/2Ω)] B (55d)

When we apply (55.a)–(55.d) to the Hamiltonian (54) we will obtain, by discarding the contributions of the form A+A + B+B, a new Hamiltonian which, in terms of the new operators A, B, has exactly the form (48). It is to be remarked that, as expected on general grounds, the choice of the coefficients of (55.a)–(55.d) is not unique. Namely they are solutions of a (nonlinear) system of algebraic equations coding the constraints allowing a satisfactory solution of the problem of transforming (52) into (48). Without reporting here, in order to save space, the complete form of this system, we will limit ourselves to mention that, among these constraints, there are the ones of fulfilment, by the involved operators, of commutation relations having the form (47), as well as the one of granting that the terms A+A and B+B have equal coefficients. Let us now apply the previous findings to a simple toy model, consisting in a (1+1) self-interacting real scalar field described by the Hamiltonian: Z © ª H = dx (1/2)[π2 + (∇ϕ)2] + V (ϕ) ,

V (ϕ) = (1/2) µ2ϕ2 + (1/4) λ ϕ4,

π = (1/c) ∂ϕ/∂ t. (56) By resorting to the usual expansion of field operators in terms of creation and annihilation operators straightforward computations show that the Hamiltonian (56), besides standard contributions of the form (52), contains further terms, coming from λ ϕ4 self-interaction, defined by products of creation and annihilation operators having the form a+a+a+a+, or a a a a, or a+a+a a. If we apply to these operators the transformations (53)–(55) described above we will obtain that the contribution of self-interaction term will contain a large number of different kinds of products of operators A+, A, B+, B. When expressing the latter in terms of momenta and coordinates of damped oscillators and their mirrors through formulae like (46) we will easily obtain that the leading order contributions will be given by terms 2 2 2 2 3 3 3 3 of the form (y py − x px), or (y py − x px), or (y py − x px). In general these terms produce a mixing of the dynamics of every damped oscillator 194 Eliano Pessa with the one of its mirror. However, when we stop our developments to the zero order in px, it becomes possible to obtain a separation of the equation describing the dynamics of a damped oscillator from the one describing the dynamics of its mirror. Within this approximation (small momenta) it is easy to derive that the field equation of the self-interacting scalar field in presence of dissipation has (by using suitable measure units) the form:

ν 2 2 3 ∂ν ∂ ϕ + µ ϕ + (γ + 3 λ ϕ )(∂ϕ/∂t) + λ ϕ = 0. (57)

This equation can be used as a starting point for the study of dynamical evolution of a self-interacting scalar field in presence of dissipation (a problem dealt with by a number of authors by using equations similar to (57); see, for instance, Laguna and Zurek 1997; Dziarmaga et al. 1999). As regards the role of doubling mechanism in QFT, a number of remarks are in order. First of all, it introduces, instead of a microscopic description of the environment, a shortened version of the latter under the form of a suitable “mirror” quantum field. Of course, this field can be always connected to a microscopic description. In any case, however, it is always entangled with the “normal” quantum field describing the original dissipative system. This entanglement characterizes the dissipation as a “quantum dissipation”, very different from a classical one. Namely, within the context of specific models, such as the dissipative quantum brain model (Vitiello 1995; Pessa and Vitiello 2004), it can be shown that the quantum dissipation is a coherence-keeping, rather than a decoherence-producing, mechanism. As a matter of fact, the existence within these models of an infinite multiplicity of different ground states, each one of which is a coherent state resulting from the entanglement of the “normal” field with its “mirror”, is just a consequence of quantum dissipation. Without entering into details for lack of space, it is to be recalled that this approach allows to prove that, within these models, all possible dynamical trajectories in the space of representations of canonical commutation relations have a classical nature and are chaotic (Vitiello 2004; Pessa and Vitiello 2004; Vitiello 2005). If the ground states quoted above are interpreted as memory states of the brain, this result implies that dissipative quantum brain model describes a chaotic memory. In the last years this kind of memory became popular for a number of different reasons. First of all, the investigations of Freeman and coworkers on rabbit olfactory bulb evidenced the prominent role played by chaotic behaviour within the operation of biological systems, and, in particular, of biological memories (Freeman 1987; 1992; 2000a; 2000b; 2005; Freeman and Vitiello 2006; Freeman et al. 2001; Kozma and Freeman 2002; Phase Transitions in Biological Matter 195

Skarda and Freeman 1987). In the second place, a number of researchers evidenced, through both computer simulations and experimental investigations, the possibility of occurrence of chaotic behaviors in single neurons (see, e.g. Aihara et al. 1984; Nakagawa and Okabe 1992). In the third place, chaotic artificial neural networks gained much popularity, being considered as the best models of human memory operation, without being affected by the heavy shortcomings of traditional attractor neural networks. Namely, while in the latter the retrieval dynamics stops in a particular attractor, which most often is a spurious one, owing to the impossibility of controlling the dynamics itself (as these models are nothing but special cases of spin glasses), in a chaotic neural network the retrieval dynamics never stops, as it consists in a never-ending wandering around the different chaotic attractors, characterized by times (generally long enough) in which the systems is in the neighbourhood of one of them, followed by shifts to the neighbours of other attractors. In other words, in a chaotic neural network the retrieval dynamics consists in a visit of the whole memory landscape, rather than of a specific attractor. It is easy to understand that, if the network would be endowed with a rule for controlling the retrieval dynamics in such a way as to transform its chaotic character into an attractive character, we should have realized a memory having just the features observed in human memory. Namely the latter acts on the basis of control strategies which let us know, for instance, if the retrieved pattern is the right or correct one, and, if this should not be the case, trigger a new retrieval attempt (see for a review Koriat 2000; Metcalfe 2000; Levy and Anderson 2002; Anderson 2003). In this regard, we recall that since the beginning of the Nineties it became evident (Ott et al. 1990) that chaotic processes can be controlled in many different ways, most of which easier than expected. Such a circumstance gave rise to a number of models of associative memories based on chaotic neural networks, differing one from another mostly on control rules adopted (see, for instance, Kushibe et al. 1996; Sinha 1996; Bondarenko 2002; Wagner and Stucki 2002; Crook et al. 2003; He et al. 2003; Yu et al. 2004). Unfortunately, so far no connection has been established between these models and the model of memory based on dissipative quantum brain theory. In any case, these remarks make evident how the adoption of a dissipative version of QFT relying on the doubling mechanism be the most promising approach for achieving a generalization of QFT able to account for change phenomena in the biological world. To conclude this section, the previous arguments showed that traditional 196 Eliano Pessa methods of describing decoherence, based on standard quantum mechanics, are useless when we deal with biological systems for two main reasons: 1) these systems are constituted by components interacting through force fields, a circumstance which need the resort to a theory of quantum fields (deeply different from quantum mechanics); 2) these systems interact with an external environment, in turn producing quantum dissipation phenomena which are outside the range of quantum mechanics. Therefore the arguments against the use of a quantum framework when modelling biological systems are reliable only as regards standard quantum mechanics but fail when we introduce QFT. Of course, it could be held that quantum fields are not important, as all interactions with environment in the biological case have to do with macroscopic entities which are fully classical. In this regard, first of all it is to be remarked that every classical feature disappears on very short distances and that these latter cannot be reached only owing to the existence of short-range repulsive interactions (like Lennard-Jones ones), which have a fully quantum origin. Then it is, possible to argue that, even in absence of this effect, the presence of a field-mediated interaction could work against decoherence, provided the field be of a particular kind. Let us suppose, for instance, to have a simple quantum system lying in an entangled state (for example the Schrödingercat state), interacting with a classical field inducing a dissipative dynamics. Then (we follow here the argument of Anglin et al. 1995), owing to the fact the different degrees of freedom of the system react in a different way to action of the field, the interference which supported the entanglement disappears and the system state reduces to the product of the single states of its components. In other words, decoherence occurred. However, as the dynamics is dissipative, the system is forced to evolve, independently from its initial conditions, towards an attractor whose dimensionality is lesser than the number of degrees of the system. Thus, after a suitable relaxation time, some degrees of freedom (or even all of them, when the attractor is an equilibrium point) fall in the same state, just as if the system were in an entangled state. Deco- herence disappeared, and recoherence took place! This apparent paradox is easily solved if we take into account that, as already stressed in this section, the dissipation induced another stronger kind of entanglement, the one between the system and the environment. And just such entanglement was responsible for the relaxation towards the equilibrium state. We could add that it has been shown that classical dissipation can be viewed as the origin of quantum spectrum of harmonic oscillator. To save space, we will not discuss here this topic, by referring the reader to the literature (Blasone et al. Phase Transitions in Biological Matter 197

2001; Blasone et al. 2005). We will go now to shortly discuss the problem of ﬁnite volume eﬀects: what really happens during a phase transition?

5. Finite Volume Effects. The Realistic Dynamics of a Phase Transition So far, QFT dealt with symmetry breaking phase transitions by resorting to infinite volume limits and focussing on instantaneous changes. While allowing to obtain a number of important results (such as the ones related to Goldstone bosons), such approach neglects what really occurs in practical cases of biological interest, where spatial domains and transition times are finite and nonzero. On the other hand, we need to know some details about what occurs inside the critical region, because only in this way we can assess the possibility of controlling and forecasting the eventual states following the transition itself. In this regard we remark that in biological domain the time extent of the critical region could even be very large. Think, for instance, of the phase transitions which, in the past, gave rise to the birth of new biological species. The fossil remnants tell us that in these cases the critical period lasted for times of the order of million years, surely enough to undertake some control action. On the other hand, in the case of processes related to human behaviour, the phase transitions can be associated to critical periods of the order of minutes, hours, days, and sometimes years, still very long if compared to the very short times characterizing phase transitions commonly dealt with by physicists. In short, the need for information about the details of processes occurring in the critical period is far greater in the biological case than in most cases of physical interest (where the critical region is often unobservable). Of course, the study of dynamics inside the critical region is a very difficult topic. Usually it is dealt with by resorting to advanced techniques of QFT and suitable simplifying hypotheses. Obviously many different questions must be answered, the first of which deals with the kind of dynamics characterizing the critical region. We recall, in this regard, that the latter is defined as the region in which the amplitude of fluctuations of physical quantities around their equilibrium points is far greater than the equilibrium values themselves. From a quantitative point of view it can be identified, provided we use temperature as a control parameter, as the region associated to the temperature interval (Tc −∆T,Tc +∆T ), where Tc is the critical temperature and ∆T = TG −Tc. As usual, TG denotes the Ginzburg temperature. We already argued in Section 2 that within this region we have a classical dynamics, as quantum fluctuations can be neglected. In order to 198 Eliano Pessa understand what could occur we can then resort to a simple example based on a (1+1)-dimensional self-interacting scalar field of the kind described by (56). Owing to previous arguments, the field can be considered as classical and we can study its behaviour by resorting to a test particle of unit mass which moves under the influence of a potential given by the scalar field itself. For reasons of computational convenience we will write this potential under the specific form: U(x) = (x − 1)2[(x − 1)2 − α] (58) where α is a control parameter which we can imagine as dependent on temperature T , and whence on time t. While this potential is nothing but a particular case of the more general form (56), it has the further advantage that the critical value of control parameter is αc = 0. When α < αc the potential has only one stable minimum (ground state), given by x = 1, while, when α > α , there are two different stable minima (ground states), c √ given by x = 1 ± α, and x = 1 becomes an unstable maximum. We will suppose that the absolute values of α be very small and that our test particle be confined within a small neighbourhood of x = 1. If we introduce an auxiliary variable ξ = x−1, the dynamical equation ruling the behaviour of test particle, in presence of dissipation, has the form: d2ξ/dt2 + γ dξ/dt = −4 ξ3 + 2 α ξ. (59) Here γ is the damping coefficient. This term has been introduced for three different reasons, the first one being the need for taking into account the dissipation produced by the interaction with the environment, discussed in the previous Section. The second reason is that, to be realistic, we must suppose that our system be confined within a finite and small volume. In this regard, to fulfil this requirement we could introduce explicit boundary conditions or explicit forces acting only when the particle is close to the boundary. However the form itself of potential function guarantees that, whatever be the value of control parameter, a strong damping prevents the particle for going too much away from the location x = 1. Thus we introduced a strong damping term as the easiest way for obtaining a confinement without resorting to explicit boundary constraints. The third reason for introducing the damping term is connected to scaling hypothesis, asserting the divergence of correlation length and of dynamical relaxation time when approaching the critical point. This amounts to introduce scaling laws of the form:

ν µ l = l0/δ , τ = τ0/δ , δ = |Tc − T | /Tc (60) Phase Transitions in Biological Matter 183 phase transition. However, as we have seen before, they cannot be studied by resorting to a QFT-based framework, as, very close to a phase transition, all phenomena are ruled by classical physics. Within this context, however, we remark that a mechanism giving rise to solitons in macromolecular systems exists even in classical physics. An example is given by the model proposed by Bhaumik et al. (1982), which introduce, within a 1-dimensional macromolecule spanned by a continuous spatial coordinate x, a polarization field P (x, t) and an acoustic field A(x, t), both of classical nature. This (1 + 1)-dimensional model is described by the following Lagrangian density: 2 2 2 2 2 2 2 2 2 2 L = (1/2)[(∂tP ) + (∂tA) − λ (∂xP ) − σ (∂xA) − ω0P − c (∂xA) P − (1/2) d2P 4] (30) The coefficients entering in this Lagrangian are nothing but parameters, to be specified on the basis of experimental data. Now, making use of Lagrange equations, and introducing new variables defined by: ∗ ϕ = ∂xA, P = P + P , P = pk(ξ) exp [i(k x − ωk t)], 2 with ξ = x − v t, ωk = λ k/v (31) some mathematical manipulations show that pk(ξ) fulfils the ordinary differential equation: 2 2 2 2 2 3 d pk/dξ − b pk + (2b /a ) pk = 0 (32) where: 2 2 2 2 2 2 2 b = [(ω0 − cϕ0)/(λ − v )] − (λ k /v ) (33a)

2 b2/a2 = [3 d2/(λ2 − v2)(σ2 − v2)][v2 − σ2 + (c2/3 d2)]. (33b) It can be shown that (32) allows for a solitonic solution:

pk = a sec h (bξ). (34) It can be easily shown that the Davydov equation (10) can be reduced to the form (32) if we introduce the hypothesis of dependence of a(x,t) only on the variable ξ = x − v t and we suppose that both the imaginary part of a(x,t) as well as the derivative of this latter with respect to ξ can be neglected in a first approximation. It is then possible to make the following identification between the parameters introduced by Davydov and the ones appearing in the model of Bhaumik et al.: b2 = Λ/J , 2 b2/a2 = G/J. (35) Phase Transitions in Biological Matter 199 where l is the correlation length and τ the dynamical relaxation time. From these relationships, we derive that, close to critical point, the first order time derivative scales as:

µ − ν l/τ = (l0/τ0) δ (61) while the second order time derivative scales as:

2 2 2µ − ν l/τ = (l0/τ0 ) δ . (62) If we adopt for the critical exponents the customary values µ = 1, ν = 1/2, it is immediate to see that the ratio between ﬁrst order and second order time derivatives scales as 1/δ. Thus close to critical point the role of ﬁrst order time derivatives becomes more and more prominent. This implies overdamping and the need to take it into account by introducing the damping term. Let us now write, for computational convenience, the control parameter α under the form:

α = α0 − z(ξ, dξ/dt, t). (63)

Here α0 is a constant parameter, which, for reasons of convenience, will be supposed positive. By introducing a specific law ruling the dynamics of the function z(ξ, dξ/dt, t) we do nothing but specifying the law of control exerted on the system by the external environment during the critical period. It is thus evident that the kind of dynamical behaviour ensuing from (59) is strongly depending on the form of adopted control law. In any case a short look at the form of (59) suggests that, even in the simplest cases, such dynamics is very complex. We remark that this complexity doesn’t depend on stochastic fluctuations induced by the environment, which we have not taken into consideration in this simple model, but is already present per se. It is not difficult to obtain chaotic deterministic behaviours. For instance, if we introduce a simple law of control of the form:

z = β0 − (1/ξ) ε cos(Ω t) (64) where β0 is a positive parameter whose value is greater than 2 α0, it is immediate to see that (59), (63) and (64) give rise to the following dynamical equation:

2 2 3 2 2 d ξ/dt + γ dξ/dt + 4 ξ + ω0 = ε cos(Ω t), ω0 = β0 − 2 α0, (65) describing a particular case of the well known Duﬃng oscillator, which, for suitable values of ε, has a chaotic attractor reached through a period- doubling route (for an elementary introduction to Duﬃng oscillator see 200 Eliano Pessa

Davis 1962, pp.386-400; this equation is dealt with by a number of textbooks, such as Tabor 1989; Ott 1993; Strogatz 1994). Another interesting example is obtained if we introduce a control law having the form of the following diﬀerential equation:

dz/dt = − [(ξ/r)2 + (ξ/σ r) ξ dξ/dt + b z] (66) where r, σ, b are suitable positive parameters. If we make use of the auxiliary dependent variable given by η = dξ/dt, so as to transform the single second order equation (59) into a pair of ﬁrst order equations, it is possible to show, through suitable algebraic manipulations, that the transformation of variables:

ξ = r ξ0, η = σ (η0 − ξ0), z = z0 (67) allows to cast the system constituted by (59), (63) and (66) in the form: dξ0/dt = ρ (η0 − ξ0), dη0/dt = − χ η0 + β ξ0 − (2r/σ) ξ0 z0, dz0/dt = ξ0η0 − b z0 (68) where the coeﬃcients:

ρ = σ/r, χ = (γ/σ) − (σ/r), β = (2 α0 r/σ) − (σ/r) (69) are all positive in the case of strong overdamping. It is easy to recognize that (68) is nothing but a particular version of the celebrated Lorenz system, one of first examples of systems of differential equations having a chaotic attractor (see, for a complete information, Sparrow 1982). However, despite these examples and other proofs of occurrence of deterministic chaos in first-order phase transitions (see Bystrai et al. 2004), we cannot assert on solid grounds that they prove the existence of a chaotic phase of dynamics close to critical point. Namely our elementary model neglects most aspects of the influence exerted on the system by the stochastic fluctuations of environment. The only way in which this influence is described in an implicit way is through damping coefficient. We should, however, add at least an explicit noise term, specifying its nature (white or coloured). In this regard we remark that, provided some form of deterministic chaos would be occurring in the critical region, the presence of noise would influence it, leading, in some cases, to its disappearing (an effect known since long time; see, for instance, Matsumoto and Tsuda 1983; Herzel and Pompe 1987; Shibata et al. 1999; Shiino and Yoshida 2001; see also Redaelli et al. 2002), and, in other cases, to a situation of stochastic chaos (Freeman et al., 2001). Phase Transitions in Biological Matter 201

As a consequence of the above the best way to describe the spatio- temporal (classical) dynamics of a field in the critical region is to make use of well known Ginzburg–Landau equation (GLE), eventually extended to the so-called cubic-quintic form (see, for instance, Crasovan et al. 2001; Akhmediev and Soto–Crespo 2003), and supplemented by suitable noise terms. Such a popular strategy has been adopted by a number of authors (among the others Laguna and Zurek 1997; Dziarmaga et al. 1999; Rivers 2000; Lythe 2001), even if alternative approaches have been used, such as the one of including the noise contributions within the definitions themselves of GLE coefficients. In any case it is to be remarked that, in absence of noise terms, GLE can have chaotic attractors as well as exhibit spatio- temporal chaos (Doering et al. 1987; Sirovich and Rodriguez 1987; Shraiman et al. 1992; Hernández–Garc´ıa et al. 1999). Once found a strategy for describing the critical dynamics, the next step is to understand what are its outcomes and whether they could or not controlled through external actions. In this regard, we begin to recall that this dynamics has two main effects: to settle the system in a suitable asymptotic state (which, in general, could be a ground state or a metastable state, static or dynamic), kept by the action of quantum mechanisms, such as Goldstone bosons, and to give rise, owing to finite volume, to the formation of suitable defects. The latter process is of paramount importance for two main reasons, the first one being that defects can be experimentally detected and this lets us test the predictions of microscopic theories of phase transitions. In the second place defects, while accounting for the macroscopic features associated to the new phase, often behave as autonomous entities which can be, in turn, viewed as the microscopic constituents of a further, and higher, macroscopic level. In short, the study of defect formation, as well as of their interactions, is the starting point for the building of a generalization of TPT, able to describe the birth of a hierarchy of different levels of organization of condensed matter. It is easy to understand that such a generalization would be just the tool required to describe the organization of biological systems. Defect formation has been the subject of a theory proposed by Kibble and Zurek (see, for instance, Kibble 1976; 1980; Zurek 1993; 1996). The theory has been refined by a number of authors who introduced a number of different microscopic models of processes taking place in the critical region (among them we quote Rivers 2000; Alfinito and Vitiello 2002; Alfinito et al. 2002; Lombardo et al. 2002; Rivers et al. 2002; Almeida et al. 2004; Antunes et al. 2005; Blasone et al. 2006). Here, without discussing the Kibble-Zurek 202 Eliano Pessa theory, we will focus on a numerical study of a simple (1+1)-dimensional toy model of the behaviour of a scalar field in the critical region, in order to evidence the possible scenarios. More precisely, we will resort to the model of a self-interacting scalar field with dissipation, whose dynamics is described by the equation (57) introduced in the previous Section. In order to derive a dynamical equation in the Ginzburg–Landau form we will simply neglect the second order time derivatives, on the basis of the arguments described above. Moreover, we will rescale the variables and add an environmental noise in such a way as to obtain a dynamical equation in the Langevin form: 2 3 (1 + 3 ϕ )(∂ϕ/∂ t) + ∂xxϕ − (α/8) ϕ + (1/2) ϕ = θ (70) In this way (we adopted exactly the same form of potential used by Laguna and Zurek 1997) the only control parameter is given by α. Elementary arguments show that the critical value is αc = 0. When α < 0, there is only one ground state ϕ = 0, while, when α > 0, we have a bifurcation √ to two stable ground states ϕ = ± (1/2) α. In all our simulations we will suppose that the absolute value of α be very small, in such a way as to lie within a close neighbourhood of the critical point. The symbol θ denotes the environmental noise, modelled as a Gaussian white noise, with zero mean and variance given by 2 T , where T is the reservoir temperature. As regard the time variation of control parameter (conceptually equivalent to time variation of temperature adopted in most models), we chose a simple law of the form:

α(t) = α0 + β0 t. (71) To perform a numerical study of (70)-(71) we replaced the finite 1- dimensional space, in which the system is supposed to be contained, with a finite 1-dimensional lattice of points, endowed with a toroidal topology so as to have periodic boundary conditions. If we substitute the second order spatial derivative with its discretized version, the equation (70) becomes a system of N ordinary differential equations (supplemented by additive noise), where N is the number of lattice points. The numerical solution of this system was achieved through a traditional fourth-order Runge–Kutta method. In the Figure 1 below we show two snapshots of ϕ, taken at time steps 50 and 980. In this simulation the adopted parameter values were: α0 = − 0.1, β0 = 0.02, T = 0.01, ∆ t = 0.01, N = 500. Moreover, the initial values of ϕ in the different lattice points were chosen at random, according to a uniform distribution between 0.1 and - 0.1. Of course, having only one spatial dimension, our model is nothing but a caricature of a phase transition. Namely the only possible defects are given by kinks. Phase Transitions in Biological Matter 203

As it is possible to see from Figure 1, after an initial stage (when α < 0) in which we have only small ﬂuctuations of ϕ around its ground state value, it is possible to observe (when α > 0) a stage characterized by the formation of kinks having a proﬁle stable enough. This method would even let us test the predictions of Kibble–Zurek theory but, as here we are interested only in proving the possibility that (70) could be a reliable model of defect formation, we will avoid a discussion of this topic. It is possible to ask

Fig. 1. Snapshots of ϕ in a dynamical evolution ruled by (70)-(71) in correspondence to 50 and 980 time steps. The parameter values are listed within the text. whether the form of damping coeﬃcient, present within (70) and depending on ϕ, be in some way responsible of the form of kinks shown in Figure 1. To answer this question we repeated the same simulation previously 204 Eliano Pessa described with the same parameters, but with a damping coeﬃcient equal to 1 (exactly like in Laguna and Zurek 1997). The snapshots of ϕ at 50 and 980 time steps are shown below in Figure 2. As it is possible to see from

Fig. 2. Snapshots of ϕ when damping is equal to 1 with the same conditions as in Figure 1.

Figure 2 the only difference consists in the fact that kink amplitudes are slightly greater than in Figure 1. This means that a damping coefficient given by (1 + 3 ϕ2) entails only a bounding of kink amplitudes, without changing the overall dynamical picture. We can now ask ourselves whether it is possible to introduce some form of control on critical dynamics. The answer is affirmative. In this regard, the simplest form of control consists in modifying the rate of time change Phase Transitions in Biological Matter 205 of control parameter, given by β0. We performed a numerical simulation of (70)-(71) with the same parameter values listed before, but with β0 = 0.2. In correspondence to this parameter value the number of time steps required to reach a value of α = 0.1 is only 100, instead of the 1000 required in the case of simulation depicted in Figure 1. Thus, for a comparison with the latter, in this case the snapshots are to be taken at time steps 50 and 98. They are shown below in the Figure 3.

Fig. 3. Snapshots of ϕ at time steps 50 and 98 when β0 = 0.2.

As we can see a high change rate of control parameter produces a suppression of kink formation. Of course, we could also interpret the snapshot at time step 98 as showing the occurrence of a large number of very 206 Eliano Pessa small kinks. Obviously this circumstance has an important influence on the nature of processes occurring when leaving the critical region. We can thus say that acting on the change rate of control parameter, we can exert some sort of control on the output of phase transition. There is, however, another simple way for controlling the output of a phase transition, by adding a suitable external force field. The latter, for instance, is responsible for the occurrence of an inner magnetization having the same direction of the external field in the paramagnetic-ferromagnetic transition. Besides, the presence of an external field changes the nature of phase transition itself, which is no longer an equilibrium phase transition but becomes a nonequilibrium phase transition. As in the latter case the Mermin–Wagner theorem no longer applies, true phase transitions become possible even in the one-dimensional case (see, for instance, Katz et al. 1983; Privman 1997). To test the influence of external force fields in our toy model we added to the left hand side of (70) a term given by −α2/8. In this case the equilibrium points of (70) are given by the roots of the cubic equation:

4 ϕ3 − α ϕ − α2 = 0. (72)

From elementary theory of cubic equations it is possible to obtain that, when α < 0, (72) has only one real root, while, when 0 < α < 4/27, there are three different real roots. Thus α = 0 is still a bifurcation point. However, as this time the two stable equilibrium points occurring after the bifurcation are associated to different values of ϕ, the dynamics of the system will be attracted by the absolute minimum. This forecasting was confirmed by another numerical study, in which, while introducing the external force described above, we used the same parameter values adopted for the case depicted in Figure 1. Even in this case the simulation was performed for 1000 time steps. The snapshot of ϕ at time step 50 is reported below in Figure 4. As it is possible to see, the dynamical evolution up to this time step doesn’t seem to differ from the one observed in other cases, as if the external force were not influential. However, as number of time steps was increasing, the value of ϕ in all lattice points was tending towards a common value, perturbed only by very small fluctuations. After 1000 time steps the value of ϕ, averaged on all lattice points, was 4.271, while the variance of fluctuations around this value was only 7.0645 × 10 − 5. In other words, the system settled, as predicted, on a particular equilibrium point, determined by the choice of external forcing term, without kink production. Phase Transitions in Biological Matter 207

Fig. 4. Snapshot of ϕ at time step 50 in presence of the external force α2/8.

Once taken into consideration the description of critical dynamics and defect production, as well as the ways for controlling these processes, we will deal with another important topic, the one of recoherence, that is of mechanisms allowing, after the occurrence of phase transition, the shift from the classical regime to a new regime ruled again by QFT. We remark, in this regard, that recoherence is not a popular topic, as few authors, so far, dealt with this subject (among them we can quote Anglin et al. 1995; Angelo et al. 2001; Mokarzel et al. 2002; the reading of Farini et al. 1996 is illuminating on what occurs in presence of deterministic chaos). In any case, the origin of recoherence is easy to understand: classical dynamics close to critical point, ruled by some form of Ginzburg–Landau equation, gives rise, after suitable time, to the occurrence of a mean-field coupling. The latter has two effects: induces a mutual synchronization between field modes and, at the same time, produces a decreasing of the influence of thermal fluctuations. For quantum fluctuations (always present even in the critical region, but neglected as overwhelmed by thermal fluctuations) it is now an easy game to take again the control of the situation and restore a quantum regime. While the details of this recoherence story are strongly dependent both on the nature of the system taken into consideration and on the kind of environment, we stress that a number of studies evidenced the capital role of mean-field coupling in driving towards a globally ordered attractor state. This occurs for different kinds of systems and even in presence of noise and chaos, which in some cases enhance the efficiency of such a mechanism (among the many contributions to this topic we quote Shibata and Kaneko 208 Eliano Pessa

1997; Shibata et al. 1999; Pikovsky et al. 2001; Topaj et al. 2001; Ito and Kaneko 2002; Mikhailov and Calenbuhr 2002, Chapp. 6 and 7; Ito and Kaneko 2003; Rosenblum and Pikovsky 2003; Willeboordse and Kaneko 2005; Morita and Kaneko 2006). We can show some features of recoherence story by resorting to our (1+1)-dimensional toy model of phase transition, described above and ruled by Ginzburg–Landau–Langevin equation (70). Without embarking on complex computations, we will resort to poor man arguments, based on rather crude approximations and on the hypothesis of dealing with very small values of ϕ and α. To begin, we will try to show that, after crossing the critical point of phase transition, the amplitude of mean field undergoes a very fast growth. To this end, we will perform a spatial discretization of (70), so that the stochastic differential equation ruling the time evolution of field value in the i-th lattice point, here denoted by ϕi, will assume the form:

2 3 (1 + 3 ϕi )(∂ϕi/∂ t) + (ϕi+1 + ϕi−1 − 2 ϕi) − (α/8) ϕi + (1/2) ϕi = θ (73) We will now suppose that (we already crossed the critical value) in both the two neighbouring sites, whose indices are i + 1 and i − 1, the ﬁeld assume a value corresponding to one of the two ground states, for instance √ (1/2) α. We will then check whether this situation can contribute to an increase of mean ﬁeld amplitude (obviously related to the chosen ground state) following a Kuramoto-like mechanism (see Kuramoto 1984; Pikovsky et al. 2002). To simplify the following computations we will assume that the 2 term 3 ϕi can be neglected with respect to unity, and that the value of ϕi 3 be so small as to let us neglect even the term (1/2) ϕi . Then the Langevin equation for ϕi will assume the simple form (hereafter we will eliminate the index i):

dϕ/dt = f(ϕ) + θ. (74)

Because we supposed that α be very small, the function f(ϕ) will be written in an approximate way as: √ f(ϕ) = α − 2 ϕ. (75)

As θ is supposed to denote a Gaussian white noise with zero mean and variance given by σ2, the probability distribution function P (ϕ, t) will fulﬁl (we remind that we are still far from a quantum regime) the following Fokker–Planck equation:

2 ∂tP = − ∂ϕ[f(ϕ)P ] + (σ /2) ∂ϕϕP. (76) Phase Transitions in Biological Matter 209

Trivial computations show that the maximum of stationary probability distribution associated to (76) is given, as expected, by: √ ϕ = (1/2) α. (77)

The mean ﬁeld amplitude at a given time t can be measured through the distance between the probability distribution occurring at t and the stationary probability distribution corresponding to (77). As we are interested in the time evolution of this distance, it is convenient to adopt, for the solution of (76), the ansatz:

P (ϕ, t) = Ps(ϕ) + Q(ϕ, 0) exp[h(ϕ) t]. (78)

Here Ps(ϕ) denotes the stationary probability distribution, while Q(ϕ, 0) is a measure of the distribution function occurring at t = 0. The (negative) function h(ϕ) is related to the rate of relaxation towards the stationary probability distribution. Its absolute value is proportional to the rate of growth of mean ﬁeld amplitude. To simplify the things, we will suppose that Q(ϕ, 0) be given by a Gaussian function of the form (here and in the following many constants will be set equal to 1 to shorten the computations):

Q(ϕ, 0) = exp(− ϕ2/2 s2). (79)

This distribution function describes a Gaussian peaked, at time t = 0, around the value ϕ = 0, as it is reasonable to suppose in correspondence to the critical point, when the values of ϕ are still in the neighbourhood of the ground state existing before the phase transition. The value of the variance s2 is supposed to be very small. However, we will assume to deal only with values of ϕ even smaller of s2 so as to neglect terms of the order of ϕ/s2 or of higher powers of this quantity. If we substitute the ansatz (78), together with (79), into (76), the previous hypotheses allow us to obtain that the function h(ϕ) fulﬁls the following diﬀerential equation:

h = K − f(ϕ)(dh/dϕ) + (σ2/2)(d2h/dϕ2) + (σ2/2)(dh/dϕ)2 (80) where

K = 2 + (σ2/2 s2). (81)

Rather than searching for a full solution of the nonlinear equation (80), we will limit ourselves to deal with a simple approximation, obtained by supposing that the first derivative dh/dϕ (as well as its higher powers) be 210 Eliano Pessa so small as to allow us to neglect it. Then we will obtain from (80) the simple linear differential equation: h = K + (σ2/2)(d2h/dϕ2). (82) If we adopt the initial conditions: h(0) = − a2, dh/dϕ (0) = 0 (83) where a is a suitable constant, elementary computations show that the solution of (82) is given by: h = K − (a2 + K) cosh[(2/σ2)1/2ϕ]. (84) Taking into account that, up to second order terms and for small values of x, hyperbolic cosine is approximated by the formula cosh(x) ≈ 1 + (x2/2), we can write (84), under the previous hypotheses, as: h = − a2 − (2/σ2)(a2 + 2) ϕ2. (85) This formula is our main result. It shows that mean field amplitude growth rate scales as ϕ2. As ϕ values undergo very fast fluctuations close to critical point this in turn entails a very fast growth of mean field amplitude itself which, in a very short time, will dominate the system dynamical evolution. In presence of a large mean field amplitude the dynamics of order parameter Φ (in our case the difference between field expectation values, in presence of fluctuations, after and before the critical point) will be described, in an effective field approximation (here we will neglect, to save space, finite volume effects) by a Ginzburg–Landau equation having the form: 2® ∂tΦ = ∂xxΦ + [(α/8) − (3/2) ϕ ]Φ. (86) ® Here ϕ2 is the average value of ϕ2 (related just to mean field amplitude). From (86) we see how the latter affects the dynamics. We remark that a similar equation describes, in a first approximation, even the dynamics of disturbances ξ around the chosen ground state. Now, if we take into √ consideration the ground state value ϕ® = (1/2) α and we assume, by making a crude approximation, that ϕ2 = α/4, we will obtain that ξ will be ruled by:

∂tξ = ∂xxξ − (α/4) ξ. (87) A linear stability analysis around the equilibrium state ξ = 0, based on an ansatz like the traditional one ξ = exp(λ t) sin(k x), leads to the following relationship between λ and k: λ(k) = −k2 − (α/4). (88) Phase Transitions in Biological Matter 211

It shows that, when α < 0 (that is before the critical point α = 0), the equilibrium state is unstable with respect to long wavelength perturbations, while, when α > 0, it becomes asymptotically stable. This entails that, in correspondence to critical point, we have λ(0) = 0. This corresponds to the occurrence of long range (here infinite range) modes, which are nothing but the classical counterpart of Goldstone bosons (Pessa 1988). The latter, thus, have a already a precursor in the classical regime. We can therefore conclude that recoherence is easy, as all features of quantum regime have already been set up in the classical stage. Let us now shortly dwell on the last important problem in phase transition dynamics: the one of interactions between defects and on the ensuing behaviour of defects themselves, each one viewed, this time, as a wholistic entity constituting the basic unit of a new level of description. This topic is of prime importance when dealing with biological matter, as we need to know whether a phase transition theory based on QFT is able to account for the hierarchical organization of levels observed in living beings and essential for their operation. As regards this subject, we recall that most research activity has been devoted to topological defects, such as vortices and solitons. On the latter there is an extensive literature, including both classical (we will limit ourselves to quote Rajaraman 1987; Lakshmanan 1988; Kivshar and Pelinovsky 2000; Nekorkin and Velarde 2002; Scott 2003) and quantum solitons (Ni et al. 1992; Friberg et al. 1996; Werner 1996; Werner and Friberg 1997; Konotop and Takeno 2001; Manton et al. 2004). How- ever in very few cases there has been some theoretical proposal about the explicit form of the interaction potential between two defects (a notable exception is Zhang et al. 2006; some related ideas were used to describe interactions between biological cells; see Schwarz and Safran 2002; Bischofs et al. 2004; in a very different context, the one of interactions between defects in quantum liquids, explicit expressions for interaction potentials have been given, for instance, in Recati et al. 2005). In other cases we have mostly numerical simulations (see, for instance, Hoyuelos et al. 2003; Podolny et al. 2005). However, almost all authors neglect the fundamental question: are the new entities (that is defects) classical or quantum? It seems that the classical nature of defects be taken for granted and that, when quantum corrections are needed (as in the case of quantum solitons), they depend on the influence of some medium in which defects themselves are embedded. In this regard we remark that the things could be more complex than suspected. Namely the existence of quantum effects is crucially dependent on the ratio between the amplitudes of quantum fluctuations and 212 Eliano Pessa thermal fluctuations. The former can be expressed through ~ω, where ω is some characteristic frequency of the system under study. But, what is ~? If we deal with atoms, molecules, or elementary particles, the answer is well known. But in other cases we could deal with some kind of “effective Planck constant”, whose numerical value could be very different from the usual one. Such a circumstance could, in principle, fully change the nature of behaviour of defects. Science fiction? In this regard we recall that from a number of years the number of researchers introducing an “effective Planck constant” is ever increasing (we can quote, for instance, Artuso and Rusconi 2001; Averbukh et al. 2002; perhaps one of clearer explanations of how to compute the effective Planck constant is contained in Angelopoulou et al. 1994). To make a further example, Fogedby (Fogedby 1998; Fogedby and Bran- denburg 2002), when studying the noisy Burgers equation: ∂u/∂t = ν ∇2u + λ u ∇u + ∇η, (89) where ν is a damping constant, or a viscosity, λ a coupling coefficient, and η a Gaussian white noise satisfying: hη(x, t) η(x0, t0)i = ∆ δ(x − x0) δ(t − t0), (90) found that the theory of this equation could be recast exactly in the form of an equivalent QFT, provided the usual Planck constant ~ be identified with the quantity ∆/ν. However, except a small number of researchers, most scientists ignore the relevance of this possibility.

6. Are we Really Describing Phase Transitions in Biological Matter? The previous sections evidenced the amazing modeling ability of QFT and statistical mechanics within the context of phase transition theory. How- ever, it seems restricted to specific (and very limited) physical contexts. On the contrary, when looking at the real world of phenomenological models of biological phenomena, we encounter things such as neural networks, cell systems, nonequilibrium phase transitions, species evolution, business cycles, without nothing resembling to Hamiltonians, QFT models, partition functions, Goldstone bosons, and like. In other words, all efforts made to remedy the shortcomings of TPT, described in the points 1)–5) listed at the end of third Section, seem to have few contacts with models of biological behaviours. The sensation is that we effectively describe all aspects of some kinds of phase transitions, but they are useless when dealing with phase transitions in biological matter. Phase Transitions in Biological Matter 213

In this regard, there are, in principle, three possible answers: r.1) phenomenological models of biological change are already QFT models; however, at present they are formulated in such a way that their similarity to physical models of phase transitions is hidden; anyway, a future suitable generalization of QFT will allow us to recognize that both phenomenological models and QFT models belong to the same general class; r.2) phenomenological models of biological change are incomplete; however, by introducing suitable quantization methods for these models, it will be possible to build more complete models, accounting in a deeper way for the observed behaviours of biological systems; r.3) phenomenological models don’t have and will never have any kind of relationships with QFT models; namely biological world is irreducible to physical world; moreover, the understanding of biological world cannot be based in an important way on modelling activity which, in the latter context, has a limited usefulness. It seems that, within the scientific community, the answer r.3) had never been given. Namely it would contradict the requirement that science be able to do predictions and design methods to control phenomena. However, it should be remarked that, in practice, many scientists work as if they were holding that r.3) is the right answer. This attitude frees them from the need of knowing alternative (and often more powerful) modelling methods and gives them the opportunity of leading a quiet life, avoiding the danger of interference of other kinds of scientists in their (personal and private) domain of study. The adoption of answer (r.1) is easier in some kinds of models, typically formulated by using a mathematical language not too different from the one used in field theories. In this regard, among the many possible classes of biological emergence models, we will take into consideration a first example given by reaction-diffusion systems. The latter have been used to model, for instance, chemical reactions, species evolution, swarm intelligence, morphogenesis. In general models of this kind are constituted by a set of basic components (which, to conform to a common usage, we will conventionally denote as particles) undergoing two types of processes: a random diffusion (which, for instance, can be modeled as a random walk on a suitable continuous or discrete space) and reactions between particles (the spontaneous decay of a particle, giving rise to new particles, being considered as a special case of a reaction). At any given time instant these models allow a suitable definition of system’s microstate, characterized by the values of microscopic variables associated to the single particles. As the 214 Eliano Pessa system evolution is stochastic, to each microstate α we can associate its probability of occurrence at time t , denoted by p(α, t). The latter must fulfill an evolution equation of the form: X X dp(α, t)/dt = Rβαp(β, t) − Rαγ p(α, t). (91) β γ

Here the symbol Rβα denotes the transition rate from the state β into α. The equation above is usually referred to as master equation. It is easy to understand that, in general, the particle number is not conserved, owing to the fact that reactions can, in principle, destroy or create particles. This circumstance lets introduce the so-called formalism of second quantization, widely used in Quantum Field Theory. This can be done (the pioneering work on this subject was done by Doi 1976; Peliti 1985) very easily, for instance, in the case of particles moving on a discrete spatial lattice characterized by suitable nodes, labeled by a coordinate i (we will use a single label to save on symbols). In this context we can introduce the numbers of particles lying in each node (the so-called occupation numbers) and two op- + erators, that is a creation operator ai and a destruction operator ai, acting on system microstate and, respectively, raising the number of particles lying in the i-th node of one unity and lowering the same number of one unity. It is possible to show that these operators fulfill the same commutation relationships holding in Quantum Field Theory. They can be used to define a state with given occupation numbers, starting from the ground state, in which no particle exists, and applying to it, in a suitable way, creation and destruction operators. If we denote by |α, ti a state characterized, at time t , by given occupation numbers (summarized through the single label α) system’s state vector can then be defined as: X |ψ(t)i = p(α, t) |α, ti. (92) α By substituting this definition into the master equation (91) it is possible to see that system’s state vector fulfills a Schrödingerequation of the form:

d |ψ(t)i /dt = −H |ψ(t)i (93) where H denotes a suitable Hamiltonian, whose explicit form depends on the specific laws adopted for transition rates. In the case of a lattice of nodes, a suitable continuum limit then gives rise to a field Hamiltonian which can be dealt with exactly with the same methods used in standard Quantum Field Theory (on this topic see Cardy, 1996). The above procedure has been applied by a number of researchers to investigate the statistical features of models of interacting biological agents (see, among the others, Cardy Phase Transitions in Biological Matter 215 and Täuber 1998; Pastor–Satorras and Solé2001). We, however, underline that this procedure has only a formal character and, as it is possible to understand from (92), doesn’t lead to a genuine quantum theory. On the other hand, it is not excluded that suitable generalizations of Doi–Peliti method can lead to a description in terms of truly quantum processes (see, for instance, Smith 2006). Other holders of answer (r.1) adopted a very different framework. For instance people interested in socio-economic models tried to formulate them in terms of some network dynamics (see, for a review, Albert and Barabási 2002). Then to each network node, characterized by a specific fitness, it was associated an energy, proportional to the logarithm of fitness, while each connection between two nodes was made to correspond to two noninteracting particles, whose energies were given by the energies associated to the two nodes. Each addition of a new node was thus viewed as equivalent to a particle creation. Suitable hypotheses on fitness distribution as well as on node connection increase rate can give rise to well known distributions of energy levels associated to the nodes, such as, for instance, Bose–Einstein distribution. Then it is possible to map essentially quantum phenomena, such as Bose–Einstein condensation, on phenomena such as network dynamics (cfr. Bianconi and Barabási2001). As regards the holders of answer (r.2) we are forced to limit our considerations to the field of quantum neural networks, so far the only one in which this approach was object of an intensive investigation. In this context, we must distinguish between two approaches: the most popular one, in which quantum neurons are introduced from the start as physical realizations of quantum systems, such as multiple slits or quantum dots (for reviews see Behrman et al. 2000; Narayanan and Menneer 2000) , and the less studied one, in which quantum neurons are quantum dynamical systems whose laws are obtained by applying a quantization procedure to dynamical laws of classical neurons (see Pessa 2004). In any case it is to be remarked that, by choosing (r.1) or (r.2), in both cases we need some form of generalization of QFT, which makes the two approaches complementary, rather than anthitetic. In order to illustrate the difficulties arising when trying to connect phenomenological models with QFT we will use a particular version of a very simple model of population evolution, introduced by Michod (see, among the others, Michod 2006; Michod et al. 2006), and applied to describe the evolutionary biology of volvocine algae. Within this model the i-th individual of a population is characterized by the values of two variables: its 216 Eliano Pessa generative ability bi and its survival ability vi. In general, as shown by biological data, the two variables b and v are not reciprocally independent, but are connected through a general law expressed by a function v(b) which is decreasing in b. In this regard, we introduced a specific form of v(b) given by: v(b) = [γ (1 + γ)/(b + γ)] − γ (94) where γ is a suitable parameter. For reasons of convenience we constrained the values of v and b within the interval [0, 1] and (94) tells us that v(0) = 1, v(1) = 0. However, as the total number of individuals is not constant (owing to the existence of a reproductive ability), but a function of time N = N(t), we supposed (Michod did not make explicitly this hypothesis), that the value of parameter γ appearing in (94) (that is the convexity of the curve) were dependent on the momentarily value of N through a law of the form:

γ = (β0/N) + α0 (95) where β0 and α0 are other parameters. What should we expect from the behaviour of this simple model? Mi- chod introduced two different kinds of fitness measure: the average individual fitness of population members, denoted byw ¯, and the total (normalized) population fitness, denoted by W . Their definitions are: X X X 2 w¯ = (1/N) bi vi,W = (1/N ) BV,B = bi,V = vi. i i i (96) From both biological observations and results of computer simulations Mi- chod recognized that, while obviously both fitness measures vary with time, however for most time they have different values. Thus he was lead to introduce a quantity, called covariance and here denoted by Cov, which measures such a difference through the simple relationship: Cov =w ¯ − W. (97) When the value of Cov is negative, the total population fitness is greater than the average individual fitness. In other words, we are in a situation in which it seems that there is some sort of cooperation (or coordination) among individuals producing an increase of total fitness. Clearly this is not quantum coherence, but recalls some aspects of the latter. How is this increased total fitness reached? The simulations show that it is sometimes due to a sort of increase in specialization of population members. In this regard, we remark that every pair of values (v, b) characterizes a single individual and that the form of the distribution of these pairs (or better, Phase Transitions in Biological Matter 217 only of b values, as (94) lets us find the value of v, once known the value of b) gives a measure of the degree of specialization present within the population at a given time instant. Namely a flattened distribution means a strong difference between the individuals, and therefore a high degree of specialization, while a strongly peaked distributions means small differences between the individuals and low degree of specialization. In order to check whether these effects are present even under the hypotheses we introduced before in (94)–(95), we performed suitable numerical simulations of the evolution of populations. They were based on a previous subdivision of the interval [0, 1] of values of continuous variable b in a suitable number of equal sub-intervals, in correspondence to each one of which the value of b was identified with the middle point of the sub- interval. The mechanism of reproduction was random, based on a uniform distribution, and such that each reproductive value was interpreted as the probability, at each generation, of producing a number of offspring which were a fraction of a maximum possible number fixed in advance by the experimenter. The produced descendants were assigned at random to the different generative ability sub-intervals. Besides, even the value of v for each individual was interpreted as the probability of its survival in the next generation. In the Figure 5 we can see a plot of both kinds of fitness vs generation number in a “life history” characterized by 100 generations, an initial total number of individuals given by 50, a number of 100 different sub-intervals of generative ability, and a maximum allowable number of descendants for each population member and for each generation given by 5. Moreover the maximum allowable number of individuals for each generative ability sub- interval was fixed to 100, and the initial value of γ was 5. The values of remaining parameters were α0 = 3, β0 = 100. As it is immediate to see, the covariance is always negative and the group fitness prevails over the average individual fitness. What has been in this case the effect of population evolution on the distribution of values of b among the individuals. We remark that at the beginning of this simulation we chose to put all individuals within the same generative ability class, corresponding to the 50-th sub-interval. This distribution is depicted in the Figure 6 a). For a comparison we show in the Figure 6 b) the final distribution obtained after 100 generations. As it is possible to see, not only the final distribution deeply differs from the initial one, but the former evidence a very high degree of specialization of single individuals. It is easy to understand that this simple 218 Eliano Pessa

Fig. 5. Group ﬁtness (upper curve) and average individual ﬁtness (lower curve) vs time.

Fig. 6. (a) Initial distribution of generative abilities, (b) Final distribution of generative abilities. model accounts not only for the evolution of populations of volvocine algae, but, more in general, of the fact that most biological organisms survive in a complex environment just owing to the fact that their components (cells or organs) are highly specialized and reciprocally cooperating. Now let us deal with the main question: can this model be dealt with through the methods of QFT? If yes, through which algorithms? If not, for which reasons? To begin, we could argue that the model was not cast under the form of a system of differential equations and, therefore, it could not be dealt with through Hamiltonian-based methods. This argument is, however, very weak as, by using suitable tricks and approximations, it would be possible, in one or in another form, to achieve a mathematical format closer to the ones popular in theoretical physics, eventually resorting to the Phase Transitions in Biological Matter 219 introduction of suitable noise (this has already been done within the research on protocells and on cellular systems; see, among the others, Furu- sawa and Kaneko 2002; Rasmussen et al. 2003; 2004; Takagi and Kaneko 2005; Fellermann and Solé2006; Lawson et al. 2006; Maciáand Solé2006). In any case, this would be a problem only of technical (mathematical) nature, and not a serious conceptual obstacle.

A stronger argument takes into consideration the nature of the environment which, as implicitly described in the previous model, has a typically reactive character. On the contrary, QFT models are placed within simpler environments, such as thermal baths, and the concept itself of fitness is absent. Besides, in biological world often the action of the environment occurs on large scales and influences the behaviour of organisms in a rather indirect way. This implies, in turn, that control of emergence cannot be implemented in the simple ways described in the previous section. We can therefore claim that QFT will never give rise to a reliable description of phase transitions in biological matter if we will not generalize it so as to include more realistic descriptions of biological environments. Of course, this generalization should also take into account the fact that, owing to previous reasons, phase transitions in biological matter are often of nonequilibrium type. The doubling mechanism quoted in Section 4 constitutes a first step towards this direction, but probably further steps are needed.

A third argument deals with the nature of system components. While QFT models can be interpreted as describing assemblies of particles, all having the same nature, the members of population previously introduced are different individuals. In other words, once translated in the language of particle creation and annihilation, the model operating according the rules (94)–(95) describes creation and annihilation, not only of particles, but even of kinds of particles. In terms of a field language it is equivalent to a theory describing the birth and the disappearing of fields. Within QFT this would require the introduction of third quantization. This is still a somewhat exotic topic, dealt with almost exclusively within the domain of quantum theories of gravitation, but so far with only very few contacts with the world of QFT models of condensed matter (for a first attempt see Maslov and Shvedov 1999). However the latter domain could be the better context for applying this kind of extension of QFT, owing to the presence of “effective force fields” which appear and disappear as a function of environmental constraints. On the contrary, it would be useless within 220 Eliano Pessa the world of elementary particle physics at high energies, where from the starting people are searching for evidence of universal force fields whose nature lasts unchanged. As a consequence of the previous arguments we can claim that, notwithstanding the theoretical efforts quoted in the previous sections, actually QFT doesn’t describe phase transitions in biological matter, except in a limited number of cases, related to low-level phenomena. However, provided the generalizations mentioned before were included within a larger theoretical framework, such a description could probably become feasible and realistic.

7. Conclusions The long path followed in this paper touched some key problems of actual theory of phase transitions, having in mind the usefulness of its application to describe changes or transitions occurring in biological matter. Owing to the complexity of this topic we neglected a number of important points, including the roles of noise and disorder within this context. However, suitable generalizations of QFT and Statistical Mechanics, already available from a number of years, allow to include these aspects within the previous framework without essential conceptual changes, if not of purely technical nature. At the end of our trip we are full of admiration for the ingenuity of a number of theoreticians who introduced so powerful tools for dealing with various aspects of phase transitions. Notwithstanding these eﬀorts, it seems that the main problems still remain unsolved. Namely it appears that, except a limited number of cases, we still lack a realistic theory of phase transitions in biological matter. Not only, we are still unable to decide whether observed biological changes can or not be described, at least in principle, within the context of actual phase transition theory, eventually generalized in a suitable way. However, despite this situation, we have reasons for being conﬁdent in the future developments of the theory. Namely the progress so far made, part of which shortly described even in this paper, gives us solid grounds for believing that in a near future the goal of achieving a general theory of phase transitions, including both physical and biological matter, will be reached. In this regard, we expect that this new theory, though including TPT as a limiting case, will include even other features, stemming from a sort of back-reaction of knowledge, deriving from biology, psychology, sociology, economics, on the physics itself. We feel that this backreaction will open new horizons even to investigation of physical world. Phase Transitions in Biological Matter 221

References 1. Aihara, K., Matsumoto, G., Ikegaya, Y. (1984). J. Theor. Biol., 109, 249. 2. Akhiezer, A., Péletminski,S. (1980). Les méthodes de la physique statistique. Mir: Moscow. 3. Akhmediev, N., Soto-Crespo, J.M. (2003). Phys. Lett. A, 317, 287. 4. Albert, R., Barabási,A.-L. (2002). Rev. Mod. Phys., 74, 47. 5. Alfinito, E., Viglione, R.G. Vitiello, G. (2001). Mod. Phys. Lett. B, 15, 127. 6. Alfinito, E., Vitiello, G. (2000). Int. J. Mod. Phys. B, 14, 853. 7. Alfinito, E., Vitiello, G. (2002). Phys. Rev. B, 65, 054105. 8. Alfinito, E., Romei, O., Vitiello, G. (2002). Mod. Phys. Lett. B, 16, 93. 9. Almeida, C.A., Bazeia, D., Losano, L., Malbouisson, J.M.C. (2004). Phys. Rev. D, 69, 067702. 10. Amit, D.J. (1984). Field Theory, the Renormalization Group, and Critical Phenomena. World Scientific: Singapore. 11. Anastopoulos, C. (1999). Int. J. Theor. Phys., 38, 2721. 12. Anderson, M.C. (2003). Journal of Memory and Language, 49, 415. 13. Angelo, R.M., Furuya, K., Nemes, M.C., Pellegrino, G.Q. (2001). Phys. Rev. A, 64, 043801. 14. Angelopoulou, P., Baskoutas, S., De Falco, L., Jannussis, A., Mignani, R., Sotiropoulou, A. (1994). J. Phys. A, 27, L605. 15. Anglin, J.R., Laflamme, R., Zurek, W.H., Paz, J.P. (1995). Phys. Rev. D, 52, 2221. 16. Antunes, N.D., Gandra, P., Rivers, R.J. (2005). Phys. Rev. D, 71, 105006. 17. Arageorgis, A., Earman, J., Ruetsche, L. (2002). Studies in the History and Philosophy of Modern Physics, 33, 151. 18. Artuso, R., Rusconi, M. (2001). Phys. Rev. E, 64, 015204(R). 19. Averbukh, V., Osovski, S., Moiseyev, N. (2002). Phys. Rev. Lett., 89, 253201. 20. Bain, J. (2000). Erkenntnis, 53, 375. 21. Bateman, H. (1931). Phys. Rev., 38, 815. 22. Bédard,C., Kröger,H., Destexhe, A. (2006). Phys.Rev.Lett., 97, 118102. 23. Beggs, J.M, Plenz, D. (2003). The Journal of Neuroscience, 23, 11167. 24. Beggs, J.M, Plenz, D. (2004). The Journal of Neuroscience, 24, 5216. 25. Behrman, E.C., Nash, L.R., Steck, J.E., Chandrashekar, V.G., Skinner, S.R. (2000). Information Sciences, 128, 257. 26. Benfatto, G., Gallavotti, G. (1995). Renormalization Group. Princeton Uni- versity Press: Princeton, NJ. 27. Bershadskii, A.,Dremencov, E., Bershadskii, J., Yadid, G. (2003). Int. J. Mod. Phys. C, 14, 897. 28. Bhaumik, D., Dutta-Roy, B., Lahiri, A. (1982). Bull. Math. Biol., 44, 705. 29. Bianconi, G., Barabási,A.-L. (2001). Phys. Rev. Lett., 86, 5632. 30. Bischofs, I., Safran, S., Schwarz, U. (2004). Phys. Rev. E, 69, 021911. 31. Blasone, M., Jizba, P., Vitiello, G. (2001). Phys. Lett. A, 287, 205. 32. Blasone, M., Jizba, P., Kleinert, H. (2005). Phys. Rev. A, 71, 052507. 222 Eliano Pessa

33. Blasone, M., Jizba, P., Vitiello, G. (2006). Quantum fields with topological defects. In J.-P. Fran¸coise,G.Naber, S.T.Tsou (Eds.). Encyclopedia of Mathematical Physics (pp. 221-229). Academic Press: New York, NY. 34. Bolivar, A.O. (1998). Phys. Rev. A, 58, 4330. 35. Bolterauer, H., Tuszyński, J.A., Satari, M.V. (1991). Phys. Rev. A, 44, 1366. 36. Bondarenko, V.E. (2002). Chaos, Solitons and Fractals, 13, 139. 37. Brandes, T. (2005). Phys. Rep., 408, 315. 38. Brizhik, L., Eremko, A., Piette, B., Zakrzewski, W. (2004). Phys. Rev. E, 70, 031914. 39. Brookes, J.C., Hartoutsiou, F., Horsfield, A.P., Stoneham, A.M. (2007). Phys. Rev. Lett., 98, 038101. 40. Bystrai, G.P., Ivanova, S.I., Studenok, S.I. (2004). Int. J. Bifurcation Chaos, 14, 3671. 41. Caldeira, A.O., Leggett, A.J. (1983). Ann. Phys., 149, 374. 42. Calzetta, E., Hu, B.L. (2000). Phys. Rev. D, 61, 025012. 43. Calzetta, E., Roura, A., Verdaguer, E. (2001). Phys. Rev. D, 64, 105008. 44. Calzetta, E., Roura, A., Verdaguer, E. (2002). Phys. Rev. Lett., 88, 010403. 45. Cardy, J. (1996). Scaling and Renormalization in Statistical Physics. Cam- bridge University Press: Cambridge, UK. 46. Cardy, J.L., Täuber, U.C. (1998). J. Stat. Phys., 90 1. 47. Celeghini, E., Rasetti, M., Vitiello, G. (1992). Ann. Phys., 215, 156. 48. Christiansen, M., Chater, N. (Eds.) (2001). Connectionist Psycholinguistics. Ablex Publishing: Westport, CT. 49. Clifton, R.K., Halvorson, H.P. (2001). Studies in the History and Philosophy of Modern Physics, 32, 1. 50. Crasovan, L-C., Malomed, B.A., Mihalache, D. (2001). PRAMANA – Jour- nal of Physics, 57, 1041. 51. Crook, N., olde Scheper, T., Pathirana, V. (2003). Information Sciences, 150, 59. 52. Cruzeiro-Hansson, L., Takeno, S. (1997). Phys. Rev. E, 56, 894. 53. Cugliandolo, L.F., Kurchan, J., Peliti, L. (1997). Phys. Rev. E, 55, 3898. 54. Davis, H.T. (1962). Introduction to nonlinear differential and integral equations. Dover: New York, NY. 55. Davydov, A.S. (1973). J. Theor. Biol. , 38, 559. 56. Davydov, A.S. (1979). Phys. Scr., 20, 387. 57. Davydov, A.S. (1982). Biology and Quantum Mechanics. Pergamon Press: Oxford, UK. 58. Davydov, A.S., Kislukha, N.I. (1973). Phys. Status Solidi, 59, 465. 59. De Boer, J., Uhlenbeck, G.E. (Eds.) (1962). Studies in statistical mechanics, vol.I. North Holland: Amsterdam. 60. Del Giudice, E., De Ninno, A., Fleischmann, M., Vitiello, G. (2005). Elec- tromagnetic Biology and Medicine, 24, 199. 61. Del Giudice, E., Doglia, S., Milani, M. (1982). Phys. Scr., 26, 232. 62. Del Giudice, E., Doglia, S., Milani, M., Vitiello, G. (1985). Nucl. Phys. B, 251, 375. Phase Transitions in Biological Matter 223

63. Del Giudice, E., Doglia, S., Milani, M., Vitiello, G. (1988). Structures, correlations and electromagnetic interactions in living matter: Theory and applications. In Fröhlich, H. (ed.). Biological coherence and response to external stimuli (pp. 49-64). Springer: Berlin. 64. Denman, H.H., Buch, L.H. (1973). J. Math. Phys., 14, 326. 65. Doering, C.R., Gibbon, J.D., Holm, D.D., Nicolaenko, B. (1987). Phys. Rev. Lett., 59, 2911. 66. Doi, M. (1976). J. Phys. A, 9, 1465. 67. Domb, C. (1996). The critical point. Taylor and Francis: London. 68. Drugowich de Fel´ıcio,J.R. Hipólito,O. (1985). Am. J. Phys., 53, 690. 69. Dziarmaga, J., Laguna P., Zurek, W.H. (1999). Phys. Rev. Lett., 82, 4749. 70. Edwards, I.K. (1979). Am. J. Phys., 47, 153. 71. Farini, A., Boccaletti, S., Arecchi, F.T. (1996). Phys. Rev. E, 53, 4447. 72. Fellermann, H., Solé,R.V. (2006). Santa Fe Institute working paper #06- 11-025. Santa Fe Institute, Santa Fe, NM. 73. Feshbach, H., Tikochinsky, Y. (1977). Trans. N.Y. Acad. Sci. Ser. II, 38, 44. 74. Fogedby, H.C. (1998). Phys. Rev. E, 57, 4943. 75. Fogedby, H.C., Brandenburg, A. (2002). Phys. Rev. E, 66, 016604. 76. Ford, G.W., O’Connell, R.F. (2001). Phys. Lett. A, 286, 87. 77. Förner,W. (1997). Int. J. Quantum Chem., 64, 351. 78. Freeman, W.J. (1987). Biological Cybernetics, 56, 139. 79. Freeman, W.J. (1992). Int. J. Bifurcation Chaos, 2, 451. 80. Freeman, W.J. (2000a). Neurodynamics. An Exploration to Mesoscopic Brain dynamics. Springer: London. 81. Freeman, W.J. (2000b). Neural Networks, 13, 11. 82. Freeman, W.J. (2005). Journal of Integrative Neuroscience, 4, 407. 83. Freeman, W.J., Vitiello, G. (2006). Physics of Life Reviews, 3, 93. 84. Freeman, W.J., Kozma, R., Werbos, P.J. (2001). BioSystems, 59, 109. 85. Friberg, S.R., Machida, S., Werner, M.J., Levanon, A., Mukai, T. (1996). Phys. Rev. Lett., 77, 3775. 86. Fröhlich, H. (1968). Int. J. Quantum Chem., 2, 641. 87. Furusawa, C., Kaneko, K. (2002).The Anatomical Record, 268, 327. 88. Georgiev, D.D. (2006). Informatica, 30, 221. 89. Giulini, D., Joos, E., Kiefer, C., Kupsch, J., Stamatescu, I.-O., Zeh, H.D. (1996). Decoherence and the appearance of a classical world in Quantum Theory. Springer: Berlin. 90. Goldenfeld, N. (1992). Lectures on phase transitions and the renormalization group. Addison-Wesley: Reading, MA. 91. Goldstone, J., Salam, A., Weinberg, S. (1962). Phys. Rev., 127, 965. 92. Greenberger, D.M. (1978). Am. J. Phys., 46, 394. 93. Haag, R. (1961). Canonical commutation relations in Quantum Field The- ory and Functional Integration. In W.E.Brittin, B.W.Downs and J.Downs (Eds.). Lectures in Theoretical Physics, vol. 3 (pp. 353-381). Wiley: New York. 94. Hagan, S., Hameroff, S.R., Tuszyński, J.A. (2002). Phys. Rev. E, 65, 061901. 224 Eliano Pessa

95. Halvorson, H.P. (2001). Journal of Philosophical Logic, 37, 27. 96. Halvorson, H.P. (2004). Studies in History and Philosophy of Modern Physics, 35, 45. 97. Hameroff, S.R., Nip, A., Porter, M.J., Tuszyński, J.A. (2002). Biosystems, 64, 149. 98. Hameroff, S.R., Penrose, R. (1996). Journal of Consciousness Studies, 3, 36. 99. He, G., Cao, Z., Zhu, P., Ogura, H. (2003). Neural Networks, 16, 1195. 100. Helms, V. (2002), Current Opinion in Structural Biology, 12, 169. 101. Hepp, K. (1972). Helvetica Physica Acta, 45, 237. 102. Hernández-Garc´ıa,E., Hoyuelos, M., Colet, P., San Miguel, M., Montagne, R. (1999). Int. J. Bifurcation Chaos, 9, 2257. 103. Herzel, H., Pompe, B. (1987). Phys. Lett. A, 122, 121. 104. Hosenfeld, B., Van der Maas, H.L.J., Van der Boom, D.C. (1997). Journal of Experimental Child Psychology, 64, 367. 105. Houghton, G. (Ed.) (2005). Connectionist models in Cognitive Psychology. Psychology Press: Hove, UK. 106. Hoyuelos, M., Hernández-Garc´ıa,E., Colet, P., San Miguel, M. (2003). Phys- ica D, 174, 176. 107. Huang, K. (1987). Statistical mechanics, 2nd edition. Wiley: New York. 108. Ito, J., Kaneko, K. (2002). Phys. Rev. Lett., 88, 028701. 109. Ito, J., Kaneko, K. (2003). Phys. Rev. E, 67, 046226. 110. Jibu, M., Yasue, K. (1995). Quantum Brain Dynamics and Consciousness: An Introduction. Benjamins: Amsterdam. 111. Jibu, M., Yasue, K. (2004). Quantum brain dynamics and Quantum Field Theory. In G.G.Globus, K.H.Pribram and G.Vitiello (Eds.). Brain and being. At the boundary between science, philosophy, language and arts (pp. 267-290). Benjamins: Amsterdam. 112. Katz, S., Lebowitz, J.L., Spohn, H. (1983). Phys. Rev. B, 28, 1655. 113. Kibble, T.W.B. (1976). J. Phys. A, 9, 1387. 114. Kibble, T.W.B. (1980). Phys. Rep., 67, 183. 115. Kivshar, Yu.S. Pelinovsky, D.E. (2000). Phys. Rep., 331, 117. 116. Kohen, A., Klinman, J. (1999). Chemistry and Biology, 6, R191. 117. Konotop, V.V., Takeno, S. (2001). Phys. Rev. E, 63, 066606. 118. Koriat, A. (2000). Control processes in remembering. In E.Tulving, F.I.M.Craik (Eds.). The Oxford Handbook of Memory (pp. 333-346). Ox- ford University Press: New York, NY. 119. Kozma, R., Freeman, W.J. (2002). Neurocomputing, 44, 1107. 120. Kuramoto, Y. (1984). Chemical Oscillations, Waves and Turbulence. Springer: Berlin. 121. Kushibe, M., Liu, Y., Ohtsubo, J. (1996). Phys. Rev. E, 53, 4502. 122. Laguna, P., Zurek, W.H. (1997). Phys. Rev. Lett., 78, 2519. 123. Lakshmanan, M. (Ed.) (1988). Solitons . Springer: Berlin. 124. Lawson, D., Jensen, H.J., Kaneko, K. (2006). J. Theor. Biol., 243, 299. 125. Leutwyler, H. (1997). Helvetica Physica Acta, 70, 275. 126. Levy, B.J., Anderson, M.C. (2002). Trends in Cognitive Science, 6, 299. Phase Transitions in Biological Matter 225

127. Lindblad, G. (1976). Commun. Math. Phys., 48, 119. 128. Lombardo, F.C., Rivers, R.J., Mazzitelli, F.D. (2002). Int. J. Theor. Phys., 41, 2121. 129. Lythe, G. (2001). Int. J. Theor. Phys., 40, 2309. 130. Maciá,J., Solé,R.V. (2006). Santa Fe Institute working paper #06-07-024. Santa Fe Institute, Santa Fe, NM. 131. Manton, N., Sutcliffe, P., Landshoff, P.V., Nelson, D.R., Sciama, D.W. Wein- berg, S. (Eds.) (2004). Topological solitons. Cambridge University Press: Cambridge, UK. 132. Masgrau, L., Roujeinikova, A., Johannissen, L.O., Hothi, P., Basran, J., Ranaghan, K.E., Mulholland, A.J., Sutcliffe, M.J., Scrutton, N.S., Leys, D. (2006). Science, 312, 237. 133. Maslov, V.P., Shvedov, O.Yu. (1999). Phys. Rev. D, 60, 105012. 134. Matsumoto, H., Sodano, P. Umezawa, H. (1979). Phys. Rev. D, 19, 511. 135. Matsumoto, K., Tsuda, I.(1983). J. Stat. Phys., 31, 87. 136. Mensky, M.B., Stenholm, S. (2003). Phys. Lett. A, 308, 243. 137. Mesquita, M.V., Vasconcellos, A.R., Luzzi, R. (1998). Int. J. Quantum Chem., 66, 177. 138. Mesquita, M.V., Vasconcellos, A.R., Luzzi, R. (2004). Braz. J. Phys., 34, 489. 139. Metcalfe, J. (2000). Metamemory. Theory and data. In E.Tulving, F.I.M.Craik (Eds.). The Oxford Handbook of Memory (pp. 197-211). Ox- ford University Press: New York, NY. 140. Michod, R.E. (2006). Proc. Nat. Acad. Sci. USA, 103, 9113. 141. Michod, R.E., Viossat, Y., Solari, C.A., Hurand, M., Nedelcu, A.M. (2006). J. Theor. Biol., 239, 257. 142. Mikhailov, A.S., Calenbuhr, V. (2002). From cells to societies. Models of complex coherent actions. Springer: Berlin. 143. Minati, G., Pessa, E. (2006). Collective Beings. Springer: Berlin. 144. Mokarzel, S.G., Salgueiro, A.N., Nemes, M.C. (2002). Phys. Rev. A, 65, 0044101. 145. Morita, H., Kaneko, K. (2006). Phys. Rev. Lett., 96, 050602. 146. Nagem, R., Rhodes, B.A., Sandri, G.V.H. (1991). Journal of Sound and Vibration, 144, 536. 147. Nakagawa, M., Okabe, M. (1992). J. Phys. Soc. Jpn., 61, 1121. 148. Narayanan, A., Menneer, T. (2000). Information Sciences, 128, 231. 149. Nekorkin, V.I., Velarde, M.G. (2002). Synergetic phenomena in active lat- tices. Patterns, waves, solitons, chaos. Springer: Berlin. 150. Nestorovic, Z., Ristovski, Lj.M., Davidovic, G.S. (1998). Phys. Scr., 58, 275. 151. Ni, G., Xu, D., Yang, J., Chen, S. (1992). J.Phys. A, 25, 679. 152. Ott, E. (1993). Chaos in Dynamical Systems. Cambridge University Press: Cambridge, UK. 153. Ott, E., Grebogi, C., Yorke, J.A. (1990). Phys. Rev. Lett., 64, 1196. 154. Pascazio, S. (2004). Journal of Modern Optics, 51, 925. 155. Pastor-Satorras, R., Solé,R.V. (2001). Phys. Rev. E, 64, 051909. 226 Eliano Pessa

156. Patashinskij, A.Z., Pokrovskij, V.L. (1979). Fluctuation theory of phase transitions. Pergamon Press: Oxford, UK. 157. Peliti, L. (1985). Journal de Physique, 46, 1469. 158. Penrose, R. (1994). Shadows of the mind. Oxford University Press: Oxford, UK. 159. Pérez-Madrid,A., Reguera, D., Rub´ı,J.M. (2003). Physica A, 329, 357. 160. Pessa, E. (1988). Biological Cybernetics, 59, 277. 161. Pessa, E. (2004). Quantum connectionism and the emergence of cognition. In G.G.Globus, K.H.Pribram and G.Vitiello (Eds.). Brain and being. At the boundary between science, philosophy, language and arts (pp. 127-145). Benjamins: Amsterdam. 162. Pessa, E., Vitiello, G. (2004). Int. J. Mod. Phys. B, 18, 841. 163. Pikovsky, A., Popovych, O., Maistrenko, Yu. (2001). Phys. Rev. Lett., 87, 044102. 164. Pikovsky, A., Rosenblum, M., Kurths, J. (2002). Synchronization: A Univer- sal Concept in Nonlinear Science. Cambridge University Press: Cambridge, UK. 165. Podolny, A., Zaks, M.A., Rubinstein, B.Y., Golovin, A.A., Nepomnyashchy, A.A. (2005). Physica D, 201, 291. 166. Privman, V. (Ed.) (1997). Nonequilibrium Statistical Mechanics in One Di- mension. Cambridge University Press: Cambridge, UK. 167. Quinlan, P. (Ed.) (2003). Connectionist models of development: Develop- mental processes in real and artificial neural networks. Psychology Press: Hove, UK. 168. Rajagopal, A.K. (1998). Physica A, 253, 271. 169. Rajagopal, A.K., Rendell, R.W. (2001). Phys. Rev. A, 63, 022116. 170. Rajaraman, R. (1987). Solitons and Instantons. North Holland: Amsterdam. 171. Rasmussen, S., Chen, L., Nilsson, M., Abe, S. (2003). Artificial Life, 9, 269. 172. Rasmussen, S., Chen, L., Stadler, B., Stadler, P. (2004). Origins of Life and Evolution of the Biosphere, 34, 171. 173. Recati, A., Fuchs, J.N., Pe¸ca,C.S., Zwerger, W. (2005). Phys. Rev. A, 72, 023616. 174. Redaelli, S., Plewczyski, D., Macek, W.M. (2002). Phys. Rev. E, 66, 035202. 175. Rédei,M., Stölzner,M. (Eds.) (2001). John Von Neumann and the Foun- dations of Quantum Physics. Kluwer: New York. 176. Ricciardi, L. M., Umezawa, H. (1967). Kybernetik, 4, 44. 177. Rivers, R.J. (2000). Int. J. Theor. Phys., 39, 1779. 178. Rivers, R.J., Lombardo, F.C., Mazzitelli, F.D. (2002). Int. J. Theor. Phys., 41, 2145. 179. Rosenblum, M., Pikovsky, A. (2003). Contemporary Physics, 44, 401. 180. Ruetsche, L. (2002). Philosophy of Science, 69, 348. 181. Rumer, Yu.B., Rivkyn, M.Sh. (1980). Thermodynamics, Statistical Physics, and Kinetics. Mir: Moscow. 182. Schwarz, U., Safran, S. (2002). Phys. Rev. Lett., 88, 245416. 183. Scott, A. (2003). Nonlinear science: Emergence and dynamics of coherent structures. Oxford University Press: Oxford, UK. Phase Transitions in Biological Matter 227

184. Scott, A.S. (1992). Phys. Rep., 217, 1. 185. Sewell, G.L. (1986). Quantum Theory of Collective Phenomena. Oxford Uni- versity Press: Oxford, UK. 186. Shi, Y. (2003). Phys. Lett. A, 309, 254. 187. Shibata, T., Kaneko, K. (1997). Europhys. Lett., 38, 417. 188. Shibata, T., Chawanya, T., Kaneko, K. (1999). Phys. Rev. Lett., 82, 4424. 189. Shiino, M., Yoshida, K. (2001). Phys. Rev. E, 63, 026210. 190. Shraiman, B.I., Pumir, A., van Saarloos, W., Hohenberg, P.C., Chaté,H., Holen, M. (1992). Physica D, 57, 241. 191. Sinha, S. (1996). Physica A, 224, 433. 192. Sirovich, L., Rodriguez, J.D. (1987). Phys. Lett. A, 120, 211. 193. Sivardière,J. (1983). Am. J. Phys., 51, 1016. 194. Skarda, C.A., Freeman, J.W. (1987). Behav. Brain Sci., 10, 161. 195. Smith, E. (2006). Santa Fe Institute working paper #06-11-40. Santa Fe Institute, Santa Fe, NM. 196. Sparrow, C. (1982). The Lorenz Equations: Bifurcations, Chaos and Strange Attractors. Springer: Berlin. 197. Stein, D.L. (1980). Journal of Chemical Physics, 72, 2869. 198. Strogatz, S.H. (1994). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Perseus Books: Cambridge, MA. 199. Stuart, C. J. I., Takahashi, Y., Umezawa, H. (1978). J. Theor. Biol., 71, 605. 200. Stuart, C. J. I., Takahashi, Y., Umezawa, H. (1979). Foundations of Physics, 9, 301. 201. Tabor, M. (1989). Chaos and integrability in nonlinear dynamics. Wiley: New York, NY. 202. Takagi, H., Kaneko, K. (2005). J. Theor. Biol., 234, 173. 203. Tang, T.Z., Beberman, R., DeRubeis, R.J., Pham T. (2005). Journal of Consulting and Clinical Psychology, 73, 168. 204. Tarasov, V.E. (2001). Phys Lett. A, 288, 173. 205. Tegmark, M. (2000). Information Sciences, 128, 155. 206. Tolédano,J.C. Tolédano,P. (1987). The Landau theory of phase transitions. World Scientific: Singapore. 207. Topaj, D., Kye, W.-H., Pikovsky, A. (2001). Phys. Rev. Lett., 87, 074101. 208. Tuszyński, J.A. (Ed.) (2006). The emerging physics of consciousness. Springer: Berlin. 209. Tuszyński, J.A., Trpiˇsová,B., Sept, D., Brown, J.A. (1997). Journal of Structural Biology, 118, 94. 210. Tuszyński, J.A., Brown, J.A., Hawrylak, P. (1998). Phil. Trans. R. Soc. Lond. A, 356, 1897. 211. Umezawa, H. (1993). Advanced Field Theory. Micro, Macro, and Thermal Physics. American Institute of Physics: New York. 212. Umezawa, H., Matsumoto, H., Tachiki, M. (1982). Thermo Field Dynamics and Condensed States. North Holland: Amsterdam. 213. Vitiello, G. (1995). Int. J. Mod. Phys. B, 9, 973. 228 Eliano Pessa

214. Vitiello, G. (2001). My double unveiled. Benjamins: Amsterdam. 215. Vitiello, G. (2004). Int. J. Mod. Phys. B, 18, 785. 216. Vitiello, G. (2005).Brazilian Journal of Physics, 35, 351. 217. Wagner, C., Stucki, J.W. (2002). J. Theor. Biol., 215, 375. 218. Werner, M.J. (1996). Phys. Rev. A, 54, R2567. 219. Werner, M.J., Friberg, S.R. (1997). Phys. Rev. Lett., 79, 4143. 220. Willeboordse, F.H., Kaneko, K. (2005). Phys. Rev. E, 72, 026207. 221. Wimmers, R.H., Savelsbergh, G.-J.P., Beek, P.J., Hopkins, B. (1998). De- velopmental Psychobiology, 32, 235. 222. Yu, H., Liu, Y., Peng, J. (2004). Chaos, Solitons and Fractals, 22, 787. 223. Zhang, X., Jiao, K., Sharma, P., Yakobson, B.I. (2006). J. Mech. Phys. Solids, 54, 2304. 224. Zurek, W.H. (1993). Acta Phys. Pol. B, 24, 1301. 225. Zurek, W.H. (1996). Phys. Rep., 276, 177. 226. Zurek, W.H. (2003). Rev. Mod. Phys., 74, 715. 229

Microcosm to Macrocosm via the Notion of a Sheaf (Observers in Terms of t-topos)

Goro Kato Mathematics Department, California Polytechnic State University San Luis Obispo, CA 93407, U.S.A. [email protected]

The fundamental approach toward matter, space and time is that particles (either objects of macrocosm or microcosm), space and time are all presheafi- fied. Namely, the concept of a presheaf is most fundamental for matter, space and time. An observation of a particle is represented by a morphism from the observed particle (its associated presheaf) to the observer (its associated presheaf) over a specified object (called a generalized time period) of a t-site (i.e. a category with a Grothendieck topology). This formulation provides a scale independent and background space-time free theory (since, for the t-topos theoretic formulation, space and time are discretely defined by the associated particle, whose particle-dependency is a consequence of quantum entanglement.). It is our basic scheme that the method of t-topos may provide a device for understanding and concrete formulation of macro-object and micro-object interconnection as morphisms in the sense of t-topos.

Keywords: t-topos; Sheaf, Space-time Free Theory; Quantum Gravity PACS(2006): 02.40.k; 04.20.Gz; 04.60.m

1. Introduction The concept of a sheaf appeared in 1940’s by Leray in algebraic topology and by K. Oka in the theory of analytic functions in several complex variables. In 1960’s, theory of sheaves (cohomology of sheaves) made tremen- dous development in algebraic geometry, especially by Grothendieck and in algebraic analysis by M. Sato and M. Kashiwara. The notion of a sheaf has been used to obtain global information from local data. The deﬁnition of a sheaf is a contravariant functor from a category with a Grothendieck topology, called a site, to a category satisfying the sheaf axiom as indicated in [1], [2], [3]. The notion of a topos, i.e. a category of (pre-) sheaves over a category with a Grothendieck topology has been used to study quantum gravity by Isham as in [4], Mallios–Rapitis as in [5], Guts–Grinkevisch as in 230 Goro Kato

[6], Kato as in [7] and [8] and others. One of the challenges for such a theory is to explain the physical reality by building a micro-macro theory, which is scale independent, (e.g. assigning real numbers or complex numbers for measuring quantities), and background spacetime free. In quantum gravity, such a theory needs to be a background spacetimeless theory. In our theory of t-topos, we have obtained certain levels of success as shown in [7], [8], [9], [10], [11]. For a general survey on the foundations of physics based on hidden variable theories, see [12], where the reports on various theories and ideas with direct experimental applications are thoroughly given.

2. The Very Small to not so Small and the Large We often divide the physical world into the microcosm where quantum physics is applied and the macrocosm where classical (or/and relativistic) mechanics is applied. One of the fundamental approaches to obtain a unified theory is to build the global (macro) object information from studying local (micro) objects. The inadequacy of the traditional continuum-based models (e.g. classical differential geometry based on either real numbers or complex numbers) as final physical theories has been pointed out by various authors. Our approach is to apply the concept of a (pre)sheaf as a candidate for such a unifying theory. Hence, the fundamental reality is described by a discrete concept. Via the notions of categories and sheaves, we need to describe the concepts of wave-particle duality, measurement and quantum entanglement in microcosm and the concepts of a light-cone and the gravitational effect on spacetime in macrocosm. (See the above references.) Here is our initial step toward such a unification theory of the very small, not so small and the large. Some of the results have been obtained toward this goal in [7], [8], [9], [10]. By definition, a category consists of objects and morphisms satisfying certain axioms as in the above references. Let us recall that a presheaf is a contravariant functor from a site S (a category with a Grothendieck topology. See [1], [2], or [3].) to a category. The basic notion is the triple of presheaves (m, κ, τ) associated with a particle, space and time, respectively. Note that presheaves (κ ,τ) are depending upon the particle m. Namely, the space and time depend upon locally m. Recall that this dependency is a consequence of quantum entanglement. (See [7] and [8] for the assertion.) Let m be a presheaf over a site S, i.e. m ∈Ob(Sˆ), i.e. m is an object of the category Sôf presheaves over S. (For definitions of presheaves, sites and toposes and the notations, see [1], [2], or [3].) Note also that in [7] and [8], the term a “t-site” is used when it is referred to in the theory of t-topos. Microcosm to Macrocosm via the Notion of a Sheaf 231

An object of t-site S is said to be a generalized time (period). There are two types of ur-wave states in the theory of t-topos : first, when the particle presheaf m is not observed, and secondly, when there are more than one choice in objects in the t-site S (as in the case of well-known double-slit experiment), then m is said to be the ur-wave state. Next, we will consider the case when m is in the ur-particle state. (For the double-slit application of t-topos and the duality, see the earlier mentioned papers [9] and [12].) That is, when m is observed by P ∈Ob(Sˆ), by definition, there exists a generalized time period V in t-site S so that there exists a morphism s(V) from m (V) to P(V), (see [7]). However, note that a particle can be in the ur-particle state as long as an object in the t-site is specified, regardless whether the particle is observed or not. The notions of microlocalizations of m and V in t-topos theory are decompositions of m and V as described in [8]. That is, write m as a direct product of finitely many sub-presheaves Q mi, namely, m = mi , and for V it is to consider a covering {Vj} of V, that is, { V← Vj}, (see [8]). When m is observed by P over V, one obtains no information of mi. However, when a microcosm-object mi is observed by P over a generalized time period Vj, there exists a morphism s(ij) from mi(Vj) to P(Vj). Note that m may not be defined over Vj. Namely, obtaining an information locally during the generalized time period does not imply that the macro-object m can be observed by P over Vj. In general, one cannot compose the morphisms s(ij) with the canonical projection π(Vj): m(Vj) → mi(Vj) to obtain the morphism from m(Vj) to P(Vj). This is one of the roots of the local-global phenomena: why studying an individual Q particle mi does not provide the global information of m = mi. When m is observed by P over generalized time periods V first and later U in S. Then there exists a morphism ρ from V to U in S. (See [7]) Then there exist morphisms from m(V) to P(V) and m(U) to P(U), ρ0 ρ00 ρ respectively For each factorization V −→ W −→ U of the morphism V −→ U in the t-site S, there corresponds a ur-particle state m(W) of m. For all the possible factorizations of the morphism of ρ, the presheaf m induces m(ρ0) m(ρ00) funtorially the corresponding factorizations m(V ) ←− m(W ) ←− m(U) m(ρ) of m(V ) ←− m(U) . Those possible “paths” via {m(W)} between m(V) and m(U) represent the Feynman paths in the classical sense (See Section 3.1 in [9]). Notice also that the directions of arrows are reversed when evaluated at the presheaf m. This is because the presheaf m is a contravariant functor by definition. One can generalize the above by formulating its relativistic version by the notion given in [8]. 232 Goro Kato

3. Conclusion The language of category theory, sheaf theory, i.e. a topos theory can give qualitative analysis in physics, especially in quantum gravity as in [4], [5], [8]. In order to obtain quantitative results, the notion of scaling needs to be introduced. Traditionally, the ﬁelds of real numbers and complex numbers have been used for scaling. Some have attempted to use the ring of p-adic integers instead of real or complex numbers. Hence, the categorical approach alone is not suﬃcient for applications to physics. However, one should have at least one theory formulating both microcosm and macrocosm, i.e., a theory on quantum gravity, with one mathematical model. Our choice is a category of presheaves from a (t-) site to a product category, which is scale independent and background spacetime free.

Acknowledgment The author would like to thank Ignazio Licata for his invitation and en- couragement with the preparation of this paper.

References 1. Kashiwara, M., and Schapira, J., (2006), Categories and Sheaves, Springer. 2. Gelfand, S., and Manin, Yu. I., (1996), Methods of Homological Algebra, Springer. 3. Kato, G., (2006), The Heart of Cohomology, Springer. 4. Isham, C., and Butterﬁeld, J., (1999) Spacetime and the philosophical challenge of quantum gravity, arXiv:gr–qc/9903072 v1. 5. Mallios, A., and Raptis, I., Finitary–Algebraic resolution of the inner Schwarzschild singularity, arXov:gr-qc/0408045 v3 20 Aug.2004. 6. Guts, A.K., and Grinkevich, E. B., (1996), Toposes in general theory of relativity, arXiv:gr–qc/9610073. 7. Kato, G., Elemental principles of t-topos, (2004), Europhys. Lett., 68 (4) pp. 467-472. 8. Kato, G., (2005), Elemental t.g. principles of relativistic t-topos, Europhys. Lett., 71(2), pp. 172–178. 9. Kato, G., and Tanaka, T., (2006), Double-slit interference and temporal topos, Foundations of Physics, Vol. 36, No.11, pp. 1681–1700. 10. Kafatos, M., Kato, G., Roy, S., and Tanaka, T., Sheaf theory and geometric approach to EPR nonlocality, to be submitted to Foundations of Physics. 11. Kato, G., Black horizon and t-topos, in preparation. 12. Genovese, M., (2005), Research on hidden variable theories: A review of recent progresses, Physics Reports, 413, pp. 319–396. 233

The Dissipative Quantum Model of Brain and Laboratory Observations

Walter J. Freemana and Giuseppe Vitiellob a Department of Molecular and Cell Biology University of California, Berkeley CA 94720-3206, USA b Dipartimento di Matematica e Informatica, Universit`adi Salerno, and Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Salerno, I-84100 Salerno, Italy [email protected] - http://sulcus.berkeley.edu [email protected] - http://www.sa.infn.it/giuseppe.vitiello

We discuss the predictions of the dissipative quantum model of brain in connection with the formation of coherent domains of synchronized neuronal oscillatory activity and macroscopic functions of brain revealed by functional brain imaging.

Keywords: Quantum Models of Brain; Biophysics; Complex Systems PACS(2006): 03.65.w; 03.67.a; 87.16.Ac; 87.15.v; 89.75.k; 89.75.Fb

1. Introduction The mesoscopic neural activity of neocortex appears consisting of the dynamical formation of spatially extended domains in which widespread cooperation supports brief epochs of patterned oscillations. These “packets of waves” have properties of location, size, duration and carrier frequencies in the beta-gamma range (12 − 80 Hz). They re-synchronize, through a sequence of repeated collective phase transitions in cortical dynamics, in frames at frame rates in the theta-alpha range (3 − 12 Hz). The formation of such patterns of amplitude modulated (AM) synchronized oscillations in neocortex has been demonstrated by imaging of scalp potentials (electroencephalograms, EEGs) and of cortical surface potentials (electro- corticograms, ECoGs) of animal and human from high-density electrode arrays 1−5. The AM patterns appear often to extend over spatial domains covering much of the hemisphere in rabbits and cats6,7 , and over the length of a 64 × 1 linear 19 cm array1 in human cortex with near zero phase dis- persion8,9 . Synchronized oscillation of large-scale neuronal assemblies in 234 Walter J. Freeman and Giuseppe Vitiello beta and gamma ranges have been detected also by magnetoencephalo- graﬁc (MEG) imaging in the resting state and in motor task-related states of the human brain10 . In this paper we compare the predictions of the dissipative quantum model of brain11,12 with these neurophysiological data5,13 . It turns out that the model explains two main features of the EEG data: the textured patterns of AM in distinct frequency bands correlated with categories of conditioned stimuli (CS), i.e. coexistence of physically distinct AM patterns, and the remarkably rapid onset of AM patterns into (irreversible) sequences that resemble cinematographic frames. Each spatial AM pattern is indeed described to be consequent to spontaneous breakdown of symmetry (SBS)14–16 triggered by external stimulus and is associated with one of the unitarily inequivalent ground states. Their sequencing is associated to the non-unitary time evolution in the dissipative model, as discussed below. Cellular and molecular models of neuronal function have been formulated in the so-called K-set theory: a hierarchical set of models based in studies of single neurons embedded in populations of increasing size and complexity17,18 . The purpose of the present paper is to show that the dissipative quantum model may describe the entire forebrain dynamics as it selects, adapts, and elaborates generic intentional actions5,13 . In our presentation we closely follow Refs. 5,13. The question concerning the mass action in the observed cortical activity as originally pointed out by Lashley is considered in Section II. The dissipative many-body model of brain is summarized in Section III and it is there discussed in relation to the evidence of a multiplicity of ground states. The formation and recurrence of AM oscillatory patterns are further discussed in Section IV. Free energy and classicality of trajectories in brain space are discussed in Section V. Final remarks and conclusions are presented in Section VI.

2. Lashley Dilemma In the first half of the 20th century Lashley was led to the hypothesis of “mass action” in the storage and retrieval of memories in the brain and observed: “...Here is the dilemma. Nerve impulses are transmitted ...form cell to cell through definite intercellular connections. Yet, all behavior seems to be determined by masses of excitation...within general fields of activity, without regard to particular nerve cells... What sort of nervous organization might be capable of responding to a pattern of excitation without limited specialized path of conduction? The problem is almost universal in the activity of the nervous system” (pp. 302–306 of Ref. 19). Many laboratory The Dissipative Quantum Model of Brain and Laboratory Observations 235 observations in the following years confirmed Lashley’s finding. Pribram extended the concept of mass action by introducing the analogy between the fields of distributed neural activity in the brain and the wave patterns in holograms20 . Observational access to real time imaging of “patterns of excitation” and dynamical formation of spatially extended domains of neuronal fields of activity has been provided by magnetoencephalogram (MEG), functional magnetic resonance imaging (fMRI), positron electron tomography (PET), blood-oxygen level deletion (BOLD) and single photon emission computed tomography (SPECT). As already mentioned in the In- troduction, observations7 show that cortex jumps abruptly from a receiving state to an active transmitting state. Spatial AM patterns with carrier frequencies in the beta and gamma ranges form during the active state and dissolve as the cortex return to its receiving state after transmission. These state transitions in cortex form frames of AM patterns in few ms, hold them for 80 − 120 ms, and repeat them at rates in alpha and theta ranges of EEG1−8. In such an activity neocortex appears to behave according to four interrelated properties21,22 : the exchangeability of its ports of sensory input; its ability to adapt rapidly and flexibly to short- and long-term changes; its reliance on large-scale organization of pattern of neural activity that mediate its perceptual functions; the incredibly small amounts of information entering each port in brief behavioral time frames that support effective and efficient intentional action and perception. Four material agencies have been then proposed to account for the processes involving large populations of neurons. None of them, however, appears to be able to explain the observed cortical activity23 : – Nonsynaptic neurotransmission21 has been proposed as the mechanism for implementation of volume transmission to answer the question on how that broad and diffuse chemical gradients might induce phase locking of neural pulse trains at ms intervals. Although nonsynaptic transmission is essential for neuromodulation, diffusion of chemical fields of metabolites providing manifestations of widespread coordinated firing is, however, too slow to explain the highly textured patterns and their rapid changes.23 The observed high rates of field modulation are not compatible with mediation of chemical diffusion such as those estimated in studies of spike timing among multiple pulse trains (see Refs. 24–26, of cerebral blood flow using fMRI or BOLD27,28 , and of spatial patterns of the distributions of radio-labeled neurotransmitters and neuromodulators as measured with PET, SPECT and optical techniques. – Weak extracellular electric currents have been postulated as the 236 Walter J. Freeman and Giuseppe Vitiello agency by which masses of neurons link together29 . They have been shown to modulate the firing of neurons in vitro. However, the current densities required in vivo to modulate cortical firing exceed by nearly two orders of magnitude those currents that are sustained by extracellular dendritic currents across the resistance of brain tissue17,30 . The passive potential differences are measured as the EEG. – The intracellular current in palisades of dendritic shafts in cortical columns generates magnetic fields of such intensity that they can be measured 4−5 cm above the scalp with MEG. The earth’s far stronger magnetic field can be detected by specialized receptors for navigation in birds and bees,31 leading to the search for magnetic receptors among cortical neurons (Refs. 32,33), so far without positive results. – Radio waves propagating from the combined agency of electric and magnetic fields has also been postulated.34 However, neuronal radio communication is unlikely, owing to the 80 : 1 disparity between electric per- mittivity and magnetic permeability of the brain tissue and to the low frequency (< 100 Hz) and kilometer wavelengths of e.m. radiation at EEG frequencies. The conclusion is that Lashley dilemma remains still to be explained: neither the chemical diffusion, which is much too slow, nor the electric field of the extracellular dendritic current nor the magnetic fields inside the dendritic shafts, which are much too weak, are the agency of the collective neuronal activity. An alternative approach is therefore necessary. The dissipative quantum model of brain has been then proposed5,11–13 to account for the observed dynamical formation of spatially extended domains of neuronal synchronized oscillations and of their rapid sequencing.

3. The Dissipative Many-Body Model and Evidence of A Multiplicity of Ground States A. The many-body model The dissipative quantum model of brain is the extension to the dissipative dynamics of the many-body model proposed in 1967 by Ricciardi and Umezawa35,36 . They suggested that the extended patterns of excitations observed in the neurophysiological research might be described by means of the SBS formalism in many-body physics. The fact that the brain is an open system in permanent interaction with the environment was not considered in the many-body model. It has been therefore extended in a way to include the intrinsically open dynamics of the brain. This led to the dissipative many-body model11,12 . Umezawa explains the motivation for using the The Dissipative Quantum Model of Brain and Laboratory Observations 237 quantum field theory (QFT) formalism of many-body physics37 : ‘In any material in condensed matter physics any particular information is carried by certain ordered pattern maintained by certain long range correlation mediated by massless quanta. It looked to me that this is the only way to memorize some information; memory is a printed pattern of order supported by long range correlations...” In QFT long range correlations among the system constituents are indeed dynamically generated through the mechanism of SBS. These correlations manifest themselves as quanta, called the Nambu–Goldstone (NG) boson particles or modes, which have zero mass and therefore able to span the whole system. At a mesoscopic/macroscopic scale, due to such correlations in this way established, the system appears in an ordered state. One expresses this fact by saying that the NG bosons are coherently condensed in the system lowest energy state (the system ground state) according to the Bose–Einstein condensation mechanism. The density of the condensed NG bosons provides a measure of the degree of ordering or coherence. One arrives thus to the definition of the order parameter, a classical field specifying the ordered patterns observed in the system ground state. This is the way the dynamical formation of extended ordered patterns is described in many-body physics. Laboratory observations allow to detect the NG quanta or particles by means of their scattering with observational probes; for example, one can use neutrons as probe to observe their scattering with phonons in a crystal. The phonons are the NG particle responsible for the crystalline ordering. They are the quanta associated to the elastic waves. Other examples of NG particles are the magnons in the ferromagnets, namely the quanta of the spin waves. In the case of the brain, the quantum variables are identified11,38,39 with the electrical dipole vibrational field of the water molecules and of other biomolecules present in the brain structures. The associated NG quanta are named the dipole wave quanta (DWQ). These do not derive from Coulomb interaction. They are dynamically generated through the breakdown of the rotational symmetry of the electrical dipole vibrational field. Water is more than the 80% of brain mass and in the many-body model it is therefore expected to be a major facilitator or constraint on brain dynamics. The theoretically and experimentally well established knowledge of condensed matter physics thus suggested to Umezawa that memory storage might be described in terms of the coherent Bose condensation process in the system lowest energy state. It has to be remarked that the neuron and the glia cells and other physiological units are not quantum objects in the many-body model of brain. 238 Walter J. Freeman and Giuseppe Vitiello

In the many-body model the external input or stimulus acts on the brain as a trigger for the spontaneous breakdown of the symmetry with the consequent long range correlation established by the coherent condensation of NG bosons. In the dissipative model it becomes clear that the stimulus action may trigger the symmetry breakdown only provided the cortex is at or near a singularity (see below), predicting indeed the laboratory observations.40 The density value of the condensation of DWQ in the ground state acts as a label classifying the memory there created. The memory thus stored is not a representation of the stimulus. This reflects a general feature of the spontaneous breakdown of symmetry in QFT: the specific ordered pattern generated through SBS by an external input does not depend on the stimulus features. It depends on the system internal dynamics. The model thus accounts for the laboratory observation of lack of invariance of the AM neuronal oscillation patterns with invariant stimuli . The recall of the recorded information occurs under the input of a stimulus able to excite DWQ out of the corresponding ground state. In the many-body model such a stimulus is called “similar” to the one responsible for the memory recording36 . In the model similarity between stimuli thus refers not to their intrinsic features, but to the reaction of the brain to them; in other words, to the possibility that under their action DWQ are condensed into or excited from the ground state carrying the same label. The many-body model, however, fails in explaining the observed coexistence of AM patterns and also their irreversible time evolution. Indeed, one shortcoming of the model is that any subsequent stimulus would cancel the previously recorded memory by renewing the SBS process with the consequent DWQ condensation, thus overprinting the ’new’ memory over the previous one (’memory capacity problem’). In order to solve these problems, considering that the brain is an open system, the original many-body model has been extended11 to the dissipative dynamics. B. The dissipative many-body model The mathematical structure of QFT provides the possibility of having different vacua with different symmetry properties. Indeed, infinitely many representations of the canonical commutation relations (CCR’s) exist in QFT. They are, with respect to each other, unitarily inequivalent (i.e there is no unitary operator transforming one representation into another one of them)41 . Thus they are physically inequivalent: they describe different physical phases of the system (it is not so in Quantum Mechanics where all the representations are unitarily (and therefore physically) equivalent)15,42,43 . The existence of infinitely many representations of the CCR’s The Dissipative Quantum Model of Brain and Laboratory Observations 239 is fully exploited in the dissipative quantum model of brain, which is therefore not a quantum mechanical model but a QFT model43 . In the dissipative model the environmental influence on the brain is taken into account by performing a suitable choice of the brain vacuum state (the brain ground state) among the infinitely many of them, each other unitarily inequivalent. The choice of the effective vacuum is done in the process of SBS triggered by the external stimulus. As explained above, the specificity of such a “choice” is fully controlled by the internal dynamics of the brain system; “making” the choice signals that the brain-environment interaction is “active”. A change in the brain–environment reciprocal influence then corresponds to a change in the choice of the brain vacuum: the brain evolution through the vacuum states thus reflects the evolution of the coupling of the brain with the surrounding world. The vacuum condensate of DWQ, consequent to SBS induced by the external stimulus, is assumed to be the quantum substrate of the AM pattern observed at a phenomenological level. In agreement with observations, the dissipative dynamics allows (quasi-)non-interfering degenerate vacua with different condensates, i.e. different values for the order parameter (AM pattern textures), and (phase) transitions among them (AM patterns sequencing). These features could not be described in the frame of the original many-body model. By exploiting the existence of the infinitely many representations of the CCR’s in QFT, the dissipative model allows a huge memory capacity. In the QFT formalism for dissipative systems44 the environment is described as the time-reversal image of the system. This is realized by ‘doubling’ the system degrees of freedom. In the dissipative model, the brain dynamics is indeed described in terms of an infinite collection of damped harmonic oscillators aκ (a simple prototype of a dissipative system) rep- 11 resenting the boson DWQ modes and by thea ˜κ modes which are the ”time-reversed mirror image” (the “mirror or doubled modes”) of the aκ modes.a ˜κ represent the environment modes. The label κ generically denotes degrees of freedom such as, e.g. spatial momentum, etc.11,44 . The breakdown of the dipole rotational symmetry is induced by the external stimulus and this leads to the dynamical generation of DWQ aκ. Their condensation in the ground state is then constrained by inclusion of the mirror modesa ˜κ in order to account for the system dissipation. The system ground state is indeed not invariant under time translation and this reveals the irreversible time evolution of the system, namely dissipation. Although the brain holds itself far from equilibrium, the balance of the energy fluxes at the system-environment interface, including heat 240 Walter J. Freeman and Giuseppe Vitiello exchanges, holds in a sequence of states through which the brain evolves. This is manifested in the regulation of mammalian brain temperature. At some arbitrary initial time t0 = 0, we denote the zero energy state (the vac- 11 P uum) of the system by |0iN . This prescribes that E0 = κ ~Ωκ(Naκ −

Na˜κ ) = 0. Here, Ωκ is the common frequency of aκ anda ˜κ, and Naκ and Na˜κ are the (non-negative integer) numbers of aκ anda ˜κ which are condensed in |0iN . This implies that the ”memory state” |0iN is a condensate of equal number of modes aκ and mirror modesa ˜κ for any κ: Naκ − Na˜κ = 0. N denotes the set of integers deﬁning the ”initial value” of the condensate,

N ≡ {Naκ = Na˜κ , ∀ κ, at t0 = 0}, namely the label, or order parameter associated to the information recorded at time t0 = 0. Clearly, balancing

Naκ −Na˜κ to be zero for any κ, does not fix the value of either Naκ or Na˜κ for any κ. It only fixes, for any κ, their difference. Therefore, at t0 we may have infinitely many perceptual states, each one in one-to-one correspondence to a given N set.

The dynamics ensures that the number (Naκ − Na˜κ ) is a constant of motion for any κ (see Ref. 11). The system ofa ˜κ mirror modes represents the system’s Double. It can be shown that aκ anda ˜κ modes satisfy the Bose–Einstein distribution. |0iN is thus recognized to be a ﬁnite temperature state and it can be shown to be a squeezed coherent state11,15,45,46 .

The important point is that the spaces {|0iN } and {|0iN 0 } are each other unitarily inequivalent for diﬀerent labels N= 6 N 0 in the inﬁnite volume limit:

0 0 N h0|0iN 0 −→ 0 ∀ N , N , N 6= N . (1) V →∞ The whole space of states thus includes infinitely many unitarily inequivalent representations {|0iN }, for all N ’s, of the CCR’s. A huge number of sequentially recorded memories may coexist without destructive interference since infinitely many vacua |0iN , ∀ N , are independently accessible. In contrast with the non-dissipative model, recording the memory N 0 does not necessarily produce destruction of previously printed memory N 6= N 0, which is the meaning of the non-overlapping in the infinite volume limit expressed by Eq. (1). Dissipation allows the possibility of a huge memory capacity by introducing the N -labeled “replicas” of the ground state. The model thus predicts the existence of textures of AM patterns, as indeed observed in the laboratory. The stability of the order parameter against quantum fluctuations is The Dissipative Quantum Model of Brain and Laboratory Observations 241 a manifestation of the coherence of the DWQ boson condensation. The memory N is thus not affected by quantum fluctuations and in this sense it is a macroscopic observable. The “change of scale” (from microscopic to macroscopic scale) is dynamically achieved through the boson condensation mechanism. The state |0iN provides an example of “macroscopic quantum state”. In “the space of the representations of the CCR’s”, each representation

{|0iN } denotes a physical phase of the system and may be thought as a “point” identified by a specific N -set. In the infinite volume limit, points corresponding to different N sets are distinct points (do not overlap, cf. Eq. (1)). We also refer to the space of the representations of the CCR’s as to the “memory space” or the brain state space. The brain in relation with the environment may occupy any one of the multiplicity of ground states, depending on how the E0 = 0 balance or equilibrium is approached. Or else, it may sit in any state that is a collection or superposition of these brain-environment equilibrium ground states. The system may shift, under the influence of one or more stimuli acting as a control parameter, from ground state to ground state in this collection (from phase to phase) namely it may undergo an extremely rich sequence of phase transitions, leading to the actualization of a sequence of dissipative structures formed by AM patterns, as indeed experimentally observed. Denote by |0(t)iN the state |0iN at time t specified by the initial value N , at t0 = 0. Time evolution of the state |0(t)iN is represented as the trajectory of ”initial condition” specified by the N -set in the space of the representations {|0(t)iN }. The state |0(t)iN provides the ’instantaneous picture’ of the system at each instant of time t, or the ’photogram’ at t in a cinematographic sequence. When the DWQ frequency is time–dependent, κ-components of the N - set with higher momentum have been found to possess longer life–time. Since the momentum is proportional to the reciprocal of the distance over which the mode can propagate, this means that modes with shorter range of propagation (more “localized” modes) survive longer. On the contrary, modes with longer range of propagation decay sooner. Correspondingly, condensation domains of different finite sizes with different degree of stability are allowed in the model51 . Finally, let us observe that coherent domains of finite size are obtained by non-homogeneous boson condensation described by the condensation function f(x) which acts as a “form factor” specific for the considered do- main15,47,48 . The important point is that such a condensation function 242 Walter J. Freeman and Giuseppe Vitiello f(x) has to carry some topological singularity in order for the condensation process to be physically detectable. A regular function f(x) would produce a condensation which could be easily “washed” away by a convenient field transformation (“gauged” away by a gauge transformation). In a similar way, the phase transition from one space to another (inequivalent) one can be only induced by use of a singular condensation function f(x). One can show that this is the reason why topologically non trivial extended objects, such as vortices, appear in the processes of phase transitions.15,47,48 Stated in different words, this means that phase transitions driven by boson condensation are always associated with some singularities in the field phase (indeterminacy of the phase at the phase transition point). The observational counterpart in the brain dynamics is the decrease in power that in the perceptual phase transition precedes the formation of a new AM pattern which reflects the reduction in the amplitude of the spontaneous background activity40 . The event that initiates a phase transition is thus an abrupt decrease in the analytic power of the background activity to near zero, depicted as a null spike. This reduction induces a brief state of indeterminacy in which the amplitude of ECoG is near to zero and phase of ECoG is undefined, as indeed should be as said above according to the dissipative model. If a stimulus arrives at or just before this state, then the cortex can be driven by the input across the phase transition process to a new AM pattern. The response amplitude depends not on the input amplitude, as the dissipative model also predicts, but on the intrinsic state of the cortex, specifically the degree of reduction in the power and order of the background brown noise. The null spike in the band pass filtered brown noise activity is conceived as a shutter that blanks the intrinsic background. At very low analytic amplitude when the analytic phase is undefined, the system, under the incoming weak sensory input, may re-set the background activity in a new AM frame, if any, formed by reorganizing the existing activity, not by the driving of the cortical activity by input (except for the small energy provided by the stimulus that is required to select an attractor and force the phase transition). The decrease (shutter) repeats in the theta or alpha range, independently of the repetitive sampling of the environment by limbic input. In conclusion, the reduction in activity constitutes a singularity in the dynamics at which the phase is undefined (in agreement with the dissipative model requiring the singularity of the boson condensation function). The aperiodic shutter allows opportunities for phase transitions. The power is not provided by the input, exactly as the dissipative model The Dissipative Quantum Model of Brain and Laboratory Observations 243 predicts, but by the pyramidal cells, which explains the lack of invariance of AM patterns with invariant stimuli40 .

4. Emergence and Recurrence of Amplitude Patterns High-density electrode arrays (typically 8 × 8 in a 2D square) fixed on the scalp or the epidural surface of cortical areas and fast Fourier transform (FFT) have been used in order to measure AM pattern textures for which high spatial resolution is required1,49 . The set of n amplitudes squared from an array of n electrodes (typically 64) defines a feature vector, A2(t), of the spatial pattern of power at each time step. The vector specifies a point on a dynamic trajectory in brain state space, conceived as the collection of all possible brain states, essentially infinite. Measurement of n EEG signals defined a finite n-dimensional subspace, so the point specified by A2(t) is unique to a spatial AM pattern of the aperiodic carrier wave. Similar AM patterns form a cluster in n-space, and multiple patterns form either multiple clusters or trajectories with large Euclidean distances between digitizing steps through n-space. A cluster with a verified behavioral correlate denotes an ordered AM pattern. The vector A2(t) is taken to be the best available numeric estimator of our order parameter, because when the trajectory of a sequence of points enters into a cluster, that location in state space signifies increased order from the perspective of an intentional state of the brain, owing to the correlation with a conditioned stimulus. We use the reciprocal of the absolute value of 2 2 the step size between successive values of De(t) = |A (t) − A (t − 1)| as a scalar index of our order parameter. Pattern amplitude stability was proved by small steps in Euclidean distances, De(t), between consecutive points (higher spikes in Fig. 1). Pattern phase stability was proved by calculating the ratio, Re(t), of the temporal standard deviation of the mean filtered EEG to the mean temporal standard deviation of the n EEGs3,4 (lower curve in Fig. 1). By these measures AM/phase-modulated patterns stabilized just after the phase transitions and before reaching the maximum in the spatial AM pattern amplitude. Re(t) = 1 when the oscillations were entirely synchronized. When n EEGs were totally desynchronized, Re(t) approached one over the square root of the number of digitizing steps in the moving window. Experimentally Re(t) rose rapidly within a few ms after a phase discontinuity and several ms before the onset of a marked increase in mean analytic amplitude, A(t). The succession of the high and low values of Re(t) revealed episodic emergence and dissolution of synchrony; therefore Re(t) was adopted as an index of 244 Walter J. Freeman and Giuseppe Vitiello cortical efficiency,50 on the premise that cortical transmission of spatial patterns was most energy-efficient when the dendritic currents were most synchronized.

3 Fig. 1. Spikes: De stability measure. Curve: Re boson density.

Re-synchronized oscillations in the beta range near zero lag commonly recurred at rates in the theta range and covered substantial portions of the left cerebral hemisphere under observation2 (exceeding the length of the recording array (19 cm) in human brain). We conclude that a specific value of the phenomenological order parameter A2(t) may be assumed to correspond to a specific value of N of the order parameter predicted by the dissipative model: the observation of the AM pattern textures and of their sequencing finds thus a description in the dissipative model. The time evolution in the brain space described by the space of the representations of the CCR’s gives the image of quantum origin of the trajectories described by the time dependent vector A2(t) in the brain state space as phenomenologically described above. A further agreement of the dissipative model with observed features is recognized by considering the common frequency Ωκ(t) for the aκ anda ˜κ The Dissipative Quantum Model of Brain and Laboratory Observations 245 modes (cf. Eq. (8) in Ref. 51). The duration, size and power of AM patterns are predicted to be a decreasing functions of the carrier wave number κ, as indeed confirmed in the observations. Carrier waves in the gamma range (30 − 80 Hz): durations seldom exceeding 100 ms, diameters seldom exceeding 15 mm; low power in the 1/f a relation. Carrier frequencies in the beta range (12 − 30 Hz): durations often exceeding 100 ms; estimated diameters large enough to include multiple primary sensory areas and the limbic system; greater power by 1/f a.

5. Free Energy and Classicality In the frame of the dissipative model one can show that, provided changes in the inverse temperature β are slow, the changes in the energy P Ea ≡ k EkNak and in the entropy Sa are related by X 1 dE = E N˙ dt = dS , (2) a k ak β a k i.e. the minimization of the free energy Fa holds at any t: 1 dF = dE − dS = 0. (3) a a β a

1 As usual heat is defined as dQ = β dSa. Moreover, the time-evolution of the state |0(t)iN at finite volume V is controlled by the entropy variations, which reflects the irreversibility of time evolution (breakdown of time-reversal symmetry) characteristic of dissipative systems, namely the choice of a privileged direction in time evolution (arrow of time)11,44 . Eq. (2) shows that the change in time of the condensate, i.e. of the order parameter, turns into heat dissipation dQ. Therefore the ratio between the rate of free energy dissipation to the rate of change in the order parameter is a good measure of the ordering stability. This is in agreement with observations. Indeed, in terms of the laboratory observables the rate of change of the order parameter is specified by the Euclidean distance De(t) between successive points in the n-space. Typically De(t) takes large steps between clusters, decreases to a low value when the trajectory enters a cluster, and remains low for tens of ms within a frame. Therefore De(t) serves as a measure of the spatial AM pattern stability. Empirically6,7 it was found that the best predictor of the onset times of ordered AM patterns was, as suggested by the dissipative model, the ratio He(t) of the rate of free energy dissipation to the rate of change in the order parameter because De(t) falls 246 Walter J. Freeman and Giuseppe Vitiello and A2(t) rises with wave packet evolution:

A2(t) He(t) = . De(t) This index is named the pragmatic information index after Atmanspacher and Scheingraber52 . Our measurements showed that typically the rate of change in the instantaneous frequency ω(t) was low in frames that coincided with low De(t) indicating stabilization of frequency as well as AM pattern. Between frames ω(t) increased often several fold or decreased even below zero in interframe breaks that repeated at rates in the theta or alpha range of the EEG2 (phase slip53). In the dissipative model as well as in the laboratory observations the time evolution of the brain state is represented as a trajectory in the brain state space. The possibility of deriving from the microscopic dynamics the classicality of such trajectories is one of the merits of the dissipative many- body field model11,12,51,54 . These trajectories are found to be deterministic chaotic trajectories54,55 , and thus the observed changes in the order parameter become susceptible to be described in terms of trajectories on attractor landscapes. One can show indeed that in the brain space the trajectories are classical and that i) they are bounded and each trajectory does not intersect itself (trajectories are not periodic). ii) there are no intersections between trajectories specified by different initial conditions. iii) trajectories of different initial conditions are diverging trajectories. The property ii) implies that no confusion or interference arises among different memories, even as time evolves. In realistic situations of finite volume, states with different N labels may have non–zero overlap (non- vanishing inner products). This means that some association of memories becomes possible. In such a case, at a “crossing” point between two, or more than two, trajectories, one can “switch” from one of these trajectories to another one which there crosses. This reminds us of the “mental switch” occurring, in the perception of ambiguous figures, or in general, while performing some perceptual and motor tasks56,57 as well as while resorting to free associations in memory tasks58 . From the property iii) one can derive54 that the difference between κ– 0 components of the sets N and N may become zero at a given time tκ. However, the difference between the sets N and N 0 does not necessarily The Dissipative Quantum Model of Brain and Laboratory Observations 247 become zero. The N -sets are made up by a large number (infinite in the continuum limit) of Naκ (θ, t) components, and they are different even if a finite number (of zero measure) of their components are equal (here θ is a convenient parameter, see Ref. 54,55). On the contrary, for very small δθκ, suppose that ∆t ≡ τmax − τmin, with τmin and τmax the minimum and the maximum, respectively, of tκ, for all κ’s, be “very small”. Then the N -sets are “recognized” to be “almost” equal in such a ∆t. Thus we see how in the “recognition” (or recall) process it is possible that “slightly different”

Naκ –patterns are “identified” (recognized to be the “same pattern” even if corresponding to slightly different inputs). Roughly, ∆t provides a measure of the “recognition time”. The relevant point, at the present stage of our research, is that the dissipative model predicts that the system trajectories through its physical phases may be chaotic54 and itinerant through a chain of ’attractor ruins’,59 embedded in a set of attractor landscapes60 accessed serially or merely approached in the coordinated dynamics of a metastable state61–64 . The manifold on which the attractor landscapes sit covers as a ”classical blan- ket” the quantum dynamics going on in each of the representations of the CCR’s (the AM patterns recurring at rates in the theta range (3 − 8 Hz)). The picture which emerges is that a CS selects a basin of attraction in the primary sensory cortex to which it converges, often with very little information as in weak scents, faint clicks, and weak flashes. The convergence constitutes the process of abstraction. The astonishingly low requirements for information in high-level perception have been amply demonstrated by recent accomplishments in sensory substitution22,65,66 . There is an indefinite number of such basins in each sensory cortical area forming a pliable and adaptive attractor landscape. Each attractor can be selected by a stimulus that is an instance of the category (generalization) that the attractor implements by its AM pattern. In this view the waking state consists of a collection of potential states, any one of which but only one at a time can be realized through a phase transition.

6. Final Remarks and Conclusions In this paper we have compared the predictions of the dissipative quantum model of brain with the dynamical formation of spatially extended domains in which widespread cooperation supports brief epochs of patterned oscillations. The model seems to explain two main features of the neurophysiological data: the coexistence of physically distinct AM patterns correlated with categories of conditioned stimuli and the remarkably rapid 248 Walter J. Freeman and Giuseppe Vitiello onset of AM patterns into irreversible sequences that resemble cinematographic frames. A relevant rôleis played by the main ingredients of the model, namely spontaneous breakdown of symmetry and dissipation: each spatial AM pattern is indeed described to be consequent to spontaneous breakdown of symmetry triggered by external stimulus and is associated with one of the QFT unitarily inequivalent ground states. Their sequencing is associated to the non-unitary time evolution implied by dissipation. Many other features predicted by the model have been found in agreement with laboratory observations. We finally observe that another possible way to break the symmetry in QFT is to modify the dynamical equations by adding one or more terms that are explicitly not consistent with the symmetry transformations (are not symmetric terms). In QFT this is called explicit breakdown of symmetry. The system is forced by the external action into a specific non-symmetric state that is determined by the imposed breaking term. Again this fits well with our observation in the response of cortex to perturbation by an im- pulse, such as an electric shock, sensory click, flash, or touch: the evoked or event-related potential (ERP). The explicit breakdown in cortical dynamics is observed by resort to stimulus-locked averaging across multiple presentations in order to remove or attenuate the background activity, so as to demonstrate that the location, intensity and detailed configuration of the ERP is predominantly determined by the stimulus, so the ERP can be used as evidence for processing by the cortex of exogenous information. Contrastingly, in SBS the AM pattern configurations are determined from information that is endogenous from the memory store. The variety of these highly textured AM patterns, their exceedingly large diameters in comparison to the small sizes of the component neurons, the long ranges of correlation despite the conduction delays among them, and the extraordinarily rapid temporal sequence in the neocortical phase transitions by which they are selected, provide the principal justification for exploring the interpretation of nonlinear brain dynamics in terms of dissipative many-body theory and multiple ground states to complement basin-attractor theory. In conclusion, much work remains to be done in many research directions, such as the analysis of the interaction between the boson condensate and the details of the electrochemical neural activity, or the problems of extending the dissipative many-body model to account for higher cognitive functions of the brain. At the present status of our research, the study of the dissipative many-body dynamics underlying the richness of the The Dissipative Quantum Model of Brain and Laboratory Observations 249 laboratory observations seems to be promising. John von Neumann noted that “...the mathematical or logical language truly used by the central nervous system is characterized by less logical and arithmetical depth than what we are normally used to. ...We require exquisite numerical precision over many logical steps to achieve what brains accomplish in very few short steps” (pp. 80–81 of Ref. 67). The dissipative quantum model describing the textured AM patterns and the sequential phase transitions observed in brain functioning perhaps opens a window on such a view.

References 1. W. J. Freeman, B. C.Burke, M. D. Holmes and S. Vanhatalo (2003) Clin. Neurophysiol. 114, 1055. 2. W. J. Freeman, B. C. Burke and M. D. Holmes (2003) Human Brain Map- ping 19 (4), 248. 3. W. J. Freeman (2004) Clin. Neurophysiol. 115, 2077. 4. W. J. Freeman (2004) Clin. Neurophysiol. 115, 2089. 5. W. J. Freeman and G. Vitiello (2006) Phys. of Life Reviews 3, 93. http://www.arxiv.org/find [Freeman] q-bio.OT/0511037 6. W. J. Freeman (2005) Clin. Neurophysiol. 116 (5), 1118. 7. W. J. Freeman (2006) Clin. Neurophysiol. 117 (3), 572. 8. W. J. Freeman, G. Gaáland R. Jornten (2003) Intern. J. Bifurc. Chaos 13, 2845. 9. W. J. Freeman and L. J. Rogers (2003) Intern. J. Bifurc. Chaos 13, 2867. 10. D. S. Bassett, A. Meyer-Lindenberg, S. Achard, T. Duke, E. Bullmore (2006) PNAS, 103, 19518. 11. G. Vitiello (1995) Int. J. Mod. Phys. B 9, 973. 12. G. Vitiello (2001) My Double Unveiled. Amsterdam: John Benjamins,. 13. W. J. Freeman and G. Vitiello, q-bio.NC/0701053v1. 14. C. Itzykson and J. Zuber (1980) Quantum field theory. New York: McGraw- Hill. 15. H. Umezawa (1993) Advanced field theory: micro, macro and thermal concepts. New York: American Institute of Physics. 16. P.W. Anderson, (1984) Basic Notions of Condensed Matter Physics. Menlo Park: Benjamin. 17. W. J. Freeman (1975) Mass Action in the Nervous System. Academic Press, New York, Reprinted 2004. 18. W. J. Freeman (2000) Neurodynamics. An Exploration of Mesoscopic Brain Dynamics. London: Springer-Verlag. 19. K. Lashley (1948) The Mechanism of Vision, XVIII, Effects of Destroying the Visual ”Associative Areas” of the Monkey, Journal Press, Provincetown MA. 250 Walter J. Freeman and Giuseppe Vitiello

20. K. H. Pribram (1971) Languages of the Brain. Engelwood Cliffs NJ: Prentice-Hall. K. H. Pribram (1991) Brain and Perception. Hillsdale, New Jersey: Lawrence Erlbaum Associates Publ. 21. P. Bach-y-Rita (1995) Nonsynaptic Diffusion Neurotransmission and Late Brain Reorganization. New York: Demos-Vermande. 22. P. Bach-y-Rita (2004) Ann. N.Y. Acad. Sci 1013, 83. P. Bach-y-Rita (2005) Neurosci 4, 183. 23. W. J. Freeman (2005) J. Integrative Neuroscience 4 (4), 407. 24. D. J. Amit (1989) Modeling Brain Function: The World of Attractor Neural Networks, Cambridge: Cambridge University Press. 25. A. L. Morrow, P. D. Suzdak and S. M. Paul(1988) Adv. Biochem. Psy- chopharmacol. 45, 247. 26. T. B. Schillen and P. König(1994) Biol. Cybern. 70, 397. 27. P. E. Roland (1993) Brain Activation. New York: Wiley-Liss. 28. F. Varela, J-P. Lachaux, E. Rodriguez and J. Marinerie (2002) Nat. Rev. Neurosci. 2, 229. 29. C. A. Terzuolo, T. H. Bullock (1961) Proc. Nat. Acad. Sci. USA 42, 687. 30. A. van Harreveld, F. I. Khattab (1968) Anat. Rec. 162 (4), 467. 31. M. M. Walker, M. E. Bitterman (1989) J. Exp. Biol. 145, 489. 32. M. J. Azanza, A. del Moral (1994) Prog. Neurobiol. 44, 517. 33. J. R. Dunn, M. Fuller, J. Zoeger, J. Dobson, F. Heller, J. Hammann, E. Caine and B. M. Moskowitz (1995) Brain Re.s Bull. 36, 149. 34. W. R. Adey (1981) Physiol. Rev. 61, 435. 35. L. M. Ricciardi and H. Umezawa (1967) Kibernetik 4, 44. 36. C. I. J. Stuart, Y. Takahashi and H. Umezawa (1978) J.Theor. Biol. 71, 605; Found. Phys. 9, 301 (1979). 37. H.Umezawa (1995) Math. Japonica 41, 109 38. E. Del Giudice, S. Doglia, M. Milani and G. Vitiello (1985) Nucl. Phys. B 251 (FS 13), 375; Nucl. Phys. B 275 (FS 17), 185 (1986). E. Del Giudice, G. Preparata and G. Vitiello (1988) Phys. Rev. Lett. 61, 1085. 39. M. Jibu and K. Yasue, (1995) Quantum brain dynamics and consciousness. Amsterdam: John Benjamins. M. Jibu, K. H. Pribram and K. Yasue (1996) Int. J. Mod. Phys. B 10, 1735. 40. W. J. Freeman (2007)in Neurodynamics of Cognition and Consciousness, R. Kozma and L. Perlovsky Eds., New York, Springer. 41. O. Bratteli and D. W. Robinson (1979) Operator Algebra and Quantum Statistical Mechanics. Berlin: Springer. 42. J. von Neumann (1955) Mathematical foundations of Quantum Mechanics. Princeton: Princeton University Press. 43. For quantum mechanical brain models see H. P. Stapp, Mind, matter an quantum mechanics, New York: Springer-Verlag 1993/2003; R. Penrose (1994) Shadows of the mind, Oxford: Oxford University Press ; S. Hameroff and R. Penrose (1996) J. Conscious. Stud. 3, 36. 44. E. Celeghini, M. Rasetti and G. Vitiello (1992) Annals Phys 215, 156. The Dissipative Quantum Model of Brain and Laboratory Observations 251

45. Y. Takahashi and H. Umezawa (1975) Collective Phenomena 2, 55; reprinted in Int. J. Mod. Phys. B 10, 1755 (1996). 46. A. Perelomov (1986) Generalized Coherent States and Their Applications. Berlin: Springer. 47. E. Alfinito, O. Romei and G. Vitiello (2002) Mod. Phys. Lett. B 16, 93. 48. E. Alfinito and G. Vitiello (2002) Phys. Rev. B 65, 054105. 49. W. J. Freeman, L. J. Rogers, M. D. Holmes, Silbergeld (2000) J. Neurosci. Meth. 95, 111. 50. H. Haken (1996) Principles of Brain Functioning: A Synergetic Approach to Brain Activity, Behavior, and Cognition, New York: Springer. H. Haken (1999) in: Analysis of Neurophysiological Brain Functioning,. C. Uhl (Ed.) Berlin: Springer-Verlag. pp. 7–40. H. Haken (2004) Synergetics: Introduction and Advanced Topics. New York: Springer. 51. E. Alfinito and G. Vitiello (2000) Int. J. Mod. Phys. B 14 853. [Erratum- ibid. B 14 1613 (2000)]. 52. H. Atmanspacher and H. Scheingraber(1990) Can. J. Phys. 68, 728. 53. A. Pikovsky, M. Rosenblum and J. Kurths (2001) Synchronization - A Uni- versal Concept in Non-linear Sciences. Cambridge UK: Cambridge Univ. Press. 54. E. Pessa and G. Vitiello (2003) Mind and Matter 1 59. E. Pessa and G. Vitiello (2004) Intern. J. Modern Physics B 18, 841. 55. G. Vitiello (2004) Int. J. Mod. Phys. B 18, 785. 56. J. A. S. Kelso (1995) Dynamic Patterns: The Self Organization of Brain and Behavior. Cambridge: MIT Press. 57. J. A. S. Kelso, P. Case, T. Holroyd, E. Horvath, J. Raczaszek, B. Tuller, M. Ding (1995) in Ambiguity in Mind and Nature, P. Kruse and M. Stadler (eds.). Berlin: Springer. 58. M. W. Eysenck (1994) Principles of Cognitive Psychology. Hillsdale, NJ: Erlbaum. 59. I. Tsuda (2001) Behav. Brain Sci. 24, 793. 60. C. A. Skarda and W. J. Freeman (1987) Brain Behav. Sci. 10, 161. 61. S. L. Bressler (2002) Current Directions in Psychological Sci. 11, 58. 62. A.A. Fingelkurts , A.A. Fingelkurts (2004) Int J Neurosci 114, 843. 63. A.A. Fingelkurts , A.A. Fingelkurts (2001) Brain and Mind 2, 261. 64. S.L. Bressler and J.A.S. Kelso (2001) Trends Cog. Sci. 5, 26. 65. L. G. Cohen, Celnik P, Pascal-Leone A, Corwell B, Faiz L, Dambrosia J, Honda M, Sadato N, Gerloff C, Catala MD, Hallett M (1997) Nature 389, 180. 66. L. Von Melchner, Pallas S.L., Sur M. (2000) Nature 404, 871. 67. J. von Neumann (1958) The Computer and the Brain. New Haven: Yale University Press. This page intentionally left blank 253

Supersymmetric Methods in the Traveling Variable: Inside Neurons and at the Brain Scale

a,c b a,d H.C. Rosu ,O.Cornejo-Pérez ,andJ.E.Pérez-Terrazas a Division of Advanced Materials, IPICyT, Apartado Postal 3-74, Tangamanga, 78231 San Luis Potos´ı, S.L.P., México b Centro de Investigación en Matemáticas (CIMAT), Apartado Postal 402, 36000 Guanajuato, Gto., México b [email protected] c [email protected] d [email protected]

We apply the mathematical technique of factorization of differential operators to two different problems. First we review our results related to the supersymmetry of the Montroll kinks moving onto the microtubule walls as well as mentioning the sine-Gordon model for the microtubule nonlinear excitations. Second, we find analytic expressions for a class of one-parameter solutions of a sort of diffusion equation of Bessel type that is obtained by supersymmetry from the homogeneous form of a simple damped wave equation derived in the works of P.A. Robinson and collaborators for the corticothalamic system. We also present a possible interpretation of the diffusion equation in the brain context.

Keywords: Biophysics; Complex Systems; Quantum Information; Soliton excitation; Nonlinear Wave Equation; Brain Dynamics PACS(2006): 87.16.Ac; 87.15.Aa; 87.15.v; 89.75.Fb; 03.67.a; 05.45.Yv; 03.65.Ge

1. Nonlinear Biological Excitations The possibility of soliton excitations in biological structures has been first 1 pointed out by Englander et al. in 1980 who speculated that the so-called ‘open states’ units made of approximately ten adjacent open pairs in long polynucleotide double helices could be thermally induced solitons of the double helix due to a coherence of the twist deformation energy. Since then a substantial amount of literature has been accumulating on the biological significance of DNA nonlinear excitations (for a recent paper, see Ref. 2). On the other hand, the idea of nonlinear excitations has emerged in 1993 3 in the context of the microtubules (MTs), the dimeric tubular polymers 254 H.C. Rosu, O. Cornejo-Pérez, and J.E. Pérez-Terrazas that contribute the main part of the eukaryotic cytoskeleton. In the case of neurons, MTs are critical for the growth and maintenance of axons. It is known that axonal MTs are spatially organized but are not under the influence of a MT-organizing center as in other cells. We also remind that in 1995 Das and Schwarz have used a two-dimensional smectic liquid crystal model to show the possibility of electrical solitary wave propagation in cell 4 membranes. Nevertheless, there is no clear experimental evidence at the moment of any of these biological solitons and kinks.

2. Supersymmetric MT Kinks 5 Based on well-established results of Collins, Blumen, Currie and Ross regarding the dynamics of domain walls in ferrodistortive materials, 3,6 Tuszyński and collaborators considered MTs to be ferrodistortive and 7 studied kinks of the Montroll type as excitations responsible for the energy transfer within this highly interesting biological context. The Euler-Lagrange dimensionless equation of motion of ferrodistortive domain walls as derived in Ref. 5 from a Ginzburg–Landau free energy with driven field and dissipation included is of the travelling reaction-diffusion type 3 ψ + ρψ − ψ + ψ + σ =0, (1) where the primes are derivatives with respect to a travelling coordinate 5 ξ = x − vt, ρ is a friction coefficient and σ is related to the driven field. There may be ferrodistortive domain walls that can be identified with the Montroll kink solution of Eq. (1) √ 2β M(ξ)=α1 + , (2) 1+exp(βξ) √ where β =(α2 − α1)/ 2 and the parameters α1 and α2 are two nonequal solutions of the cubic equation 3 (ψ − α1)(ψ − α2)(ψ − α3)=ψ − ψ − σ. (3) Rosu has noted that Montroll’s kink can be written as a typical tanh 8 kink βξ M(ξ)=γ − tanh , (4) 2 √ α1 2 where γ ≡ α1 + α2 =1+ β . The latter relationship allows one to use a simple construction method of exactly soluble double-well potentials in the Supersymmetric Methods in the Traveling Variable 255

9 Schr¨odinger equation proposed by Caticha. The scheme is a non-standard 10 application of Witten’s supersymmetric quantum mechanics having as the essential assumption the idea of considering the M kink as the switching function between the two lowest eigenstates of the Schr¨odinger equation with a double-well potential. Thus

φ1 = Mφ0 , (5) where φ0,1 are solutions of φ0,1 +[ 0,1 − u(ξ)]φ0,1(ξ) = 0, and u(ξ)isthe double-well potential to be found.

u(ξ )

R 0 ξ M

L ξ M

|0 |1

Fig. 1. Single electron within the traveling double-well potential u(ξ) as a qubit. The electron can switch from one well to the other by tunneling and the relation between the wavefunctions in the two wells is given by Eq. (5).

Substituting Eq. (5) into the Schrödinger equation for the subscript 1 and substracting the same equation multiplied by the switching function for the subscript 0, one obtains φ0 + RM φ0 =0, (6) where RM is given by M + M RM = , (7) 2M and = 1 − 0 is the lowest energy splitting in the double-well Schrödinger equation. In addition, notice that Eq. (6) is the basic equation introducing the superpotential R in Witten’s supersymmetric quantum mechanics, i.e. 256 H.C. Rosu, O. Cornejo-Pérez, and J.E. Pérez-Terrazas the Riccati solution. For Montroll’s kink the corresponding Riccati solution reads β β 2 β RM (ξ)=− tanh ξ + sinh(βξ)+2γ cosh ξ (8) 2 2 2β 2 and the ground-state Schrödinger function is found by means of Eq. (6) β − φ0,M (ξ)=φ0(0) cosh ξ exp 2 exp 2 cosh(βξ) 2 2β 2β −γβξ − γ sinh(βξ) , (9) while φ1 is obtained by switching the ground-state wave function by means of M. This ground-state wave function is of supersymmetric type ξ φ0,M (ξ)=φ0,M (0) exp − RM (y)dy , (10) 0 where φ0,M (0) is a normalization constant. The Montroll double well potential is determined up to the additive constant 0 by the ‘bosonic’ Riccati equation 2 2 2 2 β (γ − 1) uM (ξ)=RM − RM + 0 = + + + 0 4 4β2 2 2 2 2 + 4γ +2(γ +1) cosh(βξ) − 8β cosh(βξ) 8β2 2 −4γ + cosh(βξ) − 2β sinh(βξ) . (11)

If, as suggested by Caticha, one chooses the ground state energy to be 2 2 β 2 0 = − − + 1 − γ , (12) 4 2 4β2 then uM (ξ) turns into a travelling, asymmetric Morse double-well potential of depths depending on the Montroll parameters β and γ and the splitting L,R 2 2 γ U0,m = β 1 ± , (13) (2β)2 where the subscript m stands for Morse and the superscripts L and R for left and right well, respectively. Supersymmetric Methods in the Traveling Variable 257

L R The diﬀerence in depth, the bias, is ∆m ≡ U0 − U0 =2 γ, while the location of the potential minima on the traveling axis is at 2 L,R 1 (2β) ± 2 γ ξm = ∓ ln , (14) β (γ ∓ 1) that shows that γ = ±1.

An extension of the previous results is possible if one notices that RM in Eq. (8) is only the particular solution of Eq. (11). The general solution is a one-parameter function of the form d RM (ξ; λ)=RM (ξ)+ ln(IM (ξ)+λ) (15) dξ and the corresponding one-parameter Montroll potential is given by 2 d uM (ξ; λ)=uM (ξ) − 2 ln(IM (ξ)+λ) . (16) dξ2

ξ 2 In these formulas, IM (ξ)= φ0,M (ξ)dξ and λ is an integration constant that is used as a deforming parameter of the potential and is related to the irregular zero mode. The one-parameter Darboux-deformed ground state wave function can be shown to be

φ0,M φ0,M (ξ; λ)= λ(λ +1) , (17) IM (ξ)+λ where λ(λ + 1) is the normalization factor implying that λ/∈ [0, −1]. Moreover, the one-parameter potentials and wave functions display singularities at λs = −IM (ξs). For large values of ±λ the singularity moves towards ∓∞ and the potential and ground state wave function recover the shapes of the non-parametric potential and wave function. The one- parameter Morse case corresponds formally to the change of subscript M → m in Eqs. (15) and (16). For the single well Morse potential the 12 13 one-parameter procedure has been studied by Filho and Bentaiba et al. The one-parameter extension leads to singularities in the double-well potential and the corresponding wave functions. If the parameter λ is positive the singularity is to be found on the negative ξ axis, while for negative λ it is on the positive side. Potentials and wave functions with singularities are not 14 so strange as it seems and could be quite relevant even in nanotechnology where quantum singular interactions of the contact type are appropriate for describing nanoscale quantum devices. We interpret the singularity as representing the effect of an impurity moving along the MT in one direction or the other depending on the sign of the parameter λ. The impurity 258 H.C. Rosu, O. Cornejo-Pérez, and J.E. Pérez-Terrazas may represent a protein attached to the MT or a structural discontinuity in the arrangement of the tubulin molecules. This interpretation of impurities has been given by TrpiˇsováandTuszyński in non-supersymmetric models 15 of nonlinear MT excitations. Another biophysical application of this one parameter extension will be discussed in Section 5.

3. The Sine-Gordon MT solitons Almost simultaneously with Sataric, Tuszynski and Zakula, there was an- 16 other group, Chou, Zhang and Maggiora, who published a paper on the possibility of kinklike excitations of sine-Gordon type in MTs but in a biological journal. Even more, they assumed that the kink is excited by the energy released in the hydrolysis of GTP → GDP in microtubular solutions. As the kink moves forward, the individual tubulin molecules involved in the kink undergo motion that can be likened to the dislocation of atoms within the crystal lattice. They performed an energy estimation showing that a kink in the system possesses about 0.36–0.44 eV, which is quite close to the 0.49 eV of energy released from the hydrolysis of GTP. They assumed that the interaction energy U(r) between two neighboring tubulin molecules along a protoﬁlament is harmonic:

1 2 U(r) ≈ k(r − a0) , (18) 2

2 d U(a0) where k = dr2 and r = xi − xi−1. In addition to this kind of nearest neighbor interaction, a tubulin molecule is also subjected to interactions with the remaining tubulin molecules of the MT, i.e. those in the same protofilament but not nearest neighbor to it. Chou et al. cite pages 425–427 in the book of R.K. Dodd et al. (Solitons and Nonlinear Wave Equations, Academic Press 1982) for the claiming that this interaction for the ith tubulin molecule of a protofilament can be approximated by the following periodic effective potential 2πξi Ui = U0 1 − cos , (19) a0 where U0 is the half-height of the potential energy barrier and ξi is the displacement of the ith tubulin molecule from the equilibrium position within a particular protofilament. Supersymmetric Methods in the Traveling Variable 259

2π i i Introducing the new variable φ = a0 ξ the following sine-Gordon equation is obtained 2 2 2 ∂ φ 2 ∂ φ − 2π m 2 = ka0 2 U0 sin φ (20) ∂t ∂x a0 that can be reduced to the standard form of the sine-Gordon equation 2 2 ∂ φ 1 ∂ φ 1 − = sin φ (21) ∂x2 c2 ∂t2 l2 2 2 2 ka0 −2 4π U0 if one sets c = m and l = 4 .Now,itiswellknownthatthe ka0 sine-Gordon equation has the famous inverse tangent kink solution −1 γ φ =tan exp[± (x − vt)] , (22) l 1 γ where γ = 2 is an acoustic Lorentz factor and w = l is the kink 1− v c2 width. Most interestingly, the momentum of a tubulin dimer is strongly localized: d(mξ) ma0 γv γ p = = sech − (x − vt) . (23) dt π l l This momentum function possesses a very high and narrow peak at the center of the kink width implying that the corresponding tubulin molecule will have maximum momentum when it is at the top of the periodic potential. According to Chou et al. this remarkable feature occurs only in nonlinear wave mechanics. Interestingly, for purposes of illustration, these authors have assumed the width of a kink w ≈ 3a0. Therefore, with the kink moving forward, the affected region always involves three tubulin molecules. For a general case, however, the width w of a kink can be calculated from 2 a0 ka0 w = , (24) 2π U0 if the force constant k between two neighboring tubulin molecules along a protofilament, the distance a0 of their centers, and the energy barrier 2U0 of the periodic, effective potential are known. Then the number of tubulin molecules involved in a kink is given by w −1 2 =(2π) ka0/U0 . (25) a0 It is further known that the tubulin molecules in a MT are held by noncovalent bonds, therefore the interaction among them might involve 260 H.C. Rosu, O. Cornejo-Pérez, and J.E. Pérez-Terrazas hydrogen bonds, van der Waals contact, salt bridges, and hydrophobic interactions. 17 It was found by Israelachvili and Pashley that the hydrophobic force o law over the distance range 0-10 nm at 21 C is well described by

FH −D/D = Ce 0 N/m (26) R where D is the distance between tubulin molecules, D0 is a decay length, R1R2 and R = R1+R2 is a harmonic mean radius for two hydrophobic solute molecules, all in nm. R is 4 nm in the case of tubulin.

3.1. More on the Hydrolysis and Solitary Waves in MTs Inside the cell, the MTs exist in an unstable dynamic state characterized by a continuous addition and dissociation of the molecules of tubulin. The polypeptides α and β tubulin each bind one molecule of guanine nucleotide with high aﬃnity. The nucleotide binding site on α tubulin binds GTP nonexchangeably and is referred to as the N site. The binding site on β tubulin exchanges rapidly with free nucleotide in the tubulin heterodimer and is referred to as the E site. Thus, the addition of each tubulin is accompanied by the hydrolysis of GTP 5’ bound to the β monomer. In this reaction an amount of energy −21 of 6.25 × 10 J is freed that can travel along MTs as a kinklike solitary wave. The exchangeable GTP hydrolyses very soon after the tubulin binds to the MT. At pH = 7 this reaction takes place according to the formula: 4− 3− 2− + GT P + H2O → GDP + HPO4 + H +∆H E. (27)

The last mathematical formulation of the manner in which the energy ∆H E is turned into a kink excitation claims that the hydrolysis causes a 18 dynamical transition in the structure of tubulin .

4. Quantum Information in the MT Walls Biological information processing, storage, and transduction occurring by computer-like transfer and resonance among the dimer units of MTs have 19 been ﬁrst suggested by Hamerrof and Watt and enjoys much speculative 20 activity. For the case of sine-Gordon solitons, the information transport has been 21 investigated by Abdalla et al. . Supersymmetric Methods in the Traveling Variable 261

22 Recently Shi and collaborators worked out a processing scheme of quantum information along the MT walls by using previous hints of Lloyd 23 for two-level pseudospin systems. The MT wall is treated as a chain of three types of two pseudospin-state dimers. A set of appropriate resonant frequencies has been given. They conclude that specific frequencies of laser pulse excitations can be applied in order to generate quantum information processing. Lloyd’s scheme uses the driving of a quantum computer by means of a sequence of laser pulses. He assumes a 1-dimensional arrangement of atoms of two types (A and B) that could be each of them in one of two states and are affected only by nearest neighbors. Then, information processing could be performed by laser pulses of specific frequencies ωKα,β , that change the state of the atom of the K kind (A or B type) if in a pair of atoms AB, A is in α state and B is in state β.

5. Supersymmetry at the Brain Scale Neuronal activity is the result of the propagation of impulses generated at the neuron cell body and transmitted along axons to other neurons. Re- 25 cently, Robinson and collaborators obtained simple damped wave equations for the axonal pulse ﬁelds propagating at speed va between two populations, a and b, of neurons in the thalamocortical region of the brain. The explicit form of their equation is

OˆRφa(t)=S[Va(t)] , (28) where 2 1 d 2 d − 2∇2 OˆR = 2 2 + +1 ra (29) νa dt νa dt where νa = va/ra, ra is the mean range of axons a,andVa = b Vab is a so-called cell body potential which results from the ﬁltered dendritic tree inputs. Robinson has used the experimental parameters in this equation for the processing of the experimental data. In the following we concentrate on a particular mathematical aspect of this equation and refer the reader to the works of Robinson’s group for more details concerning this equation.

5.1. The Homogeneous Equation We treat first the homogeneous case, i.e. S =0andwediscardthe subindexes as being related to the phenomenology not to the mathematics. 262 H.C. Rosu, O. Cornejo-Pérez, and J.E. Pérez-Terrazas

Let us employ the change of variable z = ax + by − ct (see, Ref. 26), which is a traveling coordinate in 2+1 dimensions. This is justified because it was 24 noticed by Wilson and Cowan that distinct anatomical regions of cerebral cortex and of thalamic nuclei are functionally two-dimensional although extending to three spatial coordinates is trivial. We have the following rescal- 2 2 2 ings of functions: φt = −cφz, φtt = v φzz, φxx = a φzz, φyy = b φzz. Then, we get the ordinary differential equation corresponding to the damped wave equation in the following form 2 d d 2 2 OˆR,zφ ≡ − 2µ + µ φ = α φ, (30) dz2 dz where 4 2 2 2 νc 2 ν r (a + b ) µ = ,α= . (31) c2 − ν2r2(a2 + b2) [c2 − ν2r2(a2 + b2)]2 The simple damped oscillator equation (30) can be factorized 2 d d 2 Lµφ ≡ − µ − µ φ = α φ, (32) dz dz 2 2 2 2 2 The case c <ν r (a + b ) implies µ<0 and the general solution of (30) can be written µz αz −αz φ(z)=e (Ae + Be ) . (33) 2 2 2 2 2 Theoppositecasec >ν r (a + b ) will lead to only a change of sign in 2 2 2 2 2 front of µ in all formulas henceforth, whereas the case c = ν r (a + b ) will be considered as nonphysical. The non-uniqueness of the factorization of 27 second-order differential operators has been exploited in a previous paper on the example of the Newton classical damped oscillator, i.e. 2 d d 2 Nyˆ ≡ +2β + ω0 y =0, (34) dt2 dt which is similar to the equation (30), unless the coefficient 2β is the friction constant per unit mass, ω0 is the natural frequency of the oscillator, and the independent variable is just time not the traveling variable. Proceeding along the lines of Ref. 27, one can search for the most general isospectral factorization 2 (Dz + f(z))(Dz + g(z))φ = α φ. (35)

After simple algebraic manipulations one finds the conditions f + g = −2µ 2 λ and dg/dz + fg = µ having as general solution fλ = λz+1 − µ,whereas Supersymmetric Methods in the Traveling Variable 263 f0 = −µ is only a particular solution. Using the general solution fλ we get λ λ 2 Aˆ+λAˆ−λφ ≡ Dz + − µ Dz − − µ φ = α φ. (36) λz +1 λz +1 This equation does not provide anything new since it is just equation (31). However, a different operator, which is a supersymmetric partner of (36) is obtained by applying the factorizing λ-dependent operators in reversed order λ λ 2 Aˆ−λAˆ+λφ˜ ≡ Dz − − µ Dz + − µ φ˜ = α φ.˜ (37) λz +1 λz +1 The latter equation can be written as follows 2 2 ˆ d d 2 2 λ O˜λφ˜ ≡ − 2µ + µ − α − φ˜ =0, (38) dz2 dz (λz +1)2 or 2 d d 2 − 2µ + ω (z) φ˜ =0, (39) dz2 dz where 2 2 2 2 λ ω (z)=µ − α − (40) (λz +1)2 is a sort of parametric angular frequency with respect to the traveling coordinate. This new second-order linear damping equation contains the additional last term with respect to its initial partner, which may be thought of as 28 the Darboux transform part of the frequency. Zλ =1/λ occurs as a new traveling scale in the damped wave problem and acts as a modulation scale. If this traveling scale is infinite, the ordinary damped wave problem is recovered. The φ˜ modes can be obtained from the φ modes by operatorial 27 means . Eliminating the first derivative term in the parametric damped oscillator equation (39) one can get the following Bessel equation 2 2 − 1 d u n 4 2 − + β u =0, (41) dx2 x2 2 where x = z +1/λ, n =5/4, and β = iα. Using the latter equation, the general solution of equation (39) can be written in terms of the modified Bessel functions 1/2 √ √ µz φ˜ =(z +1/λ) [C1I 5/2(α(z +1/λ)) + C2I− 5/2(α(z +1/λ))]e . (42) 264 H.C. Rosu, O. Cornejo-Pérez, and J.E. Pérez-Terrazas

What could be a right interpretation of the supersymmetric partner equation (37) ? Since the solutions are modified Bessel functions, we consider this equation as a diffusion equation with a diffusion coefficient depending on the traveling coordinate. Noticing that the velocity in the traveling variable of this diffusion is the same as the velocity of the neuronal pulses we identify it with the diffusion of various molecules, mostly hormones, in the extracellular space (ECS) of the brain, which is known to be necessary for chemical signaling and for neurons and glia to access nutrients and 29 therapeutics occupying as much as 20 % of total brain volume in vivo .

5.2. The Nonhomogeneous Equation The source term S in Robinson’s equation (28) is a sigmoidal firing function, which despite corresponding to a realistic case led him to work out extensive numerical analyses. Analytic results have been obtained recently by 30 Troy and Shusterman by using a source term comprising a combination of discontinuous exponential coupling rate functions and Heaviside firing 31 rate functions. In addition, Brackley and Turner incorporated fluctuating 31 firing thresholds about a mean value as a source of noisy behavior . The procedure of Troy and Shusterman can be applied for the parametric damped oscillator equation as well as for the Bessel diffusion equation obtained herein in the realm of Robinson’s brain wave equation with the difference that the method of variation of parameters should be employed. The detailed mathematical analysis is left for a future work.

Acknowledgment This work was partially supported by the CONACyT Project 46980.

References 1. S.W. Englander, N.R. Kallenbach, A.J. Heeger, J.A. Krumhansl, and S. Litwin (1980) Proc. Natl. Acad. Sci. U.S.A., 77, 7222. 2. J.D. Bashford (2006) J. Biol. Phys. 32,27. 3. M.V. Satarić, J. A. Tuszyński and R.B. Zakula´ (1993) Phys. Rev. E 48, 589. 4. P. Das and W.H. Schwarz (1995) Phys. Rev. E 51, 3588 . 5. M.A. Collins, A. Blumen, J.F. Currie, and J. Ross (1979) Phys. Rev. B 19, 3630. 6. J.A. Tuszyński, S. Hameroff, M.V. Satarić, B. Trpiˇsová, and M.L.A. Nip (1995) J. Theor. Biol. 174, 371 . 7. E.W. Montroll, in Statistical Mechanics, ed. by S.A. Rice, K.F. Freed, and J.C. Light(1972) (Univ. of Chicago, Chicago). Supersymmetric Methods in the Traveling Variable 265

8. H.C. Rosu (1997) Phys. Rev. E 55, 2038. 9. A. Caticha (1995) Phys. Rev. A 51, 4264. 10. E. Witten (1981) Nucl. Phys. B 185, 513. 11. B. Mielnik (1984)J. Math. Phys. 25, 3387; D.J. Fernandez (1984) Lett. Math. Phys. 8, 337; M.M. Nieto (1984) Phys. Lett. B 145, 208 . For review see H.C. Rosu, Symmetries in Quantum Mechanics and Quantum Optics, eds. A. Ballesteros et al. (1999) (Serv. de Publ. Univ. Burgos, Burgos, Spain) pp. 301-315, quant-ph/9809056. 12. E.D. Filho, (1988) J. Phys. A 21, L1025. 13. M. Bentaiba, L. Chetouni, T.F. Hammann (1994) Phys. Lett. A 189, 433. 14. T. Cheon and T. Shigehara (1998) Phys. Lett. A 243, 111. 15. B. TrpiˇsováandJ.A.Tuszyński (1992) Phys. Rev. E 55, 3288 . 16. K-C Chou, C.-T. Zhang, and G.M. Maggiora (1994) Biopolymers 34, 143. 17. J. Israelachvili and R. Pashley (1982) Nature 300, 341. 18. M.V. Sataric and J.A. Tuszynski (2005) J. Biol. Phys. 31, 487 . 19. S. Hameroff and R.C. Watt, (1982), J. Theor. Biol. 98, 549. 20. See e.g., J. Faber, R. Portugal, L. Pinguelli Rosa, Information processing in brain MTs, contribution at the Quantum Mind 2003 Conference, q-bio.SC/0404007. 21. E. Abdalla, B. Maroufi, B. Cuadros Melgar, and M. Brahim Sedra (2001) Physica A 301, 169. 22. C. Shi, X. Qiu, T. Wu, R. Li (2006) J. Biol. Phys. 32, 413. 23. S. Lloyd (1993) Science 261, 1569. 24. H.R. Wilson and J.D. Cowan (1973) Biol. Cybernetics 13, 55. 25. P.A. Robinson, C.J. Rennie, D.L. Rowe, S.C. Connor (2004) Human Brain Mapping 23, 53; H. Wu and P.A. Robinson (2007) J. Theor. Biol. 244,1; P.A. Robinson, Phys. Rev. E, (2005), 72, 011904 . 26. P.G. Estévez, S¸. Kuru, J. Negro, L.M. Nieto (2006) J. Phys. A 39, 11441. 27. H.C. Rosu and M. Reyes(1998) Phys. Rev. E, 57, 2850. 28. G. Darboux (1882) C.R. Acad. Sci. 94, 1456. 29. R.G. Thorne and C. Nicholson (2006) Proc. Natl. Acad. Sci. U.S.A. 103, 5567 and references therein. 30. W.C. Troy and V. Shusterman (2007) SIAM J. Applied Dynamical Systems 6, 263. 31. C.A. Brackley and M.S.Turner (2007) Phys. Rev. E 75, 041913 . This page intentionally left blank 267

Turing Systems: A General Model for Complex Patterns in Nature

R.A. Barrio Instituto de F´ısica, Universidad Nacional Autónoma de México (UNAM), Apartado Postal 20-364 01000 México, D.F., México barrio@fisica.unam.mx

More than half a century ago Alan Turing showed that a system of nonlinear reaction-diffusion equations could produce spatial patterns that are stationary and robust, a phenomenon known as ”diffusion driven instability”. This remarkable fact was largely ignored for twenty years. However, in the last decade, Turing systems have been a matter of intense and active research, because they are suitable to model a wide variety of phenomena found in Nature, ranging from Turing’s original idea of describing morphogenesis from an egg, and applications to the colouring of skins of animals, to the physics of chemical reactors and catalyzers, the physiology of the heart, semiconductor devices, and even to geological formations. In this paper I review the main properties of the Turing instability using a generic reaction-diffusion model, and I give examples of recent applications of Turing models to different problems of pattern formation.

Keywords: Biophysics; Complex Systems; Nonlinear Systems; Turing Systems PACS(2006): 87.64.Bx; 87.64.t; 87.16.Ac; 89.75.k; 89.75.Fb; 82.39.Rt; 43.60.Wy

1. Introduction Nature shows many systems composed of radically diﬀerent parts. Usually these parts are self-organized in space in a remarkably precise way. Exam- ples are to be found in systems of any size: atoms and molecules in solids, nanostructured materials, living organisms, geological formations and as- tronomical systems. This organization varies in complexity, for example in living organisms it is extremely complicated at all scales and the precise spatial location of all the parts is crucial for the functioning of the whole system, that is, for life itself. In other cases the arrangement of parts is a consequence of very simple physical interactions that could be described using fundamental principles of Physics, as in simple crystals. 268 R. A. Barrio

In all cases this organization should ultimately be a consequence of the physical interactions of the components, independently of the size and complexity of the system. However, it is often extremely difficult, or even impossible, to model complicated and complex systems using only microscopic physical interactions. Coherent macroscopic behaviour is frequently not derivable from many-body interactions, and one needs to resort to phenomenological models that could capture the specific characteristics of the system at the scale one is interested in. To be specific, one needs a simple mechanism able to provide the spatial information needed to accommodate the parts of the system. For this we recall a universal principle in Physics that relates the dynamical behaviour of the system to the symmetries pos- sessed by the fields acting on it, that is, there must be a symmetry-breaking mechanism that results in a stationary spatial pattern. It is fair to say that the first one who realized that patterning in biological systems could be understood as self-organization was the naturalist 1 D’ Arcy Thompsom . The connections that he made between Biology and other fields in Science in his celebrated book of 1917 inspired a continuous stream of theoretical work in complex systems. Nonlinear systems of differential equations present universal properties that make them particularly suitable to build models of complex systems. Spontaneous symmetry-breaking is one of them, and it appears when the parameters, defining the interactions with the external world, reach certain critical values. The low-symmetry states are in general extremely robust under random perturbations, as thermal noise and variations of the initial conditions. 2 Alan Turing predicted that a system of chemical substances diffus- ing and reacting with each other in a medium, can produce concentration patterns that are stationary and robust. He attempted to describe very complicated systems and processes, as one can infer from reading the abstract of his 1952 paper: “The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomic structure of the resultant organism. The theory does not make any new hypothesis, but merely suggests that certain well known physical laws suffice 2 for explaining many facts”. Ambitious as it sounds, this phrase abridges all the basic concepts we need. He coined the word morphogenesis to give the idea of the processes that lead a system to acquire a precise form, and in consequence, the responsible substances are called morphogens. The well known physical laws are the kinetics of chemical reactions and the normal diffusion of diluted chemicals, both deriving from the statistical Turing Systems: A General Model for Complex Patterns in Nature 269 behaviour of a system with many components. The pattern results when a stationary, highly symmetrical state in the absence of diffusion becomes unstable when particles are allowed to diffuse. This symmetry-breaking mechanism, known as “diffusion driven instability” or simply Turing instability, is surprising. Usually, when a substance spreads out and diffuses one expects its concentration gradients to diminish, smearing out any structure. However, in the Turing instability, diffusion is precisely the reason for loosing a stable uniform state and for the appearance of a structured pattern. Turing models belong to a wide class of systems of differential equations known as reaction-diffusion equations. These systems could present a Turing instability, provided they fulfill a number of restrictive conditions. Reaction-diffusion systems are extremely rich and varied in behaviour, usually made ex professo for the particular phenomenon of interest. However, they all present universal characteristics that can be easily illustrated with 3 a generic model we proposed some time ago . In this paper I will briefly discuss the properties of nonlinear systems of differential equations, then I shall focus on a generic Turing model and mention some other models widely used in research at present. At the end I shall illustrate pattern modeling in systems belonging to very different fields in Science.

2. Nonlinear Systems I shall start by recalling some basic facts from the theory of nonlinear systems, needed to understand the following sections. In general the dynamical behaviour of a physical system could be represented as ∂V = Fr(V,t), (1) ∂t where V is a vector representing the fields, r are parameters describing the interaction of the fields with the external world, and F is a vector of nonlinear functions determining the dynamics and interactions between fields. Eq.1 represents an infinite set of partial differential equations if the phase space is infinite and continuous. However, one usually utilizes appropriate techniques and approximations to reduce the problem to a set of ordinary differential equations. Suppose that the system has a singular point at V0, that is, the fields do not change in time there. It is reasonable to think that if the system is driven away from this point by an infinitesimal amount, its response should be linear, independently of the form of the functions F.If 270 R. A. Barrio the perturbation is V0 + U,where is very small, one gets dU = LˆrU, (2) dt where Lˆr is a linear operator. The solution of Eq.2 is U =exp(snt)Xn, implying the eigenvalue equation

LˆrXn = snXn, (3) where n labels the normal modes X. The complex eigenvalues sn = (sn)+ i(sn) for each normal mode disclose the linear response of the system. There are various cases (1) If (sn) > 0, then Xn is unstable. (2) If (sn) ≤ 0, then Xn is stable. (3) If (sn) =0,and (sn) > 0, there are growing oscillations in time. (4) If (sn) = 0, and (sn) > 0, there might be growing spatial ﬂuctuations. If the eigenmodes form a complete set, any function can be written as a linear combination of normal modes, in particular V = n AnXn,and the time evolution of the ﬁelds is given by the behaviour of the amplitudes of each mode (An). Therefore, Eq.1 reads dAn Xn = snAnXn + N snAn . (4) n dt n n where N is a function that takes care of the nonlinear terms. Then, if the normal modes form an orthogonal basis, dAn = snAn + Xn|N( smAm). (5) dt Near a bifurcation point most of the eigenvalues sn have negative real part and die exponentially. Some modes Xc, called marginal, correspond to eigenvalues with a very small positive real part, and they govern the dynamical behaviour of the system at long times. The Center Manifold 4 Theorem states that if (sm) < 0, then

dAm 0, dt and this implies that Am = f(Ac; c =1, 2, ...) (which is the center manifold equation). Then, the important modes with sc 0, obey

dAc = scAc + N (Ac,Am) , (6) dt Turing Systems: A General Model for Complex Patterns in Nature 271

These are known as the amplitude equations, which is a set of ordinary diﬀerential equations that can be solved numerically, or analyzed linearly to investigate the eﬀects of the nonlinearities in the time evolution of the system.

3. The Turing Instability A set of reaction-diﬀusion equations contains a diﬀusion term in the entries of the functions Fr in Eq.1, that is

∂ui 2 = Di∇ ui + Fi({uj}, (7) ∂t where ui are the concentration fields of morphogens, Di are the respective diffusion constants, and the nonlinear functions Fi give the kinetics of the 2 reactions. Alan M. Turing showed that a uniform stable state in the absence of diffusion could become unstable when diffusion is acting, provided the diffusion constants are not the same. Despite the fact that a real Turing instability has been reproduced in 5 the laboratory in a chemical system , the existence of Turing patterns in Nature is still a controversial issue (see Ref. 6 for a review). There have been mainly two serious criticisms: (1) The identification of morphogens in a complex system is a very difficult task, and only few have been identified, but their function is not corroborated yet. (2) The patterns obtained with simple Turing systems are very simplistic, and hardly bear any resemblance with the patterns observed in Nature. The first objection has nothing to do with the theory, and it is a matter of technical experimental difficulties. I am sure that in the near future these shall be overcome and an explosion of identified morphogens appears. In fact, in some complex biological processes, as the chemical signaling for 7 axon growth, morphogens have been found . The second objection is more serious, since it would imply that the theory is far too simple to be applied to real systems, even Turing himself, who invented the computer, could not see the patterns that his theory would produce. Now, computer simulations make it possible to see patterns from fairly complicated models. In fact, the main purpose of this paper is to show that Turing models could render a whole zoo of complicated patterns. In recent years there has been an exponential increase of the number of papers published by a large number of researchers and a widening of the range of systems that could be modeled in this way. A magnificent review on various 8 modeling techniques in biological systems has been published recently . 272 R. A. Barrio

3.1. Generic Turing Model (BVAM) It is convenient to be more specific now and discuss the properties of Turing systems. For this I shall use a generic activator-inhibitor model that we 3 proposed in the past . I shall refer to it as the BVAM model from now on. It consists of a set of two reaction-diffusion equations for two chemicals with concentrations U,andV . Contrary to the usual procedure, we were not aiming to model a specific chemical reaction, but we wanted to find a general form for the kinetics by assuming that there is a fix point at (Uc,Vc) and Taylor-expanding around this fix point, keeping terms up to third order. The specific form I shall consider is:

∂u 2 2 = Dδ∇ u + αu(1 − r1v )+v(1 − r2u)(8) ∂t ∂v 2 αr1 = δ∇ v + βv(1 + uv)+u(γ + r2v) ∂t β where δ is a scaling factor, D is the ratio between diffusion coefficients , u = U −Uc and v = V −Vc, so there is a uniform stationary solution of Eq.(8) at the point (u, v)=(0, 0). The special arrangement of the coefficients follows from conservation relations between chemicals. There are two interaction parameters, r1 and r2, corresponding to the cubic and a quadratic terms, respectively. In zero dimensions, or in the absence of diffusion, Eq. (8) shows stationary uniform solutions at

−(α + γ) v = u. 1+β

For clarity I shall assume that α = −γ, so all the singular points collapse 11 at the origin. The analysis in the general case has been made elsewhere. In these circumstances the eigenvalues of the linear operator in Eq. (2) are 2 s =(α + β)/2 ± [(α + β)/2] − α(β +1).

The real part of s could be negative in certain regions of the parameter space (α, β) and the uniform solution is stable. When diﬀusion is added, the linear operator has eigenfunctions of the form Xk =expk · r. Therefore, there is a dispersion relation

2 s(k) − Bs(k)+C =0 (9)

2 2 2 where B = δk (1 + D) − (α + β), and C =(α − δDk )(β − δk )+α. Turing Systems: A General Model for Complex Patterns in Nature 273

There are unstable states with (s(k)) > 0and(s(k)) = 0 for certain ﬁnite values of the wave number k, if the following conditions are met    α + β<0  α(β +1)> 0 , (10)   Dβ + α>0  2 (Dβ + α) − 4Dα(β +1)> 0

This is called the Turing region, where perfectly stationary patterns appear. The maximum of the real part of the dispersion relation is situated at √ D(α − β) − (D +1) αD kc = . δD(D − 1)

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

Fig. 1. (a-l) Patterns obtained from the BVAM model with varying parameters and boundary conditions (bc): Top row shows zero ﬂux bc, in the second row there are periodic bc and in third row free bc. Columns are from left to right: First and second with parameters such that kc =0.45. The third and fourth columns with kc =0.89. In the ﬁrst and third columns r1 =3.5, r2 = 0 (favours stripes), and in the second and fourth columns r1 =0.02, r2 =0.2 (favours spots). In all twelve patterns the random initial conditions and the scale (δ = 10) were the same. 274 R. A. Barrio

In Fig. 1 there are examples of numerical calculations showing basic patterns obtained by choosing two different sets of parameters: (1) D = 0.516, α = −γ =0.899, and β = −0.91 giving kc =0.42, for the two left columns; (2) D =0.122, α =0.398, and β = −0.4, giving kc =0.84, for the two right columns. Observe that the patterns on the left half of the figure present structures with a larger wavelength (λ =2π/kc) than those on the right half, where the wavelength is smaller. With this simple model and zero flux, or periodic boundary conditions in a plane, the patterns are fairly simple. The matter of pattern selection has to do with the nonlinearities, we have established that the cubic interaction 3 r1 favors stripes and that the quadratic one r2 produces spot patterns . In three dimensions there are more possibilities, one could have ordered spheres, disordered spheres, lamellae, layered patterns or labyrinthine patterns of tunnels. In Fig. 2 we show an example taken from Ref. 10.

Fig. 2. Examples of numerical calculations on the BVAM model producing complicated patterns in three dimensions.

There are other possible bifurcations, for instance if (s) > 0and (s) =0at k = 0, there is a Hopf bifurcation. Also, there could be two regions, one at k = 0, and another at k = 0 presenting unstable states, and this is known as a Turing-Hopf bifurcation, in which the spatial patterns 9 oscillate in time. The BVAM model also admits traveling wave solutions if the condition α = −β is relaxed. In a recent paper we have shown that 11 the system presents bistability and solitons appear. The parameter space also shows regions where limit cycle solutions produce rotating spirals or targets. In Fig. 3 there are some examples of patterns that look more complicated in two dimensions. The ﬁrst two are labyrinthine stationary patterns, Turing Systems: A General Model for Complex Patterns in Nature 275

(A) (B) (C) (D)

Fig. 3. Patterns obtained by relaxing the α = −γ condition. (A) and (B) are stationary, (C) shows traveling stripes and (D) is an oscillating pattern. obtained by exploring other regions of the Turing space, the third shows a Turing pattern under anomalous diﬀusion conditions, and the last presents rotating spirals similar to the ones seen in the Belousov–Zhabotinskii reaction. It is clear that even one of the simplest Turing systems is rich enough to be applied to the most varied situations. The same could be said of other models.

3.2. Other Turing Models In this section I discuss a selected group of Turing models that have been widely used in the past. Some of these models can be considered as special cases of the BVAM model presented before. In Table 1 the kinetics of some famous models is written. By famous I mean that they are known by a speciﬁc name and that they have been used successfully to either reproduce or predict the behaviour of real systems. The notation used in the table corresponds to the convention of writing them as follows: ∂u 2 = Du∇ u + F (u, v, w, ...) (11) ∂t ∂v 2 = Dv∇ v + G(u, v, w, ...), ∂t ∂w 2 = Dw∇ w + H(u, v, w, ...), ∂t ...... etc.

As a reference, the ﬁrst row of Table 1 is the BVAM model already discussed, but conveniently written in adimensional form. Then, the simplest models have only one ﬁeld, and the simplest non-trivial kinetics would be 276 R. A. Barrio

Model Equations Ref. 2 BVAM F (u, v)=η(u + av − Cuv − uv ) 2 3 G(u, v)=η(bv + hu + Cuv + uv ) 12 Fisher equation F (u)=µu(1 − u/K) Lotka Volterra F (u, v)=au − buv 14 G(u, v)=−cu + duv 2 Gierer Meinhardt F (u, v)=ρu − µuu + ρu /v 215 G(u, v)=ρv − µvv + ρu 2 Gray-Scott F (u, v)=f(1 − u) − uv 217 G(u, v)=−(f + κ)v + uv 2 Brusselator F (u, v)=a − (b +1)u + u v 2 21 G(u, v)=bu − u v 1 Oregonator F (u, v, w)= [qv − uv + u(1 − u)] 1 G(u, v, w)= [−qv − uv + fw] − 22 H(u, v, w)=u v 2 Schnakenberg F (u, v)=γ a − u + u v 2 24 G(u, v)=γ(b − u v) 2 Lengyel-Epstein or F (u, v)=a − u − 4uv/(1 + u) 2 26 Brandeisator G(u, v)=δ u − uv/(1 + u ) 3 FitzHugh-Nagumo F (u, v)=−v + u + u 27 G(u, v)= (u − αv − β) Table 1: Famous Models that present the Turing instability. the logistic map. This is the well known Fisher equation, which presents traveling wave solutions. This equation was originally devised by R.A. Fisher to model the spread of an advantageous gene over a population. There have been recent applications of this model to study bacterial dy- 13 namics and epidemics of viruses. This model could be thought of as a special case of the BVAM model if u = v, µ = η, C =1/K and all the other parameters are zero. One of the most famous models in theoretical biology is the one due to the collaboration between the chemist Alfred Lotka and the biologist Vito Volterra, the Lotka–Volterra model, which was originally devised to describe the predator-prey interactions between two populations of animals, adding external resources for feeding. The usual result is that under certain conditions both populations oscillate in time, and when diffusion is added in two dimensions, patchy patterns appear also in space. This model and its variants has been used many times to model the most varied interactions Turing Systems: A General Model for Complex Patterns in Nature 277 between populations and it is still one of the main tools in Ecology. From the table it is obvious that this model could be also encompassed in the class of the BVAM variations. The importance of the Turing mechanism in biology was first realized 15 in 1972 when Gierer and Meinhardt proposed their activator-inhibitor model to account for patterning in biological systems. Usually the autocat- 2 2 2 alytic production u is replaced by a term with saturation u /(1 + κu ). This model has been very successful in reproducing the patterns found in 16 marine snail shells , in which the appearance of oblique lines is due to the existence of traveling waves. The Gray Scott model was first built to model glycolysis in cells. Pear- 18 son performed extensive numerical calculations that showed a wide variety of spatial patterns. This model has been used extensively in many problems due to the transparency of the autocatalytic reaction terms and their simplicity. Recently we have used this model to propose the formation of a prepattern in the interneuronal medium that provides chemical pathways 19 to promote axon growing by chemotaxis . The Brusselator model was proposed by Ilya Prigogine and co-workers to describe autocatalytic reactions, which are ubiquitous in Nature, particularly in living organisms. Despite its simplicity this model presents a rich phase diagram of bifurcations that makes it suitable for numerous applications, particulary in multilayered systems. In the Belousov–Zhabotinskii (BZ) reaction a mixture of potassium bro- mate, cerium(IV) sulfate and citric acid in dilute sulphuric acid produce oscillations of the concentrations of the cerium(III) and cerium(IV) ions. This reaction was studied by Noyes, Field and Koros at the University of Oregon. They proposed a model with eighteen chemical reactions, later simplified 23 to five reactions only. A further simplification with only three chemicals is known now as the Oregonator model written on the table. This model has been extensively used recently to produce complicated patterns. The simplest possible reaction with two variables that models the BZ reaction is the Schnackenberg model, which could be regarded as a simplification of the Oregonator, used in many occasions to study two dimensional 25 Turing patterns. The BZ reaction inspired the chlorine-iodide-malonic acid (CIMA) re- 5 action, which was the first experimental uncontestable realization of a Turing pattern in a chemical reactor. The Lengyel–Epstein model was ex- pressly created to mimic the CIMA reaction and has been universally used ever since as a basis to model many chemical systems. Finally, it is worth 278 R. A. Barrio

27 mentioning the FiztHugh–Nagumo model ,whichisamodelofamodel. Its origin is the famous Hodgkin-Huxley model for nerve membranes, which is mathematically complicated, yet the FiztHugh–Nagumo model captures the essential phenomena and it only needs three ﬁelds. Further reduction to two variables is not unreasonable for many models of excitable media. In the table we have extracted a model in a Turing form, which is one of 28 these reductions .

3.3. The Complex Ginzburg–Landau Equation The Complex Ginzburg–Landau equation (CGLE) is one of the most famous and well studied systems in applied mathematics. It describes the evolution of amplitudes of unstable modes for any process resulting from a Hopf bifurcation, and gives qualitative (and often quantitative) information on a wide variety of phenomena, including nonlinear waves, second order transitions, Rayleigh–Benard convection and superconductivity. Near a Hopf bifurcation, all reaction-diﬀusion equations could be reduced to the 20 CGLE . The equivalent reduction for ordinary diﬀerential equations is called the Stuart-Landau equation. The CGLE can be written as

2 2 ∂tA = A +(1+iα)∇ A − (1 + iβ)|A| A, (12) were the amplitudes A are complex. We now show the procedure for the BVAM model. We shall set h = −1 (see the ﬁrst row of Table 1) so all ﬁxed points collapse at the origin. A Hopf bifurcation is found in zero dimensions when b ≥ bc = −1anda>1. The bifurcation parameter is µ =(b − bc)/bc and the BVAM model could be written as

∂ρ 2 = D∇ ρ + ηLρ + CMρρ + Nρρρ, (13) ∂t where ρ is the perturbation around the ﬁxed point, L = L0 +µL1 is a linear operator, and M and N are nonlinear operators deﬁned by the kinetics. One 2 2 may write µ = ξ,whereξ =sgn(µ), and make a change of scale τ = t and e = r. The perturbation could be expanded in powers of ,thatis 2 3 ρ = ρ1 + ρ2 + ρ3 + ..., and one then substitutes everything in Eq. 13. Turing Systems: A General Model for Complex Patterns in Nature 279

Equating coeﬃcients with the same power of yields the following ∂ − L0 ρ1 = 0; (14) ∂t ∂ − L0 ρ2 = Mρ1ρ1; ∂t ∂ ∂ρ 2 − L0 ρ3 = − + ξL1ρ1 + D∇eρ1 ∂t ∂τ +2Mρ1ρ2 + Nρ1ρ1ρ1. The eigenvectors of the linear operator L0 are X1 =(1+s, −1) and ∗ X2 = X1 ,withs =+i (a − 1). The solvability condition for Eqs. 14 gives an amplitude equation analogous to Eq. 6:

∂A1 2 2 = ξλ1A1 + D∇eA1 − Γ|A1| A1, (15) ∂τ † where λ1 = ηX1 L1X1,and † −1 Γ=η 2X1 MX1(L0 − 2sI) MX1X1 † −1 +4X1 MX1L0 MX 1X2 † −3X1 NX1X1X2 (16)

Further change of variables allows to write Eq. 15 exactly as Eq. 12. This exercise is useful because the solutions of the CGLE have been analyzed in depth. It could be performed to study pattern selection under certain circumstances. The same procedure could be followed when relaxing the condition h = −1 to investigate the behaviour of the system around the other ﬁxed points.

4. Applications 4.1. The Coat Pattern of Marine Fish 29 In 1995 Kondo and Asai published an interesting paper suggesting that the coat pattern of certain marine fish of the order Pomacanthus could be explained as a Turing pattern. They based their claim on the peculiar way the wave length of the pattern is conserved as the fish grows. A one dimensional calculation in a Turing system could predict the observed phenomena. These fish in their juvenile form present a vertical or semicircular striped pattern, common to all species in the order, thought to prevent selective predation by the cannibal adults. When they grow their pattern changes dramatically to the specific colors and shapes of the different species. We 280 R. A. Barrio made extensive numerical simulations in two dimensions, using the BVAM model to obtain the skin pattern of the fish Pomacanthus imperator as it grows. In Fig.4 we show these results, taken from Ref. 30. Observe that the two mechanisms that conserve the wave length of the pattern are precisely those predicted by the Turing model, namely the splitting and insertion of modulated regions.

(a)

(b)

Fig. 4. (a) Juvenile form of the fish Pomacanthus imperator. The insets show numerical patterns obtained in a two-dimensional domain resembling the shape of the fish. (b) Adult form of the same fish. Numerical calculations on the left show patterns in domains of growing size.

These findings were corroborated by observing the changes on the coat of a real fish as it grew. After 4 years without growing (these fish stop growing if the environment does not provide enough space), suddenly the changes were produced very rapidly (when the fish was kept in a big enough tank). In Fig. 5 we show a series of snapshots of these events, spanning approximately 6 months. The patterns obtained with the original BVAM model were modified by suggesting that there must be sources of morphogen in some places that Turing Systems: A General Model for Complex Patterns in Nature 281

Fig. 5. Series of photographs taken every month of our juvenile fish. promote the alignment of the fringes in certain directions, in such a way that the combination of boundary conditions with the shape of the domain would produce the observed patterns. We also proposed that a modulation of the diffusion coefficients in some directions would produce alternate spots and stripes, as observed in other species of fish. More complicated patterns could be obtained by coupling two Turing systems by an interaction term that could be linear or not. These ideas have been useful in reproducing the 30 coats of many other fish. In Fig. 6 there are few examples comparing the numerical calculations from the coupled model with the observed patterns in fish.

4.2. Patterns in Curved Domains When a two dimensional domain changes its shape in time and space, the Turing equations have to be modiﬁed. We showed the general transforma- 32 tion of Turing patterns in growing and curved domains. Basically the diﬀerences are the appearance of an advection term in the time derivatives and the replacement of the Laplacian by the Laplace–Beltrami operator 282 R. A. Barrio

(a) (b)

(e) (f)

Fig. 6. Various marine fish whose coats were modeled. The numerical patterns are shown in the insets, and were obtained by coupling two BVAM models in different ways. (a) Zebrasoma veliferum, (b) Synchiropus picturatus, (c) Hypostomus plecostomus, (d) Pomacanthus maculatus (taken from Ref. 31), (e) Siganus vermiculatus, (f) Ostracion meleagris. that takes into account the effects of local curvature. It is interesting to observe patterns formed in growing domains when the changes due to growth take place in a time scale of the same order as the characteristic times for the pattern to form. In Fig. 7(a) there is a calculation showing a spot pattern in a linearly growing square. Notice that the two mechanisms that conserve the wavelength are in action, namely, splitting of spots and insertion of new ones in between. In Fig. 7(b) one can see a pattern of spots forming on a growing cone. The pattern is affected by the geometry of the conical domain and in certain regions appears as aligned stripes, just as it is seen in the tails of many reptiles, as lizards. Patterns in domains of constant curvature are of particular interest Turing Systems: A General Model for Complex Patterns in Nature 283

(a) (b)

Fig. 7. (a) Pattern obtained with Eq.8 on a growing square domain. (b) Pattern obtained in a growing cone.

Fig. 8. (a) and (b) Patterns obtained with the BVAM model on spheres of two sizes with only cubic nonlinearities. (c) and (d) same as before but adding quadratic nonlinearities. because they are observed in many small organisms, as radiolarians and viruses. We used the BVAM model to numerically ﬁnd the symmetries 33 of Turing patterns produced on a spherical surface . It was found that spots always accommodate themselves in regular positions, depending on the radius of the sphere. Some icosahedral symmetries were ubiquitous. Stripe patterns are very interesting because of the existence of singularities, not necessarily at the poles. When the radius of the sphere matches the wavelength of the pattern some target patterns appear. Examples of these calculations are shown in Fig. 8. 284 R. A. Barrio

4.3. Oscillating Turing Patterns Most of the systems that present a Turing instability also show other kinds of bifurcations. We already showed that the BVAM model could present Hopf and Turing–Hopf instabilities. There has been a renewed interest in studying the latter instability, since it produces patterns in which nonlinear standing waves exist. This is due to the interplay between the Turing instability, which produces spatially stable patterns, and a Hopf instability, that causes oscillations in time. These oscillating Turing patterns are quite complicated and represent a step forward modeling real processes in complex systems. I shall give two important examples.

4.3.1. The Faraday Experiment and Sea Urchins First, there is an example of an application of the BVAM model that is unexpected and its consequences far reaching. It is the famous experiment 34 made by Faraday in 1831 . This experiment is deceptively simple, it consists of agitating vertically a vessel containing a liquid. Then, when the amplitude (A) is zero, the uniform steady state corresponds to a flat surface, and it becomes unstable when A = 0, and the surface acquires a complicated shape. The theoretical treatment of this phenomenon in the linear regime was first made by Lord Rayleigh, who described surface waves that are known as Rayleigh waves ever since. In a cylindrical container with zero-flux boundary conditions the linear solutions are Bessel functions of the first kind. Much more interesting are the nonlinear solutions, and we repeated the experiment enhancing the nonlinear interactions, by using a rare liquid (fluorinert FC-75) with a high density and a very low surface tension coefficient. We observed that the nonlinear patterns are centrosymmetric, and that their symmetry axis depends on the bifurcation parameter, which is a combination of the amplitude and the frequency. We devised a model based on the Euler equation with friction, adding a nonlinear mechanism resulting from the interaction between the vertical and horizontal velocity 35 fields . We also showed that the model could be written as a reaction-diffusion system which always presents a Hopf bifurcation and that could be driven to a Turing–Hopf instability. Details are given in Ref. 35. There are other 40 models of the Faraday experiment, for example, a phenomenological one 36 based on the Swift–Hogenberg equation , or a treatment based on an 37 extension of the same idea . Turing Systems: A General Model for Complex Patterns in Nature 285

With our model one could follow the bifurcation tree of any symmetry axis. The actual photographs from the experiment of a sequence of 5-fold symmetry bifurcations are reproduced in Fig. 9, and they are not quasicrystalline (in fact, quasicrystalline patterns have been obtained in the Faraday 38 experiment ). M. Torres pointed out that these patterns are very similar to the shells of sea urchins in subsequent geological times of evolution.

Fig. 9. Photographs taken from our experiment, sequentially following the bifurcations of symmetry ﬁve.

Five-fold symmetry is remarkable, not only because of the existence of quasicrystals, but also because all known vertebrates in this planet present, while developing their limbs, the following scheme of symmetry-breaking processes 1 → 2 → 5 (see for instance Ref. 42). Furthermore, 5-fold symmetry is not only seen in sea urchins but in all the phylum echinodermata. Therefore, it is important to investigate the robustness of symmetry 5 in ﬁnite domains. We performed a series of numerical calculations, using the 39 BVAM model, in a circular domain in two dimensions varying its radius . This can be easily done by manipulating δ inEq.8.InFig.10weshowa histogram with the results. Observe that for a wide range of small radii, the patterns are 5-fold, and then they go to the more frequently found 6-fold. The results of this model allow to envisage the morphogenesis of a sea urchin: the egg can be considered as symmetry 1, then the animal 286 R. A. Barrio

Fig. 10. Results of the symmetry found in the centro-symmetric patterns calculated in disks of different sizes. undergoes metamorphosis to a larva state that is roughly 2-fold symmetric. This larva starts developing a flat circular belly that grows. At a certain size five primary podia appear on the disk as bulges, from which the symmetry of the whole animal developes to the adult state. One could say then that the podia appear as a result of a Turing mechanism, then the podia are calcified and the 5-fold symmetry is conserved in further stages. In Fig. 11 we show a numerical calculation obtained by producing a 5-fold pattern and then growing the disk assuming that the spots are morphogen sources that maintain the level of u constant. The shape is compared with the actual shell. This model predicts that if one is able to delay the appearance of the primary podia in the circular belly of the urchin, until it is larger, the animal should present 6-fold symmetry. One may ask at this point: What is the relation between the evolution of animals and the disturbed surface of a liquid?. The most suitable answer could be taken from Faraday himself, who pointed out that the universality of a Theory, as the Classical Theory of Vibrations, allows a deep understanding of the phenomena being modeled, regardless of the nature of the particular system. In the present case, we could infer that the mechanisms acting on the evolution of species could be better described following the bifurcations of a nonlinear system. The far reaching consequence is that one needs to rethink the meaning of natural selection and adaptability when Turing Systems: A General Model for Complex Patterns in Nature 287

Fig. 11. (a) Numerical calculation on a small disk while growing to a larger size, maintaining the central spots as sources of morphogen. (b) Photograph of the shell of an adult sea urchin. talking about Evolution. The nonlinear regime in the Faraday experiment is extremely rich and complicated patterns could be obtained in diﬀerent systems. For instance, Faraday patterns could be produced in quantum systems and one could harness very robust spatial arrangements of quantum states. In fact, very recently a paper in which Faraday patterns are seen in a Bose–Einstein 41 condensate has been published .

4.3.2. Layered Coupled Systems There has been a series of papers dealing with oscillating patterns. They are produced by linearly coupling two sets of Turing equations using the 43 44 Brusselator model , the Oregonator model , and the Lengyel–Epstein 45 model . These models consist of two chemically active layers, separated by an intermediate chemically inert layer that allows transport of chemicals through it. The coupling made in such a way that the stable fixed point in the absence of diffusion is preserved, and the parameters are chosen as to enhance resonating modes at two different wavelengths, resulting in superimposed patterns. For instance, spots contained in stripes or vice versa, or patterns with spots of two different sizes (wavelengths), and many other 46 47 complicated oscillating patterns . In a recent study an experimental set up of a bylayer reactor system has been realized. However, there are situations in which the intermediate layer could be chemically active (as it is the case of biological membranes), and the inter- layer interaction be more complicated than just a simple selective diffusion of chemicals, probably involving further chemical reactions or catalyzers. 288 R. A. Barrio

Such situations could be modeled by coupling two Turing systems nonlin- early, which as such serves as an interesting extension to the studies of the linear coupling case. We had proposed that in the original paper where the BVAM model appeared, but now it is interesting to explore the patterns in the region where the Turing–Hopf bifurcation takes place, and using the resonance conditions. For comparison with the linear coupling case we also 48 investigated cubic and quadratic couplings using the Brusselator model. Extensive numerical calculations, guided by linear stability analysis, were carried out to search for new complex patterns. In Fig. 12 there is an example of the comparison between linear and cubic coupling. The two patterns look very similar, although this is only true if the coupling strength is very small.

(a) (b)

Fig. 12. (a) Oscillating pattern of dots inside stripes, obtained with the linearly coupled model of Ref. 43. (b) Pattern obtained with the same model but with cubic coupling. The coupling parameter is q=0.01.

We found that the patterns are very much dependent on the coupling strength, contrary to linear coupling. If the coupling is very small, resonances appear resulting in superposition of patterns. For very strong coupling, the system with largest wavelength dominates, and the pattern becomes very simple. In the intermediate region new patterns with different symmetries are found. In this patterns there are new phenomena beyond simple superposition and resonances, as expected, because nonlinear coupling changes the kinetics of the original systems. In Fig. 13 we show two patterns obtained in the same cubic system with two different values of the coupling parameter, the figure is taken from Ref. 48. Turing Systems: A General Model for Complex Patterns in Nature 289

(a) (b)

Fig. 13. (a) Oscillating pattern of stripes inside dots, obtained with a layered system coupled cubically with strength q =0.09. (b) A pattern of “boats” obtained with q = 0.15.

4.4. Bistability and Travelling Waves In all former applications the BVAM model has been used as a true Turing system meeting all the conditions for a Turing instability. Furthermore, it has been assumed that γ = −α, which ensures that there is only one fixed point at the origin (u, v)=(0, 0). When γ = −α, two symmetric fixed points appear at v = ± (αg − 1)/αgr1,andu = −gv,whereg =(β +1)/(α + γ). Then, if these points are stable, one could have a situation analogous to a thermodynamic system driven to a spinodal decomposition situation, in 11 which two phases coexist and evolve in time. We have shown that the solutions of this bistable system are travelling wave fronts, or solitons. If one chooses C =0,D =0,η =1,andh = −2.5 in the model on the first row of Table 1, one gets the phase diagram shown in Fig. 14 In region 2 the central fixed point at (0,0) is unstable and the other two fixed points are stable. We can predict the existence of traveling wave fronts, an almost universally recognized feature of bistable reaction-diffusion sys- 42 tems . We expect heteroclinic trajectories in phase space connecting the two stable fixed points with the central hyperbolic point. This was corroborated by numerical calculations in one dimension, where one sees kinks moving in both directions with constant velocity. The velocity depends on the integral of the fields over the whole space, if this is zero, implying that the profile of a kink is symmetric around zero, the kink does not move, but of course, this situation is not stable. Another important feature is that there is a characteristic length that dictates the average separation of kinks. 290 R. A. Barrio

Fig. 14. Phase√ diagram of Eq. 8 when there are two symmetrical ﬁxed points at v0 = ± b/g − h = ± f and u = −gv. The character of the eigenvalues changes in the regions labeled by numbers. In region 2 there are oscillating stable modes and in region 5 these modes become unstable. Taken from Ref. 11.

Much more interesting is the situation in two dimensions, because the velocity of the front depends on its local curvature. This results in having rotating fronts around the point in which the curvature changes sign. This is shown in the numerical calculation depicted in Fig. 15 These patterns are very similar to the ones visualized in the electrical 49 activity of chicken hearts , and we are developing a model to tackle this problem. One of the important features of the heart is that when it does not function well it goes to ﬁbrillation, producing spiral waves. In the border between region 2 and region 5 of Fig. 14 one has limit cycle solutions and the possibility of spiral waves, as the ones appearing in the Belousov– Zhabotinskii reaction. It is then very simple to model the malfunctioning of the heart by only changing the parameter f. In Fig. 16 there is an example of this.

4.5. Anomalous Diffusion in Turing Systems There are many real situations when diffusion is anomalous, in the sense that the mean square displacement does dot grow linearly in time. This 50 51 occurs in many fields of science, as physics , biology , geology and others. Therefore, it is interesting to investigate the diffusion driven instability under anomalous conditions. Normal diffusion is a consequence of having completely random walks, which by the virtue of the Central Limit Theorem produce a normal Turing Systems: A General Model for Complex Patterns in Nature 291

(a) (b)

Fig. 15. Four snapshots of a pattern consisting of solitary wave fronts. Observe the appearance of a characteristic length and the peculiar way in which the pattern seems to rotate around some points. Taken from Ref. 11.

Fig. 16. The pattern on the left (from Fig.15(d)) was used as the initial conditions to produce the pattern on the right. The parameters were g =0.165, h = −2.5, and the critical value fc =0.5043.

Gaussian distribution of step-lengths in the thermodynamic limit, but this is not always the case. For example, diffusion could be anomalous in 292 R. A. Barrio inhomogeneous or fractal media. For instance, Levy flights present a distribution with diverging second moment. One way of describing anomalous diffusion is by replacing the normal derivatives in Fick’s equation by frac- 52 tional derivatives, either in time or in space. In the latter case one replaces ν 2 νx y νz the usual Laplacian operator by ∇ → (aDx +a Dy +a Dz ), where the fractional operators for each coordinate are defined as x νx 1 2 u(x ) aDx = dx ν −1 dx . (17) Γ(2 − νx) a (x − x ) x One of the important features of these fractional operators is that the Fourier components are eigenfunctions of these operators, for instance,

νx ikx νx ikx −∞Dx e =(ik) e . (18) This shows that the mean square displacement grows in time to a power given by the anomalous diffusion parameter ν. There is a discrete form of the Riemann–Liouville expression (Eq.17), convenient to be used in numerical 53 calculations, known as the Grunwald-Letnikov formula . We studied anomalous diffusion in the BVAM model in one and two 54 dimensions , with parameters similar to the ones producing Fig. 1. We found two important effects: (1) Asymmetric anomalous diffusion (only one sign for k in Eq. 18) produces moving patterns. In two dimensions their velocity depends on the anomalous exponents νx and νy, with direction given by the relative deviation of these exponents from 2 (when the patterns are stationary). This in consequence facilitates the healing of local defects in the pattern. The hexagonal dotted patterns are perfect, and the striped patterns are dislocation free. (2) The range of D, for which a Turing instability is possible, widens up. One can find Turing patterns even when D =1.

Fig. 17. Anomalous diﬀusion pattern rotating on a sphere. Turing Systems: A General Model for Complex Patterns in Nature 293

To illustrate the eﬀects of anomalous diﬀusion in 2 dimensions, in Fig. 17 we show a rotating pattern obtained on a spherical surface and νφ =2. Observe that as the velocity of the front is constant, curls are formed at the poles.

5. Final Remarks I hope I have shown that Turing models can be used to describe a wide range of phenomena in many fields of Science. However, I would like to stress that not all striped or spotted arrangements with a conserved wavelength need to be produced by a Turing instability. This is more noticeable in Geology and in Biology. It turns out that banded patterns of vegetation could be explained by a Turing model, but the ecological variables at play are not clear. Diffusion, chemical reactions and precipitation are important in geological formations, as the Liesegang Rings, but this does not necessarily mean that a simple Turing model would account for all facts. Particularly cumbersome is the field of mathematical biology, where the relevance of Turing mechanisms is still debatable. The general credo that organization in living organisms is dictated solely by the genes is well spread now, although not necessarily correct. The conspicuous bands in the abdomen of the fruit fly, at first glance, look very much as a Turing pattern, yet it has been demonstrated that the actual mechanism that produces these bands is purely genetic. The case of Drosophila is a shameful failure of Turing models and a warning to theorists, as Einstein put it very rightly, “a Theory should be as simple as possible, but not simpler”. However, very recently an interplay between a Turing mechanism and genes has been demonstrated 55 in the formation of the pattern of hair follicles in mice . I am sure that in the near future we will see more evidences like this, since “not yet very well known physical laws” should facilitate the genes mission to organize the enormous number of components to reach a final anatomic form without major errors. We, physicists, are living very exciting times modeling phenomena in many fields that were not touched by us before. Particularly in Biology, where qualitative models are not helpful anymore to understand the complexity of biological processes. Quantitative knowledge is needed, and this is obtainable not only from elaborate experiments, but also from mathematical models. 294 R. A. Barrio

Acknowledgments I am grateful to Ignazio Licata for the invitation to write this paper, and to the Helsinki University of Technology, Laboratory of Computational Engineering (LCE) for an adjunct professorship. Most of the work presented here was financed by the project F-40615 from CONACyT. I also thank my many collaborators, particularly C. Varea, J.L. Aragón, M. Torres (r.i.p.), K. Kaski, Teemu Leppänen, Mikko Kartunen, Klaus Kÿtta, Damián Hernández, Faustino Sánchez, and P.K. Maini.

References 1. D’ Arcy W. Thompson (1961) On Growth and Form (abridged edition), Cambridge Univ. Press, UK. 2. A. M. Turing (1952) Phil.Trans.R.Soc.Lond.B237, 37. 3. R.A. Barrio, C. Varea, J.L. Aragón, and P.K.(1999) Maini Bull. Math. Biol., 61, 483 . 4. A. Kelly (1967) J. Diff. Eqns, 3, 546. 5. V. Castets, E. Dulos, J.. Boissonade, and P.D. Kepper (1990) Phys. Rev. Lett. 64, 2953. 6. P.K. Maini, K.J. Painter, H.N.P. Chau (1997) Faraday Transactions, 93 (20), 3601-3610. 7. B.J. Dickson (2002) Science 298, 1959. 8. S. Schnell, R. Grima, P.K. Maini (2007) Multiscale modeling in biology, Amer. Sci., 95, 134. 9. R.T. Liu, S.S. Liaw and P.K. Maini (2007) J. Korean Phys. Soc., 50, 234. 10. T. Leppänen (2005) in Current Topics in Physics, R.A. Barrio, and K.K. Kaski, eds., Imperial College Press p. 199. 11. C. Varea, D. Hernández, and R.A. Barrio (2007) Soliton behaviour in a bistable reaction diffusion model, J. Math. Biol (in press). 12. R.A. Fisher, Ann. Eugenics (1937) 7, 353., see also Ref. 14 p. 237. 13. V.M. Kenkre (2004) Physica A, 342, 242. 14. J.D. Murray (1980) “Mathematical Biology”, Springer-Verlag, Berlin, pp.63– 68. 15. A. Gierer, H. Meinhardt, Kibernetic (1972) 12, 30. 16. H. Meinhardt, and M. Klingler (1987) J. Theor. Biol., 126, 63. 17. P. Gray and S.K. Scott (1985) J.Chem.Phys., 89, 22. 18. J.E. Pearson (1993) Science 261, 189. 19. O. Avila, D. Hernández, R.A. Barrio, and Limei Zhang, Modeling Neurite growth and guidance, (to be published). 20. Y. Kuramoto (2003) “Chemical Oscillations, Waves, and Turbulence”, Dover Publ. Inc., New York. 21. I. Prigogine, R. Lefever (1968) J. Chem. Phys. 48, 1695. 22. R.J. Field and R.M. Noyes (1974) J.Chem.Phys., 60, 1877. Turing Systems: A General Model for Complex Patterns in Nature 295

23. J.J. Tyson (1981) “On scaling the Oregonator equations in nonlinear Phe- nomena in Chemical Dynamics” (C. Vidal and A. Pacault, Eds.), Springer- Verlag, Berlin, pp. 222–227. 24. J. Schnackenberg (1979) J. Theor. Biol. 81, 389. 25. V. Dufiet and J. Boissonade, J.Chem.Phys., 96, 664 (1992). 26. I. Lengyel and I.R. Epstein (1991) Science, 251, 650. 27. R. FitzHugh (1961) Biophysics J., 1, 445 ; J. S. Nagumo, S. Arimoto, and S. Yoshizawa (1962) Proc. IRE, 50, 2061. 28. Kyoung J. Lee (1997) Phy. Rev. Lett., 79, 2907. 29. S. Kondo and R. Asai (1995) Nature, 376, 765–768. 30. J.L. Aragón, C. Varea, R. A. Barrio and P.K. Maini (1998) FORMA 13 154. 31. Stanislav Frank (1973) “Encyclopédie Illustrée des Poissons”, Gründ, Paris p. 389. 32. R. Plaza, F. Sanchez-Garduño, P. Padilla, R.A. Barrio, and P.K. Maini (2004) Jour. of Dynamics and Differential Equations 16, 1093 . 33. C. Varea, J.L. Aragon and R.A. Barrio (1999) Phys. Rev. E, 60, 4588. 34. M. Faraday (1831) Philos. Trans. R. Soc. London 121, 299. 35. R.A. Barrio, J.L. Aragon, C. Varea, M. Torres, I. Jimenez, and F. Montero de Espinosa (1997) Phys. Rev. E, 56, 4222. 36. J.B. Swift and P.C: Hogenberg (1977) Phys. Rev. A, 15, 319. 37. S.V. Kiyashko, L.N. Korsinov, M.I. Rabinovich, and L.S. Tsimring (1996) Phys. Rev. E, 54, 5037. 38. S. Fauve (1995) in Dynamics of nonlinear and disordered systems,(Edited by G. Martinez-Mekler and T.H. Seligman, World Scientific, Singapore) p. 67. 39. J.L. Aragón, M. Torres, D. Gil, R.A. Barrio, and P.K. Maini (2002) Phys Rev. E, 65, 051913. 40. R. Lifshitz and D.M. Petrich (1997) Physi. Rev. Lett., 79, 1271. 41. P. Engels, C. Atherton, and M.A. Hoefer(2007) Phys. Rev. Lett., 98, 095391. 42. J.D. Murray (2003) “Mathematical Biology II: Spatial Models and Biomed- ical Applications”, Springer-Verlag, Berlin. 43. L. Yang, M. Dolnik, A. Zhabotinsky, and I. Epstein (2002) Phys. Rev. Lett., 88, 208303 . 44. L. Yang and I. Epstein (2003) Phys. Rev. Lett., 90, 178303. 45. L. Yang and I. Epstein (2004) Phys. Rev. E, 69, 026211. 46. for an animated exhibition of these patterns see for instance: http://hopf.chem.brandeis.edu/yanglingfa/pattern/research frm.html. 47. I. Berenstein, M. Dolnik, L. Yang, A. Zhabotinsky, and I. Epstein (2004) Phys. Rev. E, 70, 046219. 48. K. Kyttä, K. Kaski, and R.A. Barrio, Complex Turing patterns in nonlin- early coupled systems (to be published). 49. G. Bub, A. Shrier, and L. Glass (2002) Phys. Rev. Lett. 88, 058101. 50. K.W: Gentle, et al. (1995) Phys. Plasmas, 2, 2292. 51. A.T: Winfree (1997) Int. J. Bifur. Chaos Appl. Sci. Eng., 7, 487. 52. B.I. Henry and S.L. Wearne (2002) SIAM J. Appl. Math. 62, 870. 296 R. A. Barrio

53. I. Podlubny (1999) Fractional Diﬀerential Equations, Ac. Press, (San Diego). 54. C. Varea and R.A. Barrio (2004) J. Phys.: Condens. Matter, 16, S5081. 55. S. Sick, S. Reinker, J. Timmer, and T. Schlake (2006) Science, 314, 1447. 297

Primordial Evolution in the Finitary Process Soup

Olof Görnerupa and James P. Crutchfieldb aComplex Systems Group, Department of Energy and Environment Chalmers University of Technology, 412 96 Göteborg, Sweden bCenter for Computational Science & Engineering and Physics Department, University of California, Davis, One Shields Avenue, Davis CA 95616, USA [email protected] [email protected]

A general and basic model of primordial evolution—a soup of reacting finitary and discrete processes—is employed to identify and analyze fundamental mechanisms that generate and maintain complex structures in prebiotic systems. The processes—²-machines as defined in computational mechanics—and their interaction networks both provide well defined notions of structure. This enables us to quantitatively demonstrate hierarchical self-organization in the soup in terms of complexity. We found that replicating processes evolve the strategy of successively building higher levels of organization by autocatalysis. Moreover, this is facilitated by local components that have low structural complexity, but high generality. In effect, the finitary process soup spontaneously evolves a selection pressure that favors such components. In light of the finitary process soup’s generality, these results suggest a fundamental law of hierarchical systems: global complexity requires local simplicity.

Keywords: Structural Complexity; Entropy; Information; Computational Me- chanics; Population Dynamics; Hierarchical Dynamics; Emergence; Evolution; Self-Organization; Autocatalysis; Autopoiesis PACS(2006): 89.75.k; 89.75.Fb; 82.39.Rt; 65.40.Gr; 05.70.a; 02.70.c

1. Introduction The very earliest stages of evolution—or rather, pre-evolution—remain a mystery. How did structure emerge in a system of simple interacting objects, such as molecules? How was this structure commandeered as substrate for subsequent evolution—evolution that continued to transform the objects themselves? One wonders if this recursive interplay between structure and dynamics facilitated the emergence of complex and functional organizations. 298 Olof G¨ornerupand James P. Crutchﬁeld

Since these questions concern the most fundamental properties of evolutionary systems, we explore them using principled and rigorous methods. To build a suitable model a few basic ingredients are required. First, one needs some type of elementary objects that constitute the state of the system at its finest resolution. Second, one needs rules for how the objects interact. Third, one needs an environment in which the objects interact. Fourth, one needs quantitative and calculable notions of structure and organization. These requirements led us to the finitary process soup model of primordial evolution1 . Simply stated, the soup’s ingredients are, in order, ²-machines, their functional composition, a flow reactor, and the structural complexity Cµ of ²-machines. After explaining each of these ingredients, we will relate the model to classical replicator dynamics by reducing the soup to a special case. We then move on to contrast the limited case with the full-fledged finitary process soup as a constructive, unrestricted dynamical system.

2. Objects: ²-machines Here we employ a finite-memory process called an ²-machine2–4 , as our preferred representation of an evolving information-processing individual. Using a population of ²-machines is particularly appropriate in studying self-organization and evolution from an information-theoretic perspective as they allow quantitative measurements of storage capacity and randomness. Rather than using the abstraction of a formal language—an arbitrary finite set of finite length words—we consider a discrete-valued, discrete-time stationary stochastic process described by a bi-infinite sequence of random variables St over an alphabet A: ↔ S= ...S−1S0S1.... (1) A process stores information in a set of causal states that are equivalence classes of semi-infinite histories that condition the same probability distribution for future states. More formally, the causal states S of a pro- ← ← cess are the members of the range of the map ² : S 7→ 2S from histories to sets of histories: ← ←0 → ← ← → ← ←0 ²( s ) = { s |P(S | S= s ) = P(S | S= s )} , (2) ← ← where 2S denotes the power set of S. Further, let S ∈ S be the current →1 casual state, S0 its successor, and S the next symbol in the sequence (1). The transition from one causal state Si to another Sj that emits the symbol Primordial Evolution in the Finitary Process Soup 299

(s) s ∈ A is given by a set of labeled transition matrices: T = {Tij : s ∈ A}, where →1 (s) 0 Tij ≡ P(S = Sj, S = s|S = Si). (3) The ²-machine of a process is the ordered pair {S, T }. One can show that it is the minimal, maximally predictive causal representation of the pro- cess4 . Unlike a general probabilistic ²-machine, for simplicity, here we take causal-state transitions to have equal probabilities. The ﬁnitary ²-machines can be thought of as ﬁnite-state machines with a certain properties4 : (1) All states are start states and accepting states; (2) All recurrent states form a single strongly connected component; (3) All transitions are deterministic: A causal state together with the next value observed from the process determines a unique next causal state; And (4) the set of causal states is minimal. Here we use an alphabet of input and output pairs over a binary alphabet: A = {0|0, 0|1, 1|0, 1|1}. This implies that the ²-machines work as mappings between sets of strings. In other words, they are transducers5 .

1|0 0|0 0|0 A 1|1 A B A B 0|1 1|1

TA TB TC

Fig. 1. Three examples of ²-machines. TA represents the identity function and has the causal state A. TB has two causal states (A and B), accepts the input string 1010 ... or 0101 ..., and operates by ﬂipping 0s to 1s and vice versa. TC has the same domain and range as TB , but maps input strings onto themselves.

In contrast to prior models of pre-biotic evolution, ²-machines are simply ﬁnitely-speciﬁed mappings. More to the point, they do not have two separate modes of representation (information storage) or functioning (transformation). The advantage is that there is no assumed distinction between gene and protein6,7 or between data and program8–11 . Instead, one recovers the dichotomy by projecting onto (i) the sets that an ²-machine recognizes and generates and (ii) the mapping between these sets. Examples of ²-machines are shown in Figure 1.

3. Interaction: Functional Composition The basic pairwise interaction we use in the finitary process soup is functional composition. Two machines interact and produce a third machine— 300 Olof Görnerupand James P. Crutchfield their composition. Composition is not a symmetric operation. Machine TA composed with another TB does not necessarily result in the same machine as TB composed with TA: TB ◦ TA 6= TA ◦ TB. The upper bound on the number of states of the composition is the product of the number of states of the parents: |TB ◦ TA| ≤ |TB| × |TA|. Hence, there is the possibility of exponential growth of states and machine complexity if machines are iteratively composed. A composition, though, may also result in a machine with lower complexity than those of its parents.

3.1. Interaction Networks We represent the interactions among a set of ²-machines with an interaction network G which is a a graph whose nodes correspond to ²-machines and whose transitions correspond to interactions. If Tk = Tj ◦ Ti occurs in the soup, then the edge from Ti to Tk is labeled Tj. One may represent G with the binary matrices: ( 1 if T = T ◦ T G(k) = k j i (4) ij 0 otherwise. Consider the ²-machines in Fig. 1, for example. They are related via composition according to the interaction graph shown in Fig. 2, which is given by the matrices   1 0 0 G(A) = 0 0 0 , (5) 0 0 0   0 1 0 G(B) = 1 0 1 , (6) 0 1 0 and   0 0 1 G(C) = 0 1 0 . (7) 1 0 1 Primordial Evolution in the Finitary Process Soup 301

TB TC

TA,TC TB TA,TC

Fig. 2. Interaction network of the ²-machines in Fig. 1. There is a transition, for example, that is labeled TC from the node TA to the node TC , since TA composed with TC results in TC (in fact, each transition from TA has the same label as the label of its respective sink node since TA is the identity function).

3.2. Meta-machines For a machine to survive in its environment somehow it needs to produce copies of itself. This can be done directly by self-reproduction, e.g. TA◦TA = TA, or it can be done indirectly in cooperation with other machines: e.g., TA facilitates the production of TB, which facilitates the production of TC , which, in turn, closes the loop by facilitating the production of TA. In other words, there can be sets of machines that interact with each other in such a way that they collectively self-reinforce the overall production of the set. This leads to the notion of an autonomous and self-replicating entity, which we call a meta-machine. Inspired by Maturana and Varela’s autopoietic set 12 , Eigen and Schuster’s hypercycle13 , and Fontana and Buss’ organization14 , we deﬁne a meta-machine Ω to be a connected set of ²-machines whose interaction matrix consists of all and only the members of the set. That is, a set Ω is a meta-machine if and only if (1) the composition of two ²-machines from the set is always itself a member of the set:

∀Ti,Tj ∈ Ω; Tj ◦ Ti ∈ Ω, (8) (2) all ²-machines in the set can arise from the composition of two machines in the set:

∀Tk ∈ Ω; ∃Ti,Tj ∈ Ω,Tk = Tj ◦ Ti, (9) and (3) there is a nondirected path between every pair of nodes in Ω’s interaction network GΩ. The third property ensures that there is no subset of Ω that is isolated from the rest of Ω under composition. Consider, for 302 Olof G¨ornerupand James P. Crutchﬁeld example, the union of two self-replicators, TA and TB, for which TB ◦ TA = TA ◦ TB = T∅. According to property (3), they are not a meta-machine.

4. Complexity Measures: Cµ In previous computational pre-biotic models, the objects have been represented by, for example, assembly language codes8–11 , tags15,16 , λ- expressions17 and cellular automata18 . We employ ²-machines instead mainly for one reason: there is a well developed theory (computational mechanics) of their structural properties. Assembly language programs and λ-expressions, for instance, are computational universal representations and so one knows that it is not possible to calculate their complexity5 . For finitary ²-machines, in contrast, complexity can be readily defined and analytically calculated in closed form. Define the stochastic connection P (s) matrix of an ²-machine M = {S, T } as T ≡ s∈A T . The probability distribution pS over the states in S—how often they are visited—is given by the normalized left eigenvector of T associated with eigenvalue 1. The structural complexity Cµ of M is the Shannon entropy of the distribution given by pS , X (v) (v) Cµ(M) ≡ − pS log2 pS . (10) v∈S The structural complexity of an ²-machine is the amount of information stored in the distribution over S, which is the minimum average amount of memory needed to optimally predict future configurations4 . To measure the diversity of interactions in the soup we define the interaction network complexity Cµ(G) to be the Shannon entropy of the distribution of effective transition probabilities in the graph G. We consider, in particular, only the transitions that have occurred between machine types that are present. That is, X k k Cµ(G) ≡ − υij log2 υij, (11)

pi,j,k6=0 where ( P k k pipj/ υij if Tk = Tj ◦ Ti has occurred υij = (12) 0 otherwise. and pi is the fraction of machines of type i in the population. To monitor the emergence of actual and functional reproduction paths, we consider only those interactions that occurred in the population. Primordial Evolution in the Finitary Process Soup 303

5. Framework: The Soup

T TR D

TA TB

Fig. 3. A schematic illustration of the ﬁnitary process soup. Two ²-machines, TA and TB , are composed and produce a third machine TC , or a random machine TR is introduced to the soup. In either case, another randomly selected machine TD is removed to maintain a ﬁxed population size. Note that this is a well stirred setting, and so there is no spatial relationship in the population.

The ²-machines interact in a well stirred reactor with the following iterated dynamics:

(1) Production and inﬂux:

(a) With probability Φin generate a random ²-machine TR. (b) With probability 1 − Φin (reaction):

i. Select TA and TB randomly. ii. Form the composition TC = TB ◦ TA. (2) Outﬂux:

(a) Select an ²-machine TD randomly from the population. (b) Replace TD with the ²-machine produced in the previous step— either TC or TR.

TR is uniformly sampled from the set of all two-state ²-machines in our simulations (see below). This sampling is also used when initializing the population. The insertion of TR corresponds to an influx while the removal of TD corresponds to an outflux. The latter keeps the population size constant. See Fig. 3 for a schematic illustration. There is no spatial dependence 304 Olof Görnerupand James P. Crutchfield in this version of the soup as ²-machines are sampled uniformly from the population for each replication and removal.

6. Closed Population Dynamics To familiarize ourselves with the model we first examine a simple base case: a soup with no influx that is initialized with machines taken from a finite set which is closed under composition. This case is also intended to work as a bridge between classical population dynamics and the general, constructive dynamics of the finitary process soup. The closure with respect to composition enables us to describe the system’s temporal dynamics of ²-machine concentrations by a coupled system of ordinary differential equations. In the limit of an infinite soup size, the rate equation of concentration pk of machine type Tk is given by

p˙k = ψk − Φoutpk, k = 1, ..., n, (13) where the conditional production rate ψk is the probability that Tk is produced given that two randomly sampled machines are paired: Xn k ψk = αijpipj, (14) i,j=1

k and αij is a second-order reaction rate constant: ( k 1 if Tk = Tj ◦ Ti αij = (15) 0 otherwise.

The outﬂux Φout equals the total production rate of the soup—i.e., the probability that a reaction occurs given that two ²-machines are paired. It keeps the size of the soup constant: Xn Φout(t) = ψi. (16) i=1

Given a soup with no inﬂux, Φin = 0, that hosts machines which are members of a set that is closed under composition, the distribution dynamics can alternatively be predicted from by its interaction network:

(k) (k) T −1 pt = pt−1 ·Gij · pt−1Z , (17)

(k) −1 where pt is the frequency of ²-machine type k at time t and Z is a normalization factor. This approximates the soup’s elements as updating synchronously. Primordial Evolution in the Finitary Process Soup 305

We illustrate the closed case by initiating the soup with machines that consist of only a single state—mono-machines. There are 15 mono- machines, the null (transition-less) transducer is excluded, and they form a closed set under composition. See Fig. 4 for the temporal dynamics of their respective frequencies. Nine machine types remain in the population at equilibrium. They form a meta-machine M with Cµ(M) = 5.75 bits. In this case, since Cµ(Ti) = 0 for all mono-machines Ti, the population’s structural complexity derives only from its interaction network.

0.25

0.2

0.15 p 0.1

0.05

0 0 2 4 6 8 10 t/N

Fig. 4. A simple base case: Machine type frequencies of mono-machines as functions of time. N (= 100, 000) denotes the population size. Dashed lines: simulation; solid lines: Eq. (17). (Reprinted with permission from [1])

7. Open Population Dynamics We now move on to the general case of a soup with positive influx rate consisting of ²-machines of arbitrary size. The soup then constitutes a constructive dynamical system where there is a mutual dependence between its equations of motion and the individuals. Due to the openness, Eqs. (13)-(17) do not necessarily apply. We therefore turn to simulations. In order to study dynamics that is ruled solely by compositional transformations we first set the influx rate to zero. A fine-grained description of the soup’s history on the ²-machine level is given by a genealogy—a record of descent of machine types. By studying the example in Fig. 5, a simulation with N = 100 individuals, one important observation is that nearly all the ²-machine types that are present in the soup’s initial population are replaced over time. Thus, genuine novelty emerges, in contrast to the 306 Olof Görnerupand James P. Crutchfield closed soup just described. Initially, there is a rapid innovation phase in which novel machines are introduced that displace the bulk of the initial machines. The degree of innovation flattens out, along with the diversity of the soup, and eventually vanishes as the population becomes increasingly closed under composition.

180 160 140 120 100 80

Machine Type 60 40 20

0 200 400 600 800 1000 t

Fig. 5. Genealogy of ²-machine types in a soup with 100 machines. A solid line denotes that a machine type is present in the soup. Dashed lines (drawn from the parents to the child) denote composition. Note that almost the whole set of initial ²-machine types (with one exception) is replaced by the dynamics.

To monitor the soup’s organization over time, we superimpose Cµ(G) time series from several runs in Fig. 6. One sees that plateaus are formed. These can be explained in terms of meta-machines. In addition to capturing the notion of self-replicating entities, meta-machines also describe an invariant set of the population dynamics. That is, formally, Ω = G ◦ Ω, (18) where Ω is the set of ²-machines present in the population and G is their interaction network. These invariant sets can be stable or unstable under the population dynamics. Consider, for example, the meta-machine in Fig. 2. It is unstable, since TAs are only produced by TAs, and will decay over time to the meta-machine of Fig. 7. This also illustrates, by the way, how trivial self-replication is spontaneously attenuated in the soup. The plateaus at Cµ(G) = 4 bits, Cµ(G) = 2 bits, and Cµ(G) = 0 bits Primordial Evolution in the Finitary Process Soup 307

Fig. 6. Decomposition of meta-machines in a soup with no inﬂux. Superimposed plots of Cµ(G) from 15 separate runs with N = 500. Cµ(G) is bounded by 4 bits while a 4-element meta-machine (shown), denoted Ω4, is the largest one in the soup. Ω4 decays to Ω2, a 2-element meta-machine (shown) due to ﬂuctuations, that in turn decays to Ω1, a single self-reproducing ²-machine.

TC TB TC TC TB

Fig. 7. The resulting meta-machine when the meta-machine in Fig. 2 decays under the population dynamics of Eq. (17). correspond to the largest meta-machine that is present at that time. Since a meta-machine by definition is closed under composition, it itself does not produce novel machines; thus, one has the upper bound of Cµ(G). As a meta-machine is reduced due to an internal instability or sampling fluctuations by the outflux, the upper bound of Cµ(G) is lowered. This results in a stepwise and irreversible succession of meta-machine decompositions. Fig. 6 shows only three plateaus. In principle, however, there is one plateau for every meta-machine that at some point is the largest one in the population. The diagram in Fig. 8 summarizes our results from a more systematic survey of spontaneously generated meta-machine hierarchies in simulations of soups with 500 ²-machines. We now examine the effects of influx by studying the population- averaged ²-machine complexity hCµ(T )i and the run-averaged interaction network complexity hCµ(G)i as a function of t and Φin, see Fig. 9. The average ²-machine complexity hCµ(T )i increases rapidly initially 308 Olof Görnerupand James P. Crutchfield

10 Composition 9

8 Decomposition 7

5 4 !

Fig. 8. Composition and decomposition hierarchy of meta-machines. Dots denote self- TB replicating ²-machines, solid lines denote TA −→ TC transitions and dashed lines denote TA equivalent TB −→ TC transitions. The label of the source node and the transition label are interchanged in the latter transition type. This results in a redundant representation of the interaction network, which is used to show how the meta-machines are related. The interaction networks are shown in a simplified way according to Ω4; cf. Fig. 6. before declining to a steady state. The average interaction network complexity hCµ(G)i is relatively high where the average structural complexity of the ²-machines is low, and is maximized at Φin ≈ 0.1. Higher influx rates have a destructive effect on the populations’ interaction network due to the new individuals’ low reproduction rate. hCµ(T )i is, in contrast, maximized Primordial Evolution in the Finitary Process Soup 309

4 3 2 1 100 0 -1 0 10 20 Φ 40 -2 in 60 10 t/N 80 -3 10 (a)

4 3 2 1 100 0 -1 0 10 20 Φ 40 -2 in 60 10 t/N 80 -3 10 (b)

Fig. 9. (a) Population- and run-averaged ²-machine complexity hCµ(T )i and (b) run- averaged interaction network complexity hCµ(G)i as a function of time t and inﬂux rate 1 Φin for a population of N = 100 objects. (Reprinted with permission from ).

at a relatively high inﬂux rate (Φin ≈ 0.75) at which hCµ(G)i is relatively small. The maximum network complexity Ccµ(G) of the population grows linearly at a positive rate of approximately 7.6 · 10−4 bits per replication.

8. Discussion We presented a conceptual model of pre-biotic evolution: a soup consisting of objects that make new objects1 . The objects are ²-machines and they generate new ²-machines by functional composition. The soup constitutes a constructive dynamical system since the population dynamics is not fixed and may itself evolve along with the state space it operates on. Specifically, the dimension of the state space changes over time, which is reminiscent of 310 Olof Görnerupand James P. Crutchfield the constructive population dynamics associated with punctuated equilib- ria19 . In principle, this allows for open-ended evolution. The quantitative estimate quoted above for the linear growth of the interaction network complexity supports this intriguing possibility occurring in the open finitary process soup. In the case of no influx, though, the system reaches a steady state where the soup consists of only one self-replicator. Growth and maintenance of organizational complexity requires that the system is dissipative; i.e., that there is a small, but steady inflow of random ²-machines. Notably, in this case, the soup spontaneously evolves hierarchical organizations in the population—meta-machines that in turn are organized hierarchically. These hierarchies are assembled from noncomplex, general individual ²- machines. In this way, the soup’s emergent complexity derives largely from a network of interactions, rather than from the unbounded increase in the structural complexity of individuals. It appears, therefore, that higher-order complex organization not only allows for simple local components but, in fact, requires them.

Acknowledgments This work was supported at the Santa Fe Institute under the Networks Dynamics Program funded by the Intel Corporation and under the Com- putation, Dynamics and Inference Program via SFI’s core grants from the National Science and MacArthur Foundations. Direct support was provided by NSF grants DMR-9820816 and PHY-9910217 and DARPA Agreement F30602-00-2-0583. O.G. was partially funded by PACE (Programmable Ar- tiﬁcial Cell Evolution), a European Integrated Project in the EU FP6-IST- FET Complex Systems Initiative, and by EMBIO (Emergent Organisation in Complex Biomolecular Systems), a European Project in the EU FP6- NEST Initiative.

References 1. Crutchfield, J. P. and Görnerup,O. (2006) J. Roy. Soc. Interface 3, 345–349. 2. Crutchfield, J. P. and Young, K. (1989) Phys. Rev. Let. 63, 105–108. 3. Crutchfield, J. P. (1994) Physica D 75, 11–54. 4. Crutchfield, J. P. and Shalizi, C. R. (1999) Physical Review E 59(1), 275–283. 5. Brookshear, J. G. (1989) Theory of computation: formal languages, automata, and complexity, Benjamin/Cummings, Redwood City, California. 6. Schrödinger,E. (1967) What is Life? Mind and Matter, Cambridge Univ. Press, Cambridge, United Kingdom. Primordial Evolution in the Finitary Process Soup 311

7. vonNeumann, J. (1966) Theory of Self-Reproducing Automata, University of Illinois Press, Urbana. 8. Rasmussen, S., Knudsen, C., Feldberg, P., and Hindsholm, M. (1990) In Emergent Computation : North-Holland Publishing Co. pp. 111–134. 9. Rasmussen, S., Knudsen, C., and Feldberg, R. (1992) In Artificial Life II: Pro- ceedings of an Interdisciplinary Workshop on the Synthesis and Simulation of Living Systems (Santa Fe Institute Studies in the Sciences of Complexity, Vol. 10) : Addison-Wesley. 10. Ray, T. S. (1991) volume XI, of Santa Fe Institute Stuides in the Sciences of Complexity Redwood City, California: Addison-Wesley. pp. 371–408. 11. Adami, C. and Brown, C. T. (1994) In Artificial Life 4 : MIT Press pp. 377–381. 12. Varela, F. J., Maturana, H. R., and Uribe, R. (1974) BioSystems 5(4), 187– 196. 13. Schuster, P. (1977) Naturwissenschaften 64, 541–565. 14. Fontana, W. and Buss, L. W. (1996) In S. Casti and A. Karlqvist, (ed.), Boundaries and Barriers, : Addison-Wesley pp. 56–116. 15. Farmer, J. D., Packard, N. H., and Perelson, A. S. (1986) Phys. D 2(1-3), 187–204. 16. Bagley, R. J., Farmer, J. D., Kauffman, S. A., Packard, N. H., Perelson, A. S., and Stadnyk, I. M. (1989) Biosystems 23, 113–138. 17. Fontana, W. (1991) volume XI, of Santa Fe Institute Stuides in the Sciences of Complexity Redwood City, California: Addison-Wesley. pp. 159–209. 18. Crutchfield, J. P. and Mitchell, M. (1995) Proc. Natl. Acad. Sci. 92, 10742– 10746. 19. Crutchfield, J. P. (2001) When evolution is revolution—origins of innovation In Evolutionary Dynamics—Exploring the Interplay of Selection, Neutrality, Accident, and Function Santa Fe Institute Series in the Sciences of Complex- ity pp. 101–133 Oxford University Press. This page intentionally left blank 313

Emergence of Universe from a Quantum Network

Paola A. Zizzi Dipartimento di Matematica Pura ed Applicata Via Trieste, 63 35121 Padova Italy [email protected]

We find that the very early universe can be described as a growing quantum network, where the nodes (quantum logic gates) are the quantum fluctuations of the metric, the connecting links are virtual black holes and the free links are qubits. The quantum fluctuations of the metric and the virtual black holes, which make up the quantum foam, are the connected part of the network, while the qubits are the disconnected part. At each time step (here time is quantized in Planck time units), the net grows because of the inverse relation between the quantum information and the cosmological constant. In fact, the virtual black holes carry virtual information, and a certain number of virtual states are added to the memory register. These virtual states are transformed into qubits (available quantum information) at the gates. The growth of the quantum network is responsible of the cosmic inflation. The connected part of the graph plays the role of the ”environment”, while the disconnected part plays the role of the quantum state. The N-qubits state undergoes environmental decoherence at the end of inflation but, as the quantum foam is itself part of the network, we are in fact in presence of a kind of self-decoherence. The quantum growing network of the early universe is strictly related to the problem of the cosmological constant Also, this model gives some hints to the solution of the ”information loss” puzzle. Any computational system must obey the quantum limits to computation, given by the Margolous/Levitin theorem. We show that the quantum growing network saturates these bounds, thus it can be considered the ultimate network, just as black holes can be considered ultimate computers (as claimed by Seth Lloyd). Hence, in a sense, quantum space-time itself turns out to be the most powerful tool for quantum computation and quantum communication.

Keywords: Quantum Gravity; Early Universe; Quantum Growing Network; Quantum Information; Complex Systems PACS(2006): 04.60.m; 04.60.Kz; 04.60.Pp; 04.62.+v; 04.90.+e; 98.80.Cq; 03.67.a; 89.75.k 314 Paola A. Zizzi

1. Introduction Quantum space-time is space-time at the Planck scale. It is widely believed that at that scale space-time is discrete, and has a foamy structure. How- ever, the proper structure of quantum space-time is not yet completely understood. Once we will know the structure of quantum space-time, we will also get the theory of quantum gravity, the theory which should reconcile General Relativity and Quantum Mechanics, as the quantum features of gravity should appear just at the Planck scale. The two main theories candidate to become the final theory of quantum gravity are string theory and loop quantum gravity. However, quite recently, it seems that a new strand to quantum gravity is available, that of quantum computation. As the whole universe at its very beginning (more precisely, the early inflationary universe) had a Planck size, it is strongly believed that the understanding of the early universe could give fundamental hints to formulate the theory of quantum gravity. Our approach to quantum gravity in fact consists in applying quantum information theory to the very early universe. In [1] we showed that the holographic principle [2] is strictly related to quantum information encoded in quantum space-time. However, any computational system must obey the quantum limits to computation, given by the Margolous/Levitin theorem [3]. Thus, quantum computing space-time, should obey these bounds as well. Indeed, quantum space-time saturates these bounds, as it will be illustrated in the following. Recently, Seth Lloyd [4] suggested that black holes can be regarded as ultimate computers, as they saturate the quantum limits to computation. Moreover, Ng showed [5] that space-time foam [6] provides the quantum limits to computation. Finally, in a recent paper [7], we described the early inflationary universe as a quantum growing network (QGN), which saturates the quantum limits to computation at each time step. The QGN graph is made of two parts: the connected part which is the ensemble of the connecting links and the nodes, and the disconnected part, which is made of free links out-going from the nodes. The connecting links are virtual black holes, quantum fluctuations of the vacuum, which carry virtual information (because of the inverse relation between quantum information and the quantized cosmological constant). The nodes are quantum fluctuations of the metric, which operate like quantum logic gates, and transform the virtual information into available information. The free links are the qubits. The ensemble of virtual black holes and quantum Emergence of Universe from a Quantum Network 315

fluctuations of the metric is what is called here ‘quantum foam’. Then, the ultimate network appears to be an artefact of space-time foam. If it is true that the QGN saturates the quantum limits to computation, it does that at each time step, because of its growth. This is a different case from that of the ultimate computer, where in principle the input power (which limits the speed of computation) is known once for all. In our case, it is not possible to know, for example, whether there is a maximum amount of quantum information processed by the QGN. The knowledge of that would be necessary to state a priori when the QGN decohered, giving rise to a classical universe (we recall moreover, that decoherence of the QGN would coincide with the end of inflation [7]). Nevertheless, there are a few hints (considerations on the actual entropy of our universe) which allow us to identify the speed of computation and memory space at the moment of decoherence. At this point, one could raise objections concerning decoherence of an isolated system. However, as we said, the QGN can be considered as made of two subsystems: the connected part, which is the quantum foam, and the disconnected part which is the quantum state of N qubits. We believe that decoherence is due to the quantum foam which plays the role of the environment. But, as the quantum foam is part of the QGN, the QGN as a whole undergoes self-decoherence. It should be noticed that, if the quantum state of N qubits did not decohere at some earlier time, the present amount of entropy would be huge.

2. The Inflationary Universe as A Quantum Growing Network In a recent paper [7], the early inflationary universe was described as a quantum growing network (QGN). The speed up of growth of the network (inflation) is due to the presence of virtual qubits in the vacuum state of the quantum memory register. Virtual quantum information is created by quantum vacuum fluctuations, because of the inverse relation [7] between the quantized cosmological constant Λn[8], and quantum information I: 1 Λn = 2 , ( n=0,1,2,3. . . ) , where the quantum information I is [1]: I ≡ I lP N = (n + 1)2. In this model, time is quantized in Planck time units: ∼ −43 tn = (n+1)tP , (where tP = 10 sec is the Planck time). At each time step −∆Λ tn, then, the increase of quantum information is given by: ∆I = 2 2 = lP Λ 2n + 3. That is, at each time step tn, there are 2n+3 extra bits (virtual states) in the vacuum state of the quantum memory register. 316 Paola A. Zizzi

The 2n+3 virtual states occurring at time tn, are operated on by a quantum v=2Qn+3 logic gate Un [7] at time tn0 = tn+1, where: Un = Had(j), and Had µ ¶ j=1 1 1 (j) is the Hadamard gate Had = √1 operating on bit j. Then, the 2 1 −1 virtual states, are transformed into 2n’+1 qubits at time tn0 . This results into a quantum growing network, where the nodes are the quantum logic gates, the connecting links are the virtual states, and the free links are the qubits. At each time step, the total number of qubits (free links) is N = (n + 1)2. The rules of the growing quantum network that we consider, are resumed below. At the starting time (the unphysical time t−1 = 0), there is one node, call it -1. At each time step tn, a new node is added, which links to the youngest and the oldest nodes, and also carries 2n+1 free links. Thus, at the Planck time t0 = tP , the new node 0 is added, which links to node -1 and carries one free link. At time t1 = 2tP , the new node 1 is added, which links to nodes -1 and 0, and carries three free links, and so on. In general, at time tn, there are: 1) n + 2 nodes but only n+1 of them are active, in the sense that they have outgoing free links (node -1 has no outgoing free links). 2) N = (n + 1)2 free links coming out from n+1 active nodes 3) 2n+1 links connecting pairs of nodes 4) n loops. The N free links are qubits (available quantum information), the 2n+1 connecting links are virtual states, carrying information along loops, the n+1 active nodes are quantum logic gates operating on virtual states and transforming them into qubits. In fact, notice that the number of free outgoing links at node n is 2n+1, which is also the number of virtual states (connecting links) in the loops from node -1 to node n. As it was shown in [7], the QGN saturates the quantum limits to computation [3] [4] [5], thus it can be viewed as the ultimate network. The ultimate computer [4] being a black hole, it would not be surprising that the ultimate network is quantum space-time itself.

3. Quantum Gravity Registers A quantum memory register (the memory register of a quantum computer) is a system built of qubits. In [8] we showed that the early universe can be considered as a quantum register. Emergence of Universe from a Quantum Network 317

In our case, moreover, the quantum register grows with time. In fact, at ∼ −44 each time step tn = (n+1)tP ( with n=0,1,2. . . ), where tP = 5.3×10 sec is the Planck time, a Planckian black hole, which is in fact the 1-qubit state, acts as a creation operator and supplies the quantum register with extra qubits. At time t0 = tP the quantum gravity register will consist of 1 qubit, at time t1 = 2tP the quantum gravity register will consist of 4 qubits, at time t2 = 3tP , the quantum gravity register will consist of 9 qubits, and so on. In general, at time tn, the quantum gravity register will consist of N = (n + 1)2 qubits. We will call |Ni the quantum state of N =(n + 1)2 qubits. Now, let us consider a de Sitter horizon|Ψ(tn)i at time tn, with a dis- 2 2 ∼ −33 crete area: An = (n + 1) LP of N pixels (where LP = 1.6 × 10 cm is the Planck length). By the quantum holographic principle [1] we associate N qubits to the th n de Sitter horizon: |Ni ≡ |Ψ(tn)i. The 1-qubit state is given by : |1i = Had |0i where Had is the Hadamard gate, which is a very important gate for quantum algorithms. Then, the state |Ni can be expressed as: |Ni = (Had |0i)N . As time is discrete, there will be no continuous time evolution, therefore there will not be a physical Hamiltonian which generates the time evolution according to Schrodinger’s equation. In [1] we considered discrete unitary evolution operators Enm between two Hilbert spaces Hn and Hm associated with two causally related ‘events’ |Ψni and |Ψmi. These ”events” are de Sitter horizon states at times tn and tm respectively, with ≤ the causal relation: |Ψni ≤ |Ψmi, for tn tm. The discrete evolution operators: Enm : Hn → Hm are the logic quantum gates for the quantum gravity M−N 2 2 register: Enm = (Had |0i) (with M = (m + 1) , N = (n + 1) . The discrete time evolution is: E0n |0i = |Ni = |Ψfini. As the time evolution is discrete, the quantum gravity register resembles more a quantum cellular automata than a quantum computer. Moreover, the quantum gravity register has the peculiarity to grow at each time step (it is self-producing).

4. Quantum Gravity Computation and Cybernetics The quantum gravity registers are not proper quantum computers. Basi- cally, they better resemble quantum cellular automata, as time is discrete. Quantum gravity registers do perform quantum computation, but in a rather particular way, that we shall call quantum gravity computation 318 Paola A. Zizzi

(QGC). The peculiarity of a quantum system which performs QGC, is that it shares some features of self-organizing systems. We recall that self- organization is a process of evolution taking place basically inside the system, with minimal or even null effect of the environment. In fact, the dynamical behaviour of quantum gravity registers follows some cybernetic principles: i) Autocatalytic growth At each computational time step, the presence of a Planckian black hole (which acts as a creation operator), makes the quantum gravity register grow autocatalytically. As N qubits represent here a de Sitter horizon with an area of N pixels, the autocatalytic growth, in this case, is exponential expansion, i.e., inflation. ii) Autopoiesis The quantum gravity register produces itself. The components of the quantum gravity register generate recursively the same network of processes (applications of the Hadamard gate to the vacuum state) which produced them. In this case recursion is defining the program in such a way that it may call itself:

|Ni = (Hadamard |0i)N . iii) Self-similarity This model of the early inﬂationary universe is based on the holographic principle [2] more in particular, on the quantum holographic principle [1]. But each part of a hologram carries information about the whole hologram. So, there is a physical correspondence between the parts and the whole. iv) Self-reproduction Can a quantum gravity register, as a unit, produce another unit with a similar organization? This possibility, which could be taken into account because the quantum gravity register is an autopoietic system, (and only autopoietic systems can self-reproduce), is in fact forbidden by the no- cloning theorem, i.e. quantum information cannot be copied [9]. However, there is a way out. When the selected quantum gravity register collapses to classical bits, it is not just an ordinary quantum register which collapses, but an autopoietic one. The outcomes (classical bits) carry along the autopoiesis. The resulting classical automaton is then autopoietic and, in principle, can self-reproduce. Emergence of Universe from a Quantum Network 319

5. Entanglement with the Environment This superposed state will collapse to classical bits by getting entangled with the emergent environment (radiation). This entanglement process with the environment can be interpreted [1] as the action of a XOR (or controlled NOT) gate, which gives the output of the quantum computation in terms of classical bits: the source of classical information in the post-inflationary universe. This phase can be illustrated in terms of Cellular Automata. Cellular automata (CA) were originally conceived by von Neumann [10] to provide a mathematical framework for the study of complex systems. A cellular automata is a regular spatial lattice where cells can have any of a finite number of states.The state of a cell at time tn depends only on its own state and on the states of its nerby neighbors at time tn−1 (with n ∈ Z). All the cells are identically programmed. The program is the set of rules defining how the state of a cell changes with respect of its current state, and that of its neighbours. It holds that the classical picture of holography (given in terms of classical bits) can be described by a classical CA. States: 0 or 1 (‘off’ or ‘on’). Neighbours: n at each time step tn = (n + 1)tP (this CA is autopoietic and grows With time). Rules: As there are two possible states for each cell, there are 2n+1 rules required. • 1 → 1 t = t 2 rules: 0 P 0 → 0 11 → 0 • 10 → 1 t = 2t 4 rules: 1 P ◦ 01 → 1 00 → 0 000 → 0 010 → 1 • 001 → 1 011 → 0 t = 3t 8 rules: , 2 P ◦ ◦ 100 → 1 110 → 0 101 → 1 111 → 0 • ◦ t = 4t 16 rules,. . . and so on. 3 P ◦ ◦ The rules force patterns to emerge (self-organization). By taking into account the ‘classical’ holographic principle, we are lead to believe That at the end of inflation, the universe starts to behave as a classic CA, which self -organizes and evolves complexity and structure. 320 Paola A. Zizzi

We call it ‘Classical Holographic Cellular Automata’ (CHCA). It should be noted that the CHCA is made out of the bits which are the outcomes of the collapse of the qubits of the quantum gravity register which is an autopoietic quantum system. Then, the CHCA is an autopoietic classical system. There are two important consequences. i) The CHCA, being autopoietic, undergoes autocatalytic growth, and the classical universe is still expanding. However, as classical computation is slower than quantum computation, the expansion is not anymore exponential . ii) The CHCA, being a classical autopoietic system, can self-reproduce. The produced units will be able to perform both quantum and classical computation. We conclude by saying that in our model, the post-inﬂationary universe follows the laws of classical complex adaptive systems (systems at the edge of chaos).

6. Quantum Computational Aspects of Space-time Foam It was recently shown by Jack Ng [5], that space-time foam provides limits to quantum computation. Here we investigate the quantum computational aspects of spacetime foam in the context of the QGN model. We identify the node ‘n’ as the quantum fluctuation of the metric on nt 1 the n slice [8] that is: ∆gn = n+1 . The energy of node ”n” is then the nt EP energy of the n quantum fluctuation of the metric: En = n+1 , where 19 EP ≈ 10 GeV is the Planck energy. Thus, the energy of node ‘n’ is the Planck energy divided by the number n+1 of virtual Planckian black holes (connecting links of the kind: L−1,0,L0,1,L1,2, ....Ln−1,n). The energy of the nodes decreases as the network grows. This fact is due to the increase of the number of virtual black holes. In particular, node ‘0’ has the maximum energy E0 = EP . The energy of the nodes decreases as the network grows. This fact is due to the increase of the number of virtual black holes. We define ‘quantum foam’ the ensemble of the quantum fluctuations of the metric (the nodes) and the virtual black holes (connecting links: L−1,0,L0,1,L1,2, ....Ln−1,n...). What we call here ‘quantum foam’ is based on the definition of ‘space-time foam’ due to Wheeler’ [6], where quantum fluctuations of the metric induce fluctuations of the topology. Moreover, we enclose the quantum fluctuations of the vacuum (virtual black holes) in Emergence of Universe from a Quantum Network 321 the definition of the ‘quantum foam’ as the two kinds of fluctuations are strictly related to each other in this context. Hawking [10] also considered virtual black holes as constituents of space-time foam. He showed that the topology of virtual black holes is: S2 × S2 (the 4-dimensional space-time being then a simply connected manifold). It should be noticed that these ”bubbles” are not solutions of Einstein’s field equations. However, in our context at least, this is not the whole story. In fact, at node ”0” one Eu- clidean Schwarzschild black hole of Planck size comes into existence, and it is not virtual, as it will give rise [8] to a quantum de Sitter space-time. Its 1 1 2 topology is: R(space) × S(time) × S . This micro-black hole at node ‘0’ has an horizon area of one pixel (one unit of Planck area) which, by the quantum holographic principle [1], encodes one unit of quantum information (one qubit). Our claim is that, virtual black holes carry virtual information which is transformed into real information by the quantum fluctuations of the metric (the logic gates). In the transformation from virtual to real information, the topology itself must change. Space-time foam invokes the existence of a minimum length scale [11], which is the Planck length. The Planck length is the length scale of quantum gravity. Given the Planck length as the minimum length scale, the proper distance between two events will never decrease beyond Planck length, and the uncertainty relation holds: ∆x ≥ lP . Moreover, in our case, an analogous uncertainty relation holds for time intervals: ∆t ≥ tP , as we are dealing with time quantized in Planck time units. Then we get the simultaneous bound: ∆x∆t ≥ lP tP . For quantized space- time in Planck units, it is: ∆x = xm −xn = (m−n)lP ; ∆t = tm −tn = (m− 2 n)tP , and the simultaneous bound becomes: ∆x∆t = (m−n) lP tP ≥ lP tP . This bound is saturated for m=n+1. √ −1 −1 √ Notice that we can also write: xn = ∆gn lP = I lP ; tn = ∆gn tP = −2 I tP , from which it follows: xntn = ~∆g = ~I, where I is the quantum information. Thus, the quantum information encoded in quantum space- time, is a consequence of the foamy structure of quantum space-time itself. Notice that for n=0, we get: x0t0 = ~, with I=1 (at the very Planck scale, quantum information encoded in the foam is one qubit).

7. The QGN as the Ultimate Network The speed of computation ν of a system of average energy E , is bounded as [3]: ν ≤ 2E/π~. Moreover, entropy limits the memory space I [4] as 322 Paola A. Zizzi the maximum entropy is given by: SMax = kBI ln 2, where kB = 1.38 × 10−23J/K is the Boltzmann’ s constant. As Seth Lloyd showed [4], these bounds are saturated by black holes, which can then be regarded as the ultimate computers. In what follows, however, we will use the notations in [5], which are more suitable to our purpose. As Ng showed [5], the bounds on speed of computation ν and information I can be reformulated respectively 2 P ~ as: ν ≤ ; I ≤ 2 , where P is the mean input power. ~ P tP These two bounds lead to a simultaneous bound [5] on the information I 2 1 and the speed of computation ν: Iv ≤ 2 . It is easily shown that the speed tP of computation νn and the quantum information In of the QGN, saturate the three bounds above at each time step tn:

2 Pn En 1 ~ ~ 2 2 1 v = = = ; In = = = (n + 1) ; Inv = . n 2 2 En 2 n 2 ~ ~tn tn Pnt t t P tn P P As we have seen, at each time step, the QGN saturates the quantum limits to computation. Then, it can be regarded as a ultimate Internet, in the same sense that black holes can be regarded as ultimate computers. Moreover, following Ng [5], we deﬁne the number of operations per time unit:v ¯ = Iv. In the context of the QGN, we have then:√ v ¯n = Invn. Moreover, it holds.v ¯n/ν¯0 = n + 1 = I, wherev ¯0 is the number of −1 43 −1 operations per qubit at n=0:v ¯0 = I0v0; v0 = tP = 10 sec ; I0 = 1, 43 −1 from which it follows:v ¯0 = 10 sec . 19 If the average amount of energy is the Planck energy: EP ≈ 10 GeV ≈ 9 2EP 10 J, then, a Planckian black hole saturates the bound: vP = π~ ≈ 109J 43 −1 10−34J sec ≈ 10 sec . This is the maximum speed of computation available in nature. Also, from the simultaneous bound, it follows that the memory space of −2 tP a Planckian black hole is one qubit: IP = 2 = 1. Then, the qubit, the unit νP of quantum information, is the minimum amount of information available in nature. This result is in agreement with the quantum holographic principle [1], which states that a Planckian black hole, having a horizon area of one pixel, can encode only one qubit.

8. The QGN and the Problem of Λ We calculate the present value of the quantized cosmological constant with: 60 −52 −2 nnow = 9 × 10 , and we get: Λnow = 1.25 × 10 m . Emergence of Universe from a Quantum Network 323

Λc2 1 2 2 −1 Also, we obtain: ΩΛ = 2 ≈ Λc tnow where tnow ≈ H0 ≈ 3 × 3H0 3 17 −1 10 h sec, H0 being the Hubble constant and h a dimensionless parameter in the range: 0.58 ≤ h ≤ 0.72. By choosing h=0.65, we get: ΩΛ ≈ 0.7. From ρΛ the relation: ΩΛ = , where ρΛ is the vacuum energy density and ρc is the ρc critical density, it follows: ρΛ ≈ 0.7ρc in agreement with the Type Ia SN observation data [12]. From this result, the connecting links (virtual states) of the QGN look like to be still active. But the value of the total entropy is too big, for 60 120 nnow = 10 : it would be Snow = 10 ln 2, indeed a huge amount of entropy. We believe that at some earlier time, the QGN decohered. In this scenario, the free links were not activated anymore by the nodes, since decoherence time. We can visualize the QGN after decoherence as a regular lattice, the connected part of the QGN itself. The energy of nodes at present time would be: ENODES = EP = now nnow 10−41GeV ≈ 10−51J. One would expect that at present, such a weak energy of the nodes would prevent quantum computation. But this is not true: for n → ∞, quantum information I will grow to infinity as n2, while the energy of nodes will decrease to zero as 1/n. In fact, a low energy just reflects in a low speed of computation ν but not in a low amount of information. The speed of computation at present would ∼ −17 −1 −1 be: νnow = 10 sec . It should be noticed that in our case, νnow is the −1 17 age of the universe: νnow ≈ 10 sec . Then, from the simultaneous bound to computation we get: Inow ≤ 10120, where in fact the bound is saturated because of the previous ar- 120 guments. We would get the huge total entropy Snow = 10 ln 2 if the machinery did not stop at some earlier time. In [13] Lloyd computed the maximum possible number of bits which the universe registered since the Big Bang until now, which he found to be 10120, equal to the number of elementary operations, which he calculated by 2 the use of the relation: number of bits = (t/tP ) , where in our case t is the 2 quantized time tn, and the relation above gives explicitly: (tn/tP ) = (n + 1)2 ≡ N. The main difference is that Lloyd does not consider decoherence at the end of inflation, (while we do, as it will be shown in the next section).

9. Self-decoherence The QGN can be theoretically divided into two sub-systems: the connected part, made of connecting links and nodes, (quantum fluctuations of the 324 Paola A. Zizzi vacuum and quantum fluctuations of the metric respectively) and the disconnected part: the free links (qubits). The connected part can be thought as the ‘environment’, while the disconnected part can be thought as the quantum state. The virtual black holes in between space-time slices, together with the quantum fluctuations of the metric on the slices, constitute space-time foam. Decoherence, here, can only be caused by space-time foam.The fact that virtual black holes can induce decoherence was suggested by Hawking [14]. However, as the quantum foam (virtual black holes and quantum fluctuations of the metric) resides in the connected sub-graph, which is part of the QGN, the QGN as a whole, undergoes self-decoherence. Although the details of the mechanism of self-decoherence are not yet properly understood, we can get a clue by considering entropy arguments, as it was already pointed out in [1] and [7]. The quantum entropy of N=I qubits is: S=Iln2. Thus, if the QGN did never decohere, now the maximum entropy would 120 60 be S = 10 ln 2 ( with nnow = 10 ). To get the actual entropy, one should 120 120 compute it as: Snow = 10 ln 2/Sdecoher = 10 /IMax. If one agrees with Penrose who claims [15] that the entropy now should be of order 10101, this corresponds to the maximum amount of quantum 19 9 information at the moment of decoherence: IMax = 10 (ncr ≈ 10 ). where ncr stands for the critical number of nodes which are needed to process the maximum quantum information, IMax. It follows that the early −34 quantum computational universe decohered at tdecoh ≈ 10 sec . Moreover, we find that at the moment of decoherence (n = 109) the mean energy is: 10 −13 Edecoh ≈ 10 GeV ≈ 1J (corresponding to a rest mass mdecoh ≈ 10 g) −1 and that the number of operations per qubit is:¯vdecoh = Imaxtdecoh ≈ 1053 sec−1. Finally, it should be noticed that, as the QGN describes the early inflationary universe, decoherence time corresponds to the time of the end of inflation, and the decoherence energy corresponds to the reheating energy.

References 1. P. A. Zizzi (2000) Holography, Quantum Geometry and Quantum Informa- tion Theory, gr-qc/9907063, Entropy 2 , 39. Emergence of Universe from a Quantum Network 325

2. G. ’ t Hooft, Dimensional reduction in Quantum Gravity, gr-qc/9310026; G. ’ t Hooft, ”The Holographic Principle”, hep-th/0003004; L. Susskind, ”The World as a Hologram”, hep-th/9409089. 3. N. Margolous, L. B. Levitin (1988) Physica D 120, 188. 4. S. Lloyd (2000) Ultimate physical limits to computation, quant-ph/9908043; Nature 406, 1047. 5. Y. J. Ng, Clocks, computers, black holes, spacetime foam and holographic principle, hep-th/0010234; Y. J. Ng, From computation to black holes and space-time foam, gr- qc/0006105. 6. J. A. Wheeler (1962) Geometrodynamics, Academic Press, New York. 7. P. Zizzi, The Early Universe as a Quantum Growing Network, gr- qc/0103002. 8. P. A. Zizzi (1999) Quantum Foam and de Sitter-like Universe, hep- th/9808180; Int. J. Theor. Phys., 38, 2333. 9. W.K. Wooters, W.H. Zurek (1982) Nature, 299, pp 802–819. 10. S. W. Hawking, D. N. Page and C. N. Pope (1980) Quantum gravitational bubbles, Nucl. Phys. B 170 , 283; S. W. Hawking (1996) Virtual black holes, Phys. Rev.D 53, 3099. 11. Luis J. Garay, Quantum gravity and minimum length, gr-qc/9403008. 12. S. Perlmutter et al. (1998) Nature 391, 51; B. P. Schmidt et al. (1998) Astrophys. J. 507, 46. 13. Seth Lloyd, Computational Capacity of the Universe, quant-ph/0110141. 14. S. W. Hawking (1993) Black Holes and Baby Universes and other Essays, Bantam, New York. 15. R. Penrose (1989) The Emperor ’s New Mind, Oxford University Press. This page intentionally left blank 327

Occam’s Razor Revisited: Simplicity vs. Complexity in Biology

Joseph P. Zbilut Department of Molecular Biophysics and Physiology Rush University Medical Center 1653 West Congress, Chicago, IL 60612 USA [email protected]

It would seem that the topics of complexity and emergence have become connected insofar as emergence would appear to be a unique feature of complexity. Yet it would also seem that the topic of simplicity should also be discussed as a natural counterpoint. In fact, all of these concepts are often presented as certainties without suﬃcient explication of their bases. It is suggested that there has been inadequate attention paid to these concepts vis-`a-vismodern cognitive science and the theory of mathematics. By approaching these topics from this perspective it may be that an alternate understanding can be obtained which devolves from a less mechanistic view of dynamics.

Keywords: Biophysics; Complex Systems; Nonlinear Dynamics PACS(2006): 87.15.v; 89.75.k; 82.39.Rt; 89.75.Fb; 05.45.a

1. Introduction Although the idea of complexity and emergence has found some favor in biological disciplines to describe the myriad processes found in organisms, many questions remain. The explosion of data concerning molecular biology has forced many fatigued scientists to ask if there are any over-all guiding principles, or “laws” akin to those sought by physicists and chemists. Im- plicit in this query is the additional question of how detailed must a model be for it to accurately describe a biological phenomenon. Clearly, some guiding principles could allow for reduced models which could be managed more easily. This is especially the case where mathematical models are the case. Currently, many mathematical models are high dimensional requiring considerable computational time. Yet the results of such models are often questioned for the obvious reason that even very small errors can quickly be magnified. There is also the very real “noise” problem: how does one model 328 Joseph P. Zbilut the naturally occurring noise found in biological systems? Different kinds of noise at different length scales makes the problem extremely important[1]. One can readily appreciate why many have invoked the “science of complexity” in this arena. If complexity exhibits a universal paradigm, and it is appropriate for the biological arena, then perhaps there may be a way of identifying guiding principles to reduce the dimensionalities of these biological problems. To this end, many have looked for signatures of this complexity, such as scale-free, network dynamics. Scale-free networks have been claimed for metabolic processes, organ systems, and regulatory structures. Very often these networks are determined in the context of “power laws,” whereby some feature of the observed phenomenon falls off as an exponential power law. The difficulty with such observations is that they are simply an observation of a kind of distribution, in the same way that it can be said that some variable exhibits a normal distribution, or a Poisson distribution. It says nothing in particular about the underlying mechanisms of the distribution. Indeed, such power laws are found in many phenomena and with a myriad of different mechanisms responsible for them—they are not necessarily unique.

2. Mathematics and Perceptions Implicit in such observations is the belief that the mathematics reveals some underlying universal properties not clearly seen in the empirical observations. This is not surprising given that the foundation of classical mathematics is Platonist. Universals and laws are quite at home in Plato’s cave. There is, however, a new challenge to this Platonistic foundation to mathematics, and it comes from one of the objects biologists wish to understand; namely, the human brain [2]. For a long time, it has been assumed that logic (or a logic) is independent of a thinker: “correct” reason is not dependent upon the peculiarities of a human brain–it transcends it and can be veriﬁed by any other correctly thinking human brain. However, in recent decades, cognitive neuroscientists have challenged this indirectly. Many studies have shown that the human brain often sees patterns where there are none. And as David Ruelle has pointed out, our mathematics comes out of physical reality, ﬁltered by the brain. In other words, we do the kind of mathematics we do because we “see” the mathematics in our reality—whatever this might be (see, e.g. [3]). This is an important consideration relative to the quest for biological universals, because it parallels the experience of physicists. Perhaps Occam’s Razor Revisited: Simplicity vs. Complexity in Biology 329

Heisenberg became the first widely known person to express it in his Un- certainty Principle, but the point is that an “observer” changes the thing observed by virtue of the observation. This is historically a remarkable statement which has often been glossed over by many scientists, although there have been a few who have attempted to comment upon it–mainly in the context of quantum physics. This observation puts into question the entire notion of “objectivity.” And certainly, it has been the progenitor of many so-called “weird” effects of quantum science. Although the biological experience is not quite the same as the physicist’s, it points out, to a certain extent, that the scientist as observer, “fil- ters” perceptions through the activity of the brain. It is a brain which likes to see patterns (meaning ?) even where there is none. Can this be the reason why so many scientists still pursue mechanistic mathematical models even when quantum physics has placed serious questions on such an en- deavor? As is well known, strictly speaking, Newton’s Laws are incorrect (as compared to quantum physics), but the errors are minimal and can thus be used. But does this gloss allow for a continued myopic view of various phenomena—especially in the life sciences. Consider that the Ptolemaic view of the Universe held sway because in terms of its predictions, the errors were few. The fact that biological systems must be continually adaptable, and in a sense “contingent” would tend to mitigate the universalist view. And what about the multiple causes of “noise” in the biological world. Of- ten, noise is lumped into one big “black box” of unknowns and sequestered in differential stochastic equations. Yet the biologist is confronted with the incontrovertible fact that higher biological systems coordinate many millions of processes simultaneously in a sea of noise to effect what are obviously deterministic processes. Thus neuronal synapses have a degree of randomness associated with them, as do ion channels, etc., yet they are capable of developing and coordinat- ing “messages” that can ultimately result in such things as coordinated movement and thought.

3. Simplicity While complexity has been put forward as an important paradigm for biological systems, there is still the old dictum of the medieval Franciscan philosopher, William of Occam, that states, “pluritas non est ponenda sine necessitate,” [4] otherwise known as “Occam’s razor.” And indeed, scientists of different fields are wont to quote this dictum of parsimony when deciding on multiple potential models. It was not however until the 1940s 330 Joseph P. Zbilut and 50s that there was significant attention paid to the idea of complexity [5]. To be sure, if simplicity has been such an important guiding principle, how has its complement suddenly come to the fore? Clearly, simplicity has not been abandoned, inasmuch as such luminaries as Einstein and Hawking have invoked it. But where does complexity fit in? Perhaps an analogy with medieval tapestries may clarify the issue: simple things are not necessarily more intelligible than complex ones. A tapestry is composed of numerous threads of different colors, which may be continuous throughout, but which appear at various points to trace out a figure or letter, etc. Thus apparent intricacy is really specious, and from single threads emerges a unique story. And certainly emergent properties are often cited as being a hallmark of complex systems: traits not seen in the individual parts are demonstrated only in the complete system. Yet what could be simpler than the periodic table of elements: proton number accounts for a periodicity and electron shell completeness to account for a very simple ordering of elements. From this basic arrangement emerge the elements’ visibly different characteristics. It is still not clear how all this allows for such astonishingly different elemental attributes and perceptions. These attributes would appear to be “emergent” properties. If the research on cognitive neurophysiology is to be believed, it may serve as a cautionary tale for a whole-hearted acceptance of bio-complexity. At the very least, the issue is far from decided in that a universally accepted definition of complexity is not at hand. What is known, is that it is a big problem based on massive amounts of data. Still, it should be recalled that just because something is big, does not necessarily mean that it is complex; nor does it mean that just because something is small does not mean it is not complex. The noted artificial intelligence pioneer and Nobel laureate, Herbert Si- mon, in his dissertation work discovered that when confronted with massive amounts of data, the human mind was inclined to pull back to a smaller subset of data which was well appreciated to make decisions [6]. A similar tendency may be operative with the suddenness of massive amounts of molecular data. There may be a psychological comfort in looking for “basic principles.” Still there may yet be another approach. Given that there is at least some determinism, and some “noise,” an approach which favors “piece-wise determinism may provide a way which recognizes the essential “building block” nature of many processes, while at the same time recognizing that building blocks are not always easy to assemble. Occam’s Razor Revisited: Simplicity vs. Complexity in Biology 331

It has been argued that if the Lipschitz conditions for diﬀerential equations are relaxed, this allows for a simpler way to construct processes. At the same time it can more easily model the massive asynchronous parallelism found in biological processes [7,8].

4. Piece-wise Determinism The governing equations of classical dynamics based upon Newton’s laws: d ∂L ∂L ∂R = − , i = 1, 2, ..., n (1) dt ∂˙qi ∂qi ∂˙q i where L is the Lagrangian, and q,q˙i are the generalized coordinates and velocities, include a dissipation function R(q ˙iq˙j) which is associated with the friction forces: ∂R Fi(q ˙1, q˙2, ··· q˙n) = − . (2) ∂˙qi But the functions (2) do not necessarily follow from Newton’s laws, and, strictly speaking, some additional assumptions need to be made in order to deﬁne it. The “natural” assumption (which has been never challenged) is that these functions can be expanded in Taylor series with respect to an equilibrium state

q˙i = 0. (3) Obviously this requires the existence of derivatives:

∂Fi | |< ∞ atq ˙j → 0 (4) ∂˙qj i.e. Fi must satisfy the Lipschitz condition. This condition allows one to describe the Newtonian dynamics within the mathematical framework of classical theory of differential equations. However, there is a certain price to be paid for such a mathematical “convenience”: the Newtonian dynamics with dissipative forces remain fully reversible in the sense that the time- backward motion can be obtained from the governing equations by time inversion, t → −t. In this view, future and past play the same role: nothing can appear in future which could not already exist in past since the trajectories followed by particles can never cross (unless t → ±∞). This means that classical dynamics cannot explain the emergence of new dynamical patterns in nature. To simplify the exposition, consider a one-dimensional motion of a particle decelerated by a friction force: mv˙ = F (v) (5) 332 Joseph P. Zbilut in which m is mass, and v is velocity. Invoking the assumption (4 ) one can linearize the force F with respect to the equilibrium v = 0: ∂F F → −αv at v → 0, α = −( ) > 0 (6) ∂v v=0 and the solution to (5) for v → 0 is: − α t v = v0e m → 0 at t → ∞, v0 = v(0). (7) As follows from (7), the equilibrium v = 0 cannot be approached in finite time. The usual explanation of such an effect is that, to accuracy of our limited scale of observation, the particle “actually” approaches the equilibrium in finite time. In other words, eventually the trajectories (7) and v = 0 become so close that we cannot distinguish them. The same type of explanation is used for the emergence of chaos: if two trajectories originally are “very close,” and then they diverge exponentially, the same initial conditions can be applied to either of them, and therefore, the motion cannot be traced. Hence, there are variety of phenomena whose explanations cannot be based directly upon classical dynamics: in addition, they require some additional qualifications or distinctions, e.g. about a scale of observation, “very close” trajectories, etc. A different structure of dissipation forces can be described which eliminates the paradox discussed above and makes the Newtonian dynamics irreversible. The main properties of the new structure are based upon a violation of the Lipschitz condition (4). Turning to the example (5), let us assume that p F = −αv − α vk, α ¿ α, k = < 1, p À 1 (8) 1 1 p + 2 in which p is an odd number. By selecting large p, one can make k close to 1 so that (6) and (8) will be almost identical everywhere excluding a small neighborhood of the equilibrium point v = 0, while, as follows from (8), at this point: ∂F | |= (α + kα vk−1) → ∞ at v → 0, i.e.F → −α1vk at v → 0. (9) ∂v 1 Consequently, the condition (4) is violated, the friction force grows sharply at the equilibrium point, and then it gradually approaches the straight line (6). This effect can be interpreted as a jump from static to kinetic friction. It appears that this “small” difference between the friction forces (6), (8) leads to fundamental changes in Newtonian dynamics. Occam’s Razor Revisited: Simplicity vs. Complexity in Biology 333

Firstly, the time for approaching the equilibrium v = 0 becomes ﬁnite. Indeed, as follows from (5) and (9):

Z 0 1 −k mdv mv 0 t0 = − − k = < ∞. (10) v0 α1v α1(1 − k) Obviously this integral diverges in the classical case. Secondly, the motion described by (5), (8) has a singular solution v ≡ 0 and a regular solution

1−k α1 1 v = [v − (1 − k)t] 1−k . (11) 0 m In a finite time the motion can reach the equilibrium and switch to the singular solution, and this switch is irreversible. It is interesting to note that the time-backward motion α v = {[v1−k − (1 − k)(−t)]p+2}1/2 (12) − 0 m is imaginary. [One can verify that the classical version of this motion (7) is fully reversible if t < ∞]. The equilibrium point v = 0 of (8) represents a “terminal” attractor which is “infinitely” stable and is intersected by all the attracted transients. Therefore, the uniqueness of the solution at v = 0 is violated, and the motion for t < t0 is totally “forgotten.” This is a mathematical implication of irreversibility of the dynamics (8). So far we were concerned with stabilizing effects of dissipative forces. However, as well-known from dynamics of non-conservative systems, these forces can destabilize the motion when they feed the external energy into the system (e.g. the transmission of energy from laminar to turbulent flow in fluid dynamics, or from rotations to oscillations in dynamics of flexible systems). In order to capture the fundamental properties of these effects in the case of a “terminal” dissipative force (8) by using the simplest mathematical model, consider(5) and assume that now the friction force feeds energy into the system: p mv˙ = α vk, k = < 1, v → 0. (13) 1 p + 2 One can verify that for (13) the equilibrium point v = 0 becomes a terminal repeller, and since dv˙ kα = 1 vk−1 → ∞ at v → 0 (14) dv m 334 Joseph P. Zbilut it is “infinitely” unstable. If the initial condition is infinitely close to this repeller, the transient solution will escape it during a finite time period: Z vo 1−k mdv mv o t0 = k = < ∞ (15) ε→0 α1v α1(1 − k) while for a regular repeller, the time would be infinite. As in the case of a terminal attractor, here the motion is also irreversible: the solution α1 1 v = ±[ (1 − k)t] 1−k (16) m and the solution (11) are always separated by the singular solution v ≡ 0, and each of them cannot be obtained from another by time inversion: the trajectory of attraction and repulsion never coincide. But in addition to this, terminal repellers possess even more surprising characteristics: the solution (16) becomes totally unpredictable. Indeed, two different motions described by the solution (16) are possible for “almost the same” (v0 = +ε → 0, or v0 = −ε → 0 at t → 0) initial conditions. The most essential property of this result is that the divergence of these two solutions is characterized by an unbounded rate: 1 t 1/1−k σ = lim ( ln ) → ∞ at | v0 |→ 0. (17) t→t0 t | v0 |

In contrast to the classical case where t0 → ∞, here σ can be defined in an arbitrarily small time interval t0, since during this interval the initial infinitesimal distance between the solutions becomes finite. Thus, a terminal repeller represents a vanishingly short, but infinitely powerful “pulse of unpredictability” which is pumped into the system via terminal dissipative forces. Obviously failure of the uniqueness of the solution here results from the violation of the Lipschitz condition (4) at v = 0.

5. Neural Example Traditional “neural net” models of cognition have some diﬃculties, including diﬃculties dealing with noise, parallelism, etc. A “piece-wise deterministic,” or “terminal” model of neurodynamics can exhibit a stochastic attractor as a universal tool for generalization. Indeed, in contradistinction to chaotic attractors of deterministic dynamics, stochastic attractors can provide any arbitrarily prescribed probability distributions by an appropriate choice of (fully deterministic!) synaptic weights Tij. Occam’s Razor Revisited: Simplicity vs. Complexity in Biology 335

The information stored in a stochastic attractor can be measured by the entropy H via the probabilistic structure of this attractor: X X H(Xi,X2, ..., Xn) = − ... f(x1, ..., xn) log f(x1, ..., xn) (18)

X1 Xn where the joint density f(x1, ..., xn) is uniquely deﬁned by the synaptic weights Tij. Random neurodynamical systems can have several, or even, an

Fig. 1. Piece-wise dynamics with multiplicative probabilities (top) result in a stochastic attractor (bottom). inﬁnite number of stochastic attractors (Fig. 1). For instance, a dynamical system

k √ x˙ 1 = γ1 sin [ ω sin(x1 + x2)] sin ωt (19)

k √ x˙ 2 = γ2 sin [ ω sin(x1 + x2)] sin ωt (20) has stochastic attractors with the following densities:

f(x1, x2) = 0.5 | cos(x1 + x2) cos(x1 − x2) | (21) 336 Joseph P. Zbilut

πm π(m + 2) πm π(m + 2) 1 < x + x < 1 , 2 < x − x < 2 (22) 2 1 2 2 2 1 2 2

m1 = · · · −7, −3, 1, 5, 9, etc.; m2 = · · · −5, −1, 3, 5, 7, ··· etc. (23)

The solution (21) represents a stationary stochastic process, which at- tracts all the solutions with initial conditions within the area (22). Each pair m1 and m2 from the sequences (23) deﬁnes a corresponding stochastic attractor with the joint density (21). Hence, the dynamical system (19–20) is capable of discrimination between diﬀerent stochastic patterns, and therefore, it performs pattern recognition on the level of classes. There is some experimental evidence that this may be the case for real neural systems. Since this would be a multiplicative process, one would expect power law behaviour of the observed variables as well as a lognormal distribution. In fact, this is what has been routinely observed in neural investigations [9].

Table 1.

Non-Lipschitz Dynamics Dependent upon singularity No attractor in the usual sense Uncertainty at singular points Dynamics is well behaved away from singular points Random spread of points in region of phase space (stochastic attractor) Information inﬁnite at singularity (knowledge of past is zero) Countable combinatorial set Predictability based on probability evolution Easily controlled at singularity (not on trajectory)

6. Metabolic Nets Considerable interest has been paid to metabolic scale free metabolic networks. The architecture of these metabolic nets is in many ways similar to neural nets. Indeed, several studies based on real data point to lognormal distributions and power laws [10,11]. Thus there is a good reason to consider the possibility of underlying multiplicative processes. Furthermore it should be stressed that the beneﬁt of piece-wise modeling is that such dynamics are much more forgiving of “errors” (Table 1). Occam’s Razor Revisited: Simplicity vs. Complexity in Biology 337

7. Conclusion What is notable about piece-wise determinism is that it acknowledges that a given corpus can constitute a very large amount of inter-related data, while at the same time posing the linkages in a way which does not tie the deterministic pieces to the past. A further benefit is that “control” of the dynamics is exquisite at the singularities, while being relatively impervious to the effects of noise once a trajectory is chosen. The stochastics of one level evolve into a progressively “deterministic” process at a higher level as can be seen through lognormal distributions. Thus a very large problem can be reduced to one of combinatorics and probabilities. This last point needs to be emphasized since there is a persistent tendency to view many of the implied dynamics through the legacy of Laplace. This is to say that there is a tendency to model these dynamics with traditional differential equations. As has been suggested, however, this need not be so. Piece- wise deterministic dynamics can produce similar results, but are relatively simpler, and perhaps heuristically more satisfying, given the underlying stochastics [12]. It should be also noted that although the singularities have been modeled with Lagrangians, different types of mathematics may be more appropriate with other systems. We note that topology admits numerous singularities without metrics. The experimental indications are that in intransitive situations (dynamical systems with fixed points of rotation and translation) 3 spatial dimensions could best represent physical situations if the metric signature is not Euclidean geometry, but instead of the Minkowski type Non-Euclidean geometry [13].

Acknowledgments Discussions with Sandro Giuliani, Robert Kiehn, David Ruelle, Chuck Web- ber, and Michail Zak have been useful and greatly appreciated.

References 1. E.F. Keller (2007) Nature, 445, 603. 2. D. Ruelle (2000) Nieuw Archief voor Wiskunde, 5/1: 30-33. 3. P. Sterzer, P. A. Kleinschmidt (2007) PNAS, 104, 323-328. 4. W.M. Thorburn (1918) Mind, 217, 345–353. 5. J.P. Zbilut, A. Giuliani (2005) Encyclopedia of Nonlinear Science, A. Scott (Ed.), New York and London, Routledge, pp 7–9 6. H. Simon (1997) Administrative Behavior: A Study of Decision-Making Pro- cesses in Administrative Organizations. 4th Ed. New York, The Free Press. 338 Joseph P. Zbilut

7. M. Zak, J.P. Zbilut, R.E. Meyers (1997) From Instability to Intelligence: Complexity and Predictability in Nonlinear Dynamics, (Lecture Notes in Physics: New Series m 49). Springer Verlag, Berlin Heidelberg New York. 8. J.P. Zbilut (2004) Unstable Singularities and Randomness. Elsevier, Ams- terdam. 9. P. Cao, Y. Gu, S-R. Wang (2004) J Neurosci 24, pp 7690–7698. 10. K. Kaneko (2003) Phys Rev E 68, 031909. 11. C. Furusawa, K. Kaneko (2006) Phys Rev E 73, 011912. 12. D. Mumford (1999) Dawing of the Age of Stochasticity [Lecture delivered to the Accademia Nazionale dei Lincei, May], in Mathematics: Frontiers and Perspectives, (V. Arnold, M. Atiyah, P. Lax, B. Mazur, Eds.)(2000) AMS. 13. R. Kiehn, Non-Equilibirum Thermodynamic Processess, Vol. 1: Non- Equilibrium Systems and Irreversible Processes. Copyright R. Kiehn, 2003– 2006. Lulu Enterprises Inc., Morrisville, NC; (http//:www.lulu.com/kiehn). 339

Order in the Nothing: Autopoiesis and the Organizational Characterization of the Living

Leonardo Bicha and Luisa Damianob aCE.R..CO.-Center for Research on the Anthropology and Epistemology of Complexity, University of Bergamo, Piazzale S. Agostino 2, 24129 Bergamo, Italy b Biology Department, University of RomaTre, V.le G. Marconi 446, 00146 Rome, Italy [email protected]

An approach which has the purpose to catch what characterizes the specificity of a living system, pointing out what makes it different with respect to physical and artificial systems, needs to find a new point of view — new descriptive modalities. In particular it needs to be able to describe not only the single processes which can be observed in an organism, but what integrates them in a unitary system. In order to do so, it is necessary to consider a higher level of description which takes into consideration the relations between these processes, that is the organization rather than the structure of the system. Once on this level of analysis we can focus on an abstract relational order that does not belong to the individual components and does not show itself as a pattern, but is realized and maintained in the continuous flux of processes of transformation of the constituents. Using Tibor Ganti’s words we call it “Order in the Nothing”. In order to explain this approach we analyse the historical path that generated the distinction between organization and structure and produced its most mature theoretical expression in the autopoietic biology of Humberto Maturana and Francisco Varela. We then briefly analyse Robert Rosen’s (M,R)-Systems, a formal model conceptually built with the aim to catch the organization of living beings, and which can be considered coherent with the autopoietic theory. In conclusion we will propose some remarks on these relational descriptions, pointing out their limits and their possible developments with respect to the structural thermodynamical description.

Keywords: Autonomy; Autopoiesis; (M,R)-Systems; Organization; Relational Biology PACS(2006): 89.75.k; 89.75.Fb; 82.39.Rt; 87.10.+e; 87.16.Ac 340 Leonardo Bich and Luisa Damiano

1. Introduction: Epistemological Remarks

“More is different” (P.W. Anderson) In this paper we analyse, conceptually and historically, a descriptive modality which could allow to identify, even if not yet to explain at its present state of development, what is specific of a living being. It is a relational qualitative description based on the systemic thesis that what is peculiar of an organism is not the single physical processes which take place in it, but the way these are related in order to produce and maintain the integrated biological unity they belong to. The main characteristic of the descriptive approach we derive from this hypothesis is to focus on the organization of the system. With this term we mean the topology of relations which allow us as scientific observers to identify a system as a unity belonging to a certain class, that is the class of living systems. Such a theoretical definition expresses one of the main epistemological outcomes of contemporary science: the impossibility to consider the scientific knowledge as independent from the activity of observation and catego- rization performed by the observer. With this acknowledgment comes the collapse of the classical idea which makes of the scientist an “absolute witness” of nature, whose cognitive point of view, neutral and external from the natural world, guarantees the power of intelligibility that able to access and represent a reality in itself. In the epistemology of modern science this amounts to capturing and representing the laws of nature, which are considered as pre-existent to the observer and discovered through an act of apprehension of a reality external to him (Prigogine and Stengers, 1979). A first breaking of this classical epistemology was operated by quantum physics. By showing the inseparability of the activity of observing from the object observed, it caused the subsiding of the objectivistic view of modern science. This crisis of the scientific objectivity was amplified by some branches of the contemporary science, which have been led to re-orientate the traditional epistemological axis from the classic representationism into a radical constructivism. Among them we can find a twentieth century tradition of research dedicated to the study of the biological organization, and known for having given birth to the concept of self-organization (Stengers, 1985). This branch of science has the peculiarity of having developed a scientific epistemology alternative to the modern tradition (Damiano, 2007), according to which the scientific observer does not have a direct access to reality, for he has an active role in the determination of the object he inves- tigates. He interacts with the natural world through theoretical categories which instead of representing pre-existing and pre-defined objects build Autopoiesis and the Organizational Characterization of the Living 341 reality as a set of defined objects of research. From this perspective no model or theoretical apparatus can express the reality as independent from the activity of the scientific observer (von Foerster and Zopf, 1962; Pri- gogine and Stengers, 1979; Maturana, 1988; 2000). The object treated by the science is co-construed: the observer gives it an objectual form through the categories he resorts to, while the reality, limiting the range of their applicability, defines the area in which nature can be handled as made of objects categorised by the observer. This epistemological thesis is specifically connected to the problem of the correlations between scientific disciplines, in particular those specifically concerned by this tradition: physics and biology. It is a non-reductionistic approach oriented to the establishing of communicative circuits between these disciplines: bidirectional transfers of models, questions, theoretical structures (Morin and Piattelli-Palmarini, 1974; Domouchel and Dupuy, 1983). The thesis of this paper, about the necessity of the construction of new kinds of models specific for biology, goes in fact in this direction, rejecting the reductionistic approaches which move from physics to biology such as the one promoted by molecular biology, which tried to find what is pertinent in order to explain the living on a different level from the one where life manifest itself, missing the crucial problem of the systemic unity, as the name itself shows. Biology instead can undergo a development that follows its own modalities, derived from the kind of problems it faces, that is the necessity of producing models able to explain the experience of the observer interacting with a living system. And also, the attention given to the problems which emerge from biological research can lead to new approaches to physics and consequently to open new lines of research. The first to understand this opportunity was Erwin Schrödinger,a physicist that explicitly faced the main question of biology, what is life?, in his seminal essay on the living (Schrödinger,1944). He tried to characterize the order proper of living systems, that he called “order from order” embedded in a particular rigid structure, in opposition with the statistic order of physical systems. The new descriptive modality was not supposed to be due to a new sort of force, but to the attention focused on the way living systems are built, expressed by a different concept of order. Even if he for he didn’t face the problem of the unity that constitutes the living system, he had the merit to introduce the hypothesis that looking for the laws of biology could have opened the way to the development of a new physics. This was in fact what Ilya Prigogine did (Prigogine and Stengers, 1979). 342 Leonardo Bich and Luisa Damiano

Starting from the concept of open systems proposed by von Bertalanffy’s organicistic biology in the twenties (von Bertalanffy, 1949) as one of the properties characterizing the living, he opened the way to the development of a new branch of thermodynamics, that of dissipative structures. Coherently with this theoretical line, and referring explicitly to Schrödinger’sinsight (Rosen, 2000), Robert Rosen faced the problem of the integration of the biological unity (Rosen, 1958a; 1958b, 1959, 1972), putting into evidence the epistemological consequences of his approach for scientific knowledge (Rosen, 1991). One main example is the inclusion into science of circular causal loops, considered as the characteristic peculiar to biological systems, with the purpose to widen the range of scientific explanation from the specificity of mechanistic systems to the higher generality of complex ones (Mikulecky, 2001a). Following this line of research he built new kinds of models able to deal with phenomena not explainable by the means of the physical description (Stewart 2002) and he developed new approaches to the study of complex systems also in disciplines other than biology.

2. Order in the Nothing The distinction of a unity from its background is the primitive epistemic action which an observer performs in order to study a natural object. It is the procedure which separates, according to some criteria, the object under study from the medium with which it interacts and so specifies its domain of existence. As George Spencer Brown wrote: “a universe comes into being when a space is severed or taken apart [. . . ] The act is [. . . ] our first attempt to distinguish different things in a world where, in the first place, the boundaries can be drawn anywhere we please. At this stage the universe cannot be distinguished from how we act upon it”(Spencer Brown, 1969).The kind of unity that is distinguished depends on the observer’s operation of distinction, which relies on his purpose and his point of view. Consequently it is an epistemological action, which puts some boundaries that can be topological or functional. An example is the study of thermodynamical physical systems, where the boundary can be the container used for the experiment. What do we distinguish while interacting with a living system? The study of living systems has a particular aspect. When we interact with them we can perform distinctions according to a manifold criteria, like the production of a certain enzyme or of waste products or the identification of a certain reaction pathway inside of it. But in order to identify them as Autopoiesis and the Organizational Characterization of the Living 343 living unities, which is the main problem faced in this work, we perform a special kind of distinction. In fact the identification of these systems operated by the observer depends on an operation of distinction that defines their identity in the same domain where they specify it through their internal operations. The organism produces itself and by so doing specifies its topology in such a way that its topological and functional boundaries coincide. In the interaction with a living system we can observe that it is characterized by the maintenance of the unity as such in a continuous process of transformation of components. A special kind of homeostasis is involved here which does not concern single processes but the whole system which is maintained stable. What makes the set of processes which occur in a living organism a unity, or better a system, needs to be found at a level different from that of the physical and chemical description. Material components indeed are continuously produced, transformed and degraded in such a way that what characterizes the identity of the system, is that what is maintained in this change, does not belong to their epistemological domain. Starting from these remarks and following Schrödinger’sinsight about the necessity of characterizing the order peculiar of living systems, we pose the following question: which kind of order characterizes the continuous and intertwined flux of processes of production and transformation of components which a scientific observer sees as realizing a living being? The difficulty in catching what makes a living system a unity of some kind consists exactly in this: we are dealing with a system which is characterised by a continuous change and nevertheless maintains itself. Unlike a machine, where the material parts are in an order visually observable, in the organism the chemical components are mixed in a continuous flux of processes that occurs at the same time in a fluid network of reactions. The order is not positional, it is not localized, as everything is dissolved in a field-like way. Also, there is a difference from some physical systems like for example a twister where there is an order in spite of the movements of components. In organisms indeed components are not pre-given but they are produced and transformed by the system itself. The first step towards a characterization of the order proper from the living is to differentiate it from purely statistical order or order from disorder (Schrödinger,1944). This way of producing regularities in natural system is not absent from biology, but it is not the factor which catches what happens in a living system and that differentiates it from other natural and artificial systems. Jean Jacques Kupiec and Pierre Sonigo (Kupiec and 344 Leonardo Bich and Luisa Damiano

Sonigo, 2000), in their critics of the concept of gene, refer to statistic order as the crucial element to describe the living. They propose a sort of “molecular mechanics” which interiorizes the Darwinian selection at the molecular level together with chance. Implicitly, they develop the line of research of the tradition of Francis Galton, Wilhelm Roux, and August Weissmann who used the term “intrabiontic selection” (Pichot, 1999). Kupiec and Sonigo consider organization as an epiphenomenon, the result of the dynamics of the ontogenesis, in a way similar to the processes of pattern formation, without any sort of causal role in conserving and realizing the system. Their theoretical model is more similar to the twister referred above, and depends directly on the materiality of the parts contained in the system observed. They have the great merit to put into evidence some problems in the point of view on metabolism assumed by molecular biology, especially about instructive functional interactions operated by the genes or by the environment, which are typical of a machine-like approach. But a living system is not only a visible shape, an ordered pattern. On the contrary it shows autonomy and produces its own components. The difficulty of explaining its difference from a phenomenon like the one giving rise to a self-organizing (or more correctly a self-ordering) system as a twister or Bernard’s cells, comes out because the starting point is not the basic question “What is life”, but just a description of some chemical processes happening in the organism. The problem of the integrated unity is not taken into consideration. Maturana and Varela instead, belong to a systemic tradition focused on the problem of the relational unity of the living, which has its origins in Claude Bernard’s concept of milieu intérieur (Bernard, 1865). In one of their early papers on the problem of the description proper to the living (Varela and Maturana, 1972), they propose the thesis according to which machines and organisms are different from physical systems in that they depend on how their elements are connected: “what makes physics peculiar is the fact that the materiality per se is implied; thus, the structures described embody concepts which are derived from materiality itself, and don’t make sense without it. [...] the definitory element in the living organization [and in machines] is a certain structure independent of the materiality that em- bodies it; not the nature of the components but their interrelations” (Varela and Maturana, 1972). The problem of how things are connected in a living system is also at the core of Schrödinger’sconceptual model. But another step is needed, that is the distinction between organisms and artificial systems. Both are Autopoiesis and the Organizational Characterization of the Living 345 organized systems whose qualitative properties are generated by the way their components are interconnected: that is by their internal organization. But machine are characterized by a positional order. Hard automata in fact are organized by geometrical spatial constrains which unlike some physical objects like crystals, are not characterized by symmetries, but have a dynamical functioning. Von Neumann’s self-reproducing automata (von Neumann, 1966) constitute an example. They don’t really produce themselves, but just recreate a certain spatial disposition. They are nevertheless at the basis of the metaphor of the organism as an “information processing machine” that conceptually influenced molecular biology. The order characterizing the machine metaphor is of the same kind of the one proposed by Schrödingeras fundamental in the living. His order from order is in fact realized by an aperiodic solid characterized by a positional structure. The living processes are controlled by an ordered group of atoms that maintains itself and transmits his structural order to other molecular structures. Schrödinger’sconceptual model is that of a clock- like system with mechanisms interconnected in a way that the positions are what is relevant (Schrödinger,1944). He solidify the biochemical flux in a sort of chemical mechanism in which the microscopic order of the genetic code is transmitted positionally to the macroscopic order of the living. Both in organisms and in mechanisms the shape, the structure, is what is conserved and transmitted in ontogenesis and phylogenies. This idea is conceptually very close to von Neumann’s one, as the aperiodic solid performs the role of the program in a computationalist view. But living systems are not solid, they are a flux of processes and transformations in a solution that is almost homogeneous from the point of view of matter and energy. That is the reason why considering processes and especially interactions between processes is crucial. The macroscopic order is not due to the shape of the microscopic structure of components but depends on the shape of the interconnections between the processes of transformations. Their proper order differentiates itself from the other two kinds and can be found on a different level of abstractions over the flux of change. It is an epistemologically higher level of analysis where we can find a kind of order which with Tibor Ganti’s words we call “order in the nothing” (Ganti, 2003). What is produced and maintained in the continuous change of components that we observe in living systems, and what produces and maintain this same change is the shape of the relations between the processes. Consequently, in order to understand the living as a systemic unity we need to put ourselves on a relational level of description. 346 Leonardo Bich and Luisa Damiano

Generative Interactions Class of systems mechanism Order from Statistical Non-positional(between Physico.chemical disorder given components) systems Order from Relational Positional (between given Artiﬁcial order spatial(between com- components) systems(machines) ponents) Order in the Relational Non-positional(between Living systems nothing abstract(between components that are pro- processes) duced by the system itself)

The order in the nothing is different from the first kind of order because it is independent from material parts. But it is also different from the machine kind because of its fluidity and because the geometry of relations is not spatial but abstract. It doesn’t connect solid components but processes, characterizing itself as a meta level of relations which emerges when these ones assume a shape that allows self-production and self-maintenance to be instantiated. It doesn’t belong to the domain of the material structural interactions but to a relational abstract independent but complementary domain: that of organization.

3. Structure and Organization The origin of the organizational description of the living, considered as complementary to the structural one, can be ascribed to a scientific tradition developed between the 30’s and the 70’s. It is a trend of research constituted by a variety of lines of the contemporary science which faced the same problem along some theoretical pathways which, although different, converged on many aspects. This crucial problem was to structurize a categorical approach pertinent to the specificity of the biological domain. In spite of the different disciplinary origins, these lines shared the same theoretical assumption. It consists in the idea that the distinctive property of the biological domain is autonomy: a relative independence from the environment which cannot be referred to the physico-chemical components, but to the integrated totality in which these components are supposed to be dynamically connected. The preliminary definition of autonomy provided by this trends of research refers to an endogenous determination, a self- determination, which reveals itself primarily in the scientific exploration Autopoiesis and the Organizational Characterization of the Living 347 of the biological behaviours of self-stabilization, that is, of active reaction to exogenous perturbations, consisting in compensative movements which cannot be ascribed to the individual perturbed components. Typically they appear as global regulative processes, distributed on the whole biological unities to which the components belong. The development of the investigations performed by these branches of the scientific research is usually gathered under the denomination “scientific genealogy of the notion of self-organization” (Stengers, 1985; Ceruti, 1989; Damiano, 2007). Its characteristic is to be a theoretical movement which, although plural and differentiated, is strictly coherent and oriented. Born from the attempts to modelise the internal functional scheme which allow stability in living systems, this wide theoretical process developed up to the production of a general modelistic which has the ambition to provide a specific conceptual definition of the biological domain. It has been realized through the succession of different descriptive models, which shared some common methodological and theoretical characteristics: (a) the assumption of a qualitative and relational level of analysis, focused not on the specific physico-chemical components and processes, but rather on their organization, that is the functional interconnection that integrates them in the biological unities; (b) the characterization of the organizational scheme which allows the expression of the property of the biological autonomy as a relational network with a circular character. The development of this kind of modelistic has been carried out through the progressive formal specification of the concept of organizational circularity, which allowed the so-called “tradition of self-organization” to gradually approach the rigorous scientific definition of the general dynamical mechanism at the basis of the biological phenomenology. This theoretical movement, accomplished through a progressive distinction of the organization from the structure of biological systems, can be traced back to three main phases: (1) the production of the first theories of self-organization, performed by three independent lines of scientific research which shared a close attention to the biological level of nature: (a) the wienerian cybernetics (Wiener, 1948) and its derivations, such as second order cybernetics (von Foerster, Zopf, 1962) and French neo-connectionism (Atlan, 1972); (b) the organicistic embryology (Weiss, 1974); (c) the thermodynamics of dissipative structures (Prigogine, Stengers, 1979); 348 Leonardo Bich and Luisa Damiano

(2) the rigorous synthesis of the ﬁrst conceptualizations on self- organization, operated by Piaget (Piaget, 1967) through the elaboration of a general theory of the biological organization based on the concept of “organizational closure” (Ceruti, 1989); (3) the critical revision of the theories of the biological self-organization performed by Maturana and Varela, which led them to an original theoretical deﬁnition of the dynamical mechanism proper of the biological systems expressed through the concept of “autopoietic organization”.

3.1. The Origins of the Concept of Self-Organization: Biological Autonomy and Organizational Circularity The concept of biological self-organization has its origin in the problem of the modelization of one of the most relevant and evident properties of the living. Biology at first called it homeostasis and defined it as the capability proper to the organisms to maintain their internal environment relatively stable against the external perturbations (Cannon, 1932). It is a biological property that the tradition of self-organization re-conceptualized as autonomy, conceiving it as the active control performed by the organism on its same processes: the capability to react to the external destabilizing perturbations through self-determined variations which tend to cancel the effects of the exogenous destabilizations. This interpretation of the homoeostasis of living beings is at the core of the fundamental theoretical hypothesis of the biological modelistic provided by the tradition of self-organization: the assumption which traces the autonomous behaviour of self-stabilization back to the organizational circularity. A schematic reconstruction of the development of this axis of research can be briefly outlined showing the main contributions of the three directions of investigation that characterize the first period of the research on self-organization: (a) the model of the “feedback circuit” or “retroaction” elaborated by Norbert Wiener (Wiener, 1948); (b) the theoretical scheme of the “organized hierarchic system” produced by the embryologist Paul Weiss (Weiss, 1969; 1974); (c) the characterization of the “dissipative systems” proposed by Ilya Prigogine (Prigogine and Stengers, 1979; Nicolis and Prigogine, 1989). The first of these theoretical models already shows the methodological and theoretical definitory elements of the descriptive hypothesis which associates autonomy to a circular scheme of organization. The model of the feedback circuit was developed by Wiener coherently with the general Autopoiesis and the Organizational Characterization of the Living 349 approach of cybernetic, a discipline based on the recognition of the convergence between the domain of the technological research and that of biology with respect to the problem of stability. The common referring by these two scientific areas to objects exhibiting the capability of self-stabilization (servo-mechanisms and organisms), led Wiener to the elaboration of cybernetics as the locus of bidirectional theoretical exchanges between technology and biology. The idea was to use the analysis of the technological mechanisms of stabilization to advance hypotheses on the functioning of biological ones and vice versa to exploit the knowledge of the latter for the implementation of the formers. Such a procedure of reciprocal exchange of knowledge has an important consequence. It requires a theorization able to set aside properties of the specific components of the two kind of objects explored. It forces the research to move to an explorative level susceptible to give an ac- ces to the general functional relations of the object investigated, focalizing not on the structure but on the organization. This methodological option is at the origin of the model of the feedback circuit elaborated by Wiener along the axis that goes from technology to biology: defining the organizational scheme of the servomechanisms and proposing it as the hypothetical model of the general functional organization that allows the living systems to carry out self-stabilizing behaviours. As it is well known, the peculiarity of the wienerian model is to delineate a circular causal relation between a sensor and an effector, mediated by a regulator able to make the sensorial recording of the perturbative deviation and to act on the devices which perform compensative actions. Once applied to living systems, this scheme traces their general functional organization as a ring: a functional circle which, interconnecting the sensorial and effectorial apparatus, implies that the effects of perturbations trigger compensative activities expressed by motorial interactions which are effective in the environmental context. This model, associated by Wiener to the idea of “self-regulation” of biological systems, became the cybernetic prototype of the notion of “self-organizing system” developed in the modelization of biological autonomy by the heterodox lines of research of the Biological Computer Laboratory (Von Foerster and Zopf, 1962; von Foerster, 1980) and of the French neo-connectionism (Atlan, 1972). The fundamental methodological and theoretical lines of Wiener’s cybernetic modelization of the autonomy of the living, were independently introduced in the context of the organicistic embryology, where they were immediately associated to the qualitative description of biological systems as “self-organizing systems”. An exemplary model of the conceptual 350 Leonardo Bich and Luisa Damiano production characteristic of this line of research is the weissian theoretical scheme of the hierarchical organized system which develops the theoretical line opened by the wienerian modelistic. This descriptive scheme was elaborated by Paul Weiss (Weiss, 1969; 1974) from the experimental investigations on the self-stabilizing behaviour of biological systems, that led him to identify their characteristic dynamics. Typically the presence of local alterations in the internal dynamics of a living system do not induce the activation of a localized and specific stabilizing centre, but triggers a series of correlated modifications in the elementary processes. If the destabilisation is lower that the stability threshold of the system, it causes a compensation. In Weiss’s exploration this remark becomes primarily a methodological option. It consists in the refusal of the approaches to the study of biological stability focalized on the individual physico-chemical components and their processes, in favour of the assumption of an explorative procedure centred on the functional relations that dynamically correlate the elements in the biological global unities constituted by the living systems. With the adoption of this methodological perspective of an organizational character, Weiss aligned his modellistic production to the wienerian one not only from the procedural point of view but also from the theoretical one. The hypothesis he developed to modelise the organization allowing the biological stability proposes the idea of an organizational circle. From the seminal insights of the founders of the organicistic embryology,∗ Weiss conceptualized it as a strict functional correlation with the shape of a “closed network”: every element of the system is functionally correlated to another one so tightly that a behavioural deviation of it entails a compensative distributed reaction in the whole network. This characterization of the biological circularity of organization distinguishes weissian modelistic from the wienerian one. It leads to the explicitation of the idea that in the living systems the compensative reaction to the perturbations is a collective action. The capability to self-stabilize does not belong to the individual components of a biological system, but to their organizational reticular correlation—to the totality. On the basis of this remark Weiss produced a general descriptive scheme that distinguishes in the living systems two qualitatively different levels: that of the individual parts, subjected to continuous alterations, and that

∗The reference is to the research team which founded the organicistic embriology, that is the so called Group of Cambridge (Stengers, 1985). Autopoiesis and the Organizational Characterization of the Living 351 of the unity which comprehend them, strongly and actively conservative: “The variability of the complete system V is much lower then the sum of variabilities v of its components. V<<[v(a)+ v(b) + v(c) + ... + v(n)] This formula contains the essential aspect of systems dynamics with respect to the composition of elements. It could not be respected if the components were free and independent.” (Weiss, 1969). This is the model of a biological hierarchical organization. It describes a relational global unity which interconnects its subunities to one another through functional reticular links. By so doing the totality subjects its components to its own global dynamics—a collective and coordinated dynamics of its elements—and reacts to the elementary local deviations by regulat- ing the components’ singular behaviours to achieve its own conservation. The theoretical idea of “self-organizing system” was associated by Weiss to this concept of biological system: the notion of a totality that, through the organizational reticular constraints that constitute it, produces its own constitutive dynamics, imposing it to its own elements and stabilizes itself in presence of exogenous perturbations, acting on its individual elementary processes. By doing this, the weissian model leads to a decisive enrichment of the descriptive hypothesis that associates the autonomous behaviour to the organizational circularity. First of all it extends the meaning of the notion of living autonomy, connecting it not only to the property of self-stabilization but to a more fundamental one, that of self-production with respect to which the first is characterized as derived. Secondly, he connects the thesis of the circular organizational scheme to that of the stratification of the living systems into al least two interdependent and qualitatively different levels—the individual parts and the totality. Both these developments are now acquired by the scientific theory of self-organization. The last one was conceptualized by Weiss through the controversial notion of “emergence”, which describes the capability of the organizational circle to generate a level of reality characterized by qualities that are not present in the lower level. It is the hypothesis of a qualitative difference between the parts and the totality , that Weiss made intelligible through an argumentation widespread today: the reticular organizational connections, constraining the components to each other in a circular way, inhibite the expression of some of the properties of the individual components and make possible the expression of global properties. 352 Leonardo Bich and Luisa Damiano

From this notion, inseparably associated to the model of the hierarchical organized system, Weiss built a complex theoretical characterization of the living. It consists of a perspective which identifies biological systems as “molecular ecologies”, describing them as structurized in a plurality of levels of “molecular organization” of increasing complexity and stability—cells, groups of cells, tissues, organs, apparatus etc. It is a theoretical idea that played a decisive role in the development of the tradition of self-organization. It gave a significant contribution to the construction of the third of the explorative direction of the research on natural self- organization (Stengers, 1985). The thermodynamics of dissipative structures was founded by Prigogine in the context of the development of a program of research finalized to deal experimentally with the crucial problem of the relation between physics and biology: the antagonism between the cosmological scenario pointed out by the principle of the entropic growth and the evidence of the biological evolution towards complexity. The ambition to provide a solution to this issue led Prigogine to develop the weissian hypothesis of the “molecular ecologies”: the idea that a molecular population with a high degree of freedom can generate a level of stable integration and able to evolve towards higher levels of complexity. The assumption of this thesis oriented the prigoginian research program to the exploration of the behaviour of thermodynamically open molecular systems, an investigation characterized by relevant results. It put into evidence that these systems, under certain conditions (Prigogine and Stengers, 1979; Nicolis and Prigogine, 1989), are subject to phenomena of “super- molecular aggregation” which generate persistent macroscopic dynamical structures-ordered patterns. They are irreversible physico-chemical processes that topologically organize themselves: collective coordinated movements of the molecular population which constitute active and organized global unities that self-generate and self-maintain exploiting the exogenous flux of matter and energy and by doing so producing entropy which is released in the environment. Prigogine called them “dissipative structures” to make explicit the compatibility with the second principle of thermodynamics. He provided a characterization of them in terms of self-organizing systems, taking the weissian theoretical framework of the circular and stratified organizational scheme. He explained these natural forms as emergent organized unities. He attributed to them the minimal and incomplete forms—“ancestral” (Prigogine and Stengers, 1979)—of some of the properties shown at the biological level. Among these Autopoiesis and the Organizational Characterization of the Living 353 properties he included autonomy: the capability of these systems to self- produce their own constitutive dynamics, to stabilize against a wide range of perturbations and to exploit external perturbations to evolve—by an endogenous re-organizational control, that is by positive retroaction (Nicolis and Prigogine, 1989)—towards higher levels of complexity and stability. Such a theoretical characterization puts the biological autonomy inside an evolutive scenario which grounds it in the physical domain. It proposes an image of the prebiotic evolution that generates more and more stable and complex molecular ecologies, which plausibly are able to develop until transcending the physico-chemical domain and proceeding into the biological one. It thus opens the possibility of a connection between the physical and biological domain, not a reductionistic, but an emergentistic one. It offers the opportunity of an exchange of knowledge between these two domains of science which is not unidirectional and reductive, but founded on a bilateral “dialogue” (Prigogine and Stengers, 1979; Stengers, 2003).

3.2. The Piagetian Synthesis: The Notion of “Organizational Closure” Besides these first lines of research, the scientific genealogy of the notions of autonomy and self-organization includes some trends of investigation definable as of first derivation. They are scientific orientations generally related to programs of research developed in the context of the human sciences and directed to bridge the cartesian cut: to overcome the theoretical gap that separates the natural and the human sciences, by producing a naturalistic notion of human being which avoids to reduce him to physical and biological aspects. The peculiarity of these lines of investigation it that they acknowledged in the evolutive scenario opened by the first research on self- organization the possibility to re-articulate the anthropo-social sphere on the biological and the physical ones, according to a non-reductionistic but emergentistic theoretical approach. The aim of this operation was realized through the investigation of the first research on self-organization, in order to perform theoretical syntheses able to produce a complex and multidimensional notion of the human being. (Morin, 1973; Morin and Piattelli- Palmarini, 1974; Jantcsh, 1980; Dumouchel and Dupuy, 1983). Among these programs of research the piagetian genetic epistemology assumes a particular relevance, due to its decisive contribution to the scientific characterization of the biological organization (Ceruti, 1989). This line of the contemporary epistemology was directed by Jean Piaget towards the construction of a natural science of cognition based on a specific 354 Leonardo Bich and Luisa Damiano interpretation of the general assumption of cognitive sciences that identify life with cognition. It consists in a minority theoretical option that refuses the computationalist modelization of living systems in favour of the identification of autonomy as their definitory property. The piagetian work of exploration and integration of the scientific production on the natural self-organization led the genetic epistemology to the elaboration of an innovative theoretical element, quickly acquired by the scientific research and still at the core of the investigations on biological autonomy. It is a concept that rigorizes the previous achievements on the organizational circularity of living systems: the notion of “organizational closure”, proposed by Piaget in Biologie et Connaissance as a concept complementary to that of thermodynamical openness, earlier emphasized by von Bertalanffy’s systemic biology (Piaget, 1969). “The central ambiguity is that of the ‘open system’, for, if system exist, then something like a closure intervenes, which has to be reconcilied with the ‘opening’. The opening is certainly justified and is founded on the basic idea that ‘in biology there is not rigid organic form carrying out vital processes but a stream of processes which are revealed as forms of a seemly persistent kind’ (von Bertalanffy). The opening then is the system of exchanges with environment, but this in no way excludes a closure, in the sense of a cyclic rather than a linear order. This cyclic closure and the opening of exchanges are, therefore, not on the same plan, and they are reconcilied in the following way, which may be entirely abstract but will suffice for a analysis of a very general kind. (AxA’)→ (BxB’)→(CxC’)→...→ (ZxZ’)→(AxA’)→ ecc. A,B,C ...: the material or dynamic elements of a structure with cyclical order A’, B’, C’...: the material or dynamic elements necessary for their maintenance: the interaction of the terms of the first range with those of the second→: the end points of these interactions In a case like this we are confronted by a closed cycle, which expresses the permanent reconstitutions of the elements A,B,C ... Z, A, and which is characteristic of the organism; but each interaction (AxA’), (BxB’), etc., at the same time represents an opening into the environment as a source of aliment.” (Piaget, 1967). The piagetian notion of closure can be considered as a development of the weissian formulation of the theoretical assumption that associates the biological autonomy to the organizational circularity. The concept formulated by Piaget presents biological autonomy as an endogenous Autopoiesis and the Organizational Characterization of the Living 355 determination of the living systems that has the character of the self- production and correlates it to the idea of an organizational scheme realized by the circular functional interconnection of the components. The specific contribution of the piagetian notion of closure consists in the idea that the organizational circularity corresponds to a concatenation of processes which continuously re-constitute the components of the living systems. The theoretical picture is that of a closed chain of elementary transformative operations which, by realizing themselves, trigger each other, giving rise to a self-determined recursive and cyclical process that, producing the components, produces the organism itself. Piaget associated this conceptualization to the theoretical explicitation of the difference between organization and structure (Ceruti, 1989). He defines the first as the general relational scheme of all the living systems and the second as its materialization in specific processes and components. Piaget underlined that the two aspects are distinct because, as the idea of organizational closure points out, the peculiarity of the living systems is that their materialization keeps changing: what persists in them are the functional relations that integrates the components in the global unity, while the specific components are permanently in flux. By this theoretical distinction Piaget laid the basis for what can be considered the most mature development of the genealogy of self-organization: the formulation of the autopoietic biology, oriented by Maturana and Varela towards the definition of the dynamical mechanism of living systems.

3.3. The Autopoietic Biology: The Duality of Structure and Organization in the Scientific General Definition of Living Systems The basic theoretical project of the autopoietic biology was to provide a new kind of solution to the problem of the identification of the living. Maturana and Varela wanted to develop a criterion of identification which does not consist in the classical enumeration of properties we can recognise in the living from our external point of view. Instead of undertaking an analytic descriptive procedure, doomed to go on indefinitely, Maturana and Varela conceived a criterion which identify the living by specifying a mechanism able to produce these properties, or, more accurately, a mechanism able to produce all the living phenomenology. “Our aim is to proceed scientifically: if it is not possible to produce an inventory which characterizes a living being, 356 Leonardo Bich and Luisa Damiano

why not conceiving a system which, while operating, produces all its phenomenology?” (Maturana and Varela, 1987) This can be considered the main aim of the autopoietic biology: deﬁning the living from the inside, by specifying its deep “dynamical mechanism”, the inner mechanism able to generate the living phenomenology. The development of the theory of autopoiesis relies on two fundamental thesis:

(1) the most basic property of the living is autonomy, understood as the capability that such natural systems have to produce and maintain, all by themselves, their own identity, by a triple endogenous action: a) self-production; b) permanent topological self-distinction in an environment; c) self-stabilisation against endogenous and exogenous perturbations; (2) autonomy is a property which does not belong to the individual physico- chemical components of the living, but to their organizational correlation, that is the functional “organization” which integrates them in the relational unities usually called “organisms” (Maturana and Varela, 1973).

The adoption of these hypotheses coincided with the alignment of the autopoietic biology with the tradition of self-organization, a connection which remained implicit and offered the ground for a transformative development of the production of this tradition. Here is what allows us to consider autopoietic biology as a critical reform of the first research on self- organization: although it shared with the latter some theoretical presuppo- sitions and some conceptual elements, this biological school spent more time stressing the flaws of the tradition of self-organization than acknowledging its theoretical debts towards it. The main shortcoming that autopoietic biology attribute to this tradition of research consists in a insufficiency. The genealogy of self- organization has postulated that the biological level of reality is populated by relational unities of elements whose primarily property is autonomy. It has postulated biological self-organization, but it didn’t try to scientifically explain it, that is to define the mechanism able to generate these unities and their phenomenology (Maturana and Varela 1973). In order to do it, one essential consideration, substantially missed by the research on self-organization, has to be taken in account. It consists in the idea that the organization which functionally supports the biological autonomy constitutes the invariant of the biological phenomenology—both at Autopoiesis and the Organizational Characterization of the Living 357 the ontogenetic and at the phylogenetic levels. The relational unity of components is what maintain itself in the permanent flux of physico-chemical elements peculiar to the organism’s life. This unity is what is permanent within the strong transformations that can make the individual living being un-recognisable from one observation to another. The relational unity is the biological element which is transmitted through reproduction: it is the feature of the living that remains unchanged generation after generation. The global unity is what is constant in the evolutive differentiation, to such an extent it constitutes the feature shared by all the living: being a relational unity of components. It is from this consideration that Maturana and Varela infer the generative hypothesis of the theoretical conceptual structure of their autopoietic biology, according to which the organization is the identity that the living permanently produces and keeps, thanks to a permanent and self-determined change the physico-chemical components (Maturana and Varela, 1973). It is a thesis that implies the refuse of the term “self-organization”, which can seem to express the idea of a self-determined change of the living organization. It can seem to miss the specificity of the dynamics of the living, namely, the conservation of the organizational invariance coupled with the permanent flux of physico-chemical elements. Here relies the need for a new term, namely, “autopoiesis”, which can avoid the confusion between the invariant and the variant aspects of the living dynamics. This term expresses the acknowledgment that biological systems constitute dynamical systems in which the relation between invariance and change has a peculiar aspect: in these systems what is conserved is the relational unity of components, which, in themselves and in their specific functional relations, permanently change. Maturana and Varela elaborated this point through the implicit recov- ering and strict scientific treatment of the conceptual dyad introduced a few years before by Piaget: the duality of organization and structure, which within the autopoietic biology coincides with the duality between the invariant element of biological phenomenology and the variant one. Here “organization” refers to the stable and permanent relations which define a biological individual as a unity, while “structure” refers to the particular and transient materializations of a living unity. These two notions are distinct but inseparable for they express two complementary aspects of the living. In this kind of systems the relational unity of the components –the organization- cannot be without a concrete and transient realization into specifics 358 Leonardo Bich and Luisa Damiano elements and relations between them –a structure. Conversely there cannot be a concrete and transient unity of elements in flux—a structure without the stable relations—the organization which integrates them into a persistent unity. The formulation of such a conceptual complementarity has constituted the core of the development of autopoietic biology’s program, aiming at defining the dynamical mechanisms of living systems through the individu- ation of the interplay between the invariant element of their dynamics and the variant one. Maturana and Varela carried out this theoretical operation on the basis of a specific hypothesis concerning the relation between organization and structure: the continuous structural change, due to the transformative interactions of the components, produces and maintains the organization which, in turns, enables the structural change. This thesis allowed autopoietic biology to provide the definition of a plausible mechanism for the living dynamics, elaborated by Maturana and Varela in relation to the minimal and “fundamental” living system: the cellular system, present in all living forms, evolutively anteceding them, and therefore able to generate them. The dynamical mechanism of the cellular unity has been explained by Maturana and Varela through the definition of cellular organization, called by them “autopoietic organization”. It is a theoretical elaboration that implicitly recovers the piagetian concept of organizational closure. It corresponds to the notion of a circular dynamic mechanism: a close chain of operations of elements transformations in which the realization of one operation triggers and integrates another one, in such a way that the global cyclical process that emerges is essentially characterized by the property to determinate and regenerate itself. It is easy to recognize the influence of the piagetian idea in the definition of autopoietic organization, according to which: “ [The autopoetic organization] (...) is a network of production processes (transformation and destruction) of components which produces the components which :

(1) Through their interactions and transformations, permanently regenerate and realise the network of processes (relations) which produces the components ; and (2) Constitute a concrete unity in space, within which they (the components) exist by specifying the topological domain of its realisation in that network. ” (Maturana and Varela, 1973) Autopoiesis and the Organizational Characterization of the Living 359

The more relevant aspect of the autopoietic model is that this theoretical scheme considers relations between processes that involve components. It opens a large range of possible materializations of the autopoietic organization. It does not impose conditions on the specific basic elementary constitution of autopoietic systems (Bich and Damiano, 2007). It points out the crucial aspect of elements not in their intrinsic properties, but in they interactive specificity: the relational properties that define the forms of functional correlation that the elements can develop (Bich, 2005). The constitution of the elementary level is allowed to change, but within in a well defined range of variability: the space of all the elementary com- positions able to generate a recursive chain of functional relations of reciprocal production. It is by imposing this only relational constraint onto the level of components that the descriptive scheme is able to generate the two dimensions that constitute the dynamics of the living unity: it poses the organization as the invariant element and the structure as the variable one. In doing so it succeeds in expressing theoretically the idea that in this kind of system the conservation of organization is obtained through the continuous structural variation.

4. Relational Biology and Rosen’s (M,R)-Systems A parallel line of thought that goes in the same direction inaugurated by the studies of cybernetic and continued by genetic epistemology and autopoiesis is the one opened by Nicholas Rashevsky and carried on by Robert Rosen. It has the merit not only to focus on the integrated organization of the living systemic unity but also to develop specific formal tools in order to describe it. Rashevsky developed mathematical biology since the early 30’s (Abra- ham, 2004), giving a great contribution to the mathematical study of the biological processes of self-organization which led to Alan Turing’s and Ilya Prigogine’s models of pattern generation. His first approaches dealt with separate biological phenomena from a physical point of view. In his seminal paper “Topology and Life” in 1954 Rashevsky shifted his approach to the study of living systems from the construction of structural models of the biological processes to the modelization of the relational properties of organism, that is, to the study or their organization. His purpose was to develop a mathematical theory able to treat the integrated activity of the organism as a whole: “we must look for a principle which connects the different physical phenomena involved and express the biological unity of the organism and of the organic world as a whole” (Rashevsky, 1954). 360 Leonardo Bich and Luisa Damiano

Following the idea, he reflected upon the difference between physical and biological phenomena. According to him, the formers are characterized by metrical aspects and the latter ones by relations. Also, while the structural physical description focuses on the differences between systems, the relational one considers their similarity, embedded in the common minimal organization that defines their class, in this case the organization of the minimal living system, shared by all the other more complex organisms. The identity is the starting point of his research. The differences are secondary to the theoretical characterization of the organisms, which is primary with regards to the investigation of the variety of their phenomenology. In looking for what makes biology special we notice in fact not only quantitative aspects but, above all, the presence of complex relations. These ones, according to Rashevsky, do not concern shape, like in the approaches a la D’Arcy Thompson, but consist in relations between processes or properties. His new line of research, called relational biology, focused on the abstract description of the relations that connect the fundamental properties of organisms up to letting us recognize that network of relations as characteristic of the minimal living being. He opened the way to the development of a mathematical theory able to treat not the metric aspect of these systems but the integrated properties realizing the organism. His work in relational biology thus, was based on two related theoretical assumptions:

(1) the ﬁrst concerns the importance of relations in identifying and characterizing the living: “we postulate that the highly complex biological structures are due to relational forces, that is, that they are formed because as a result of this formation certain qualitative relations appear, relations which make us recognise the structure as a living organism” (Rashevsky, 1972). (2) The second concerns the possibility to study living systems starting from their common organization. It is the Principle of Bio-topological mapping, according to which “all organisms can be mapped on each other in such a manner that certain basic relations are preserved in this mapping” (Rashevsky, 1960).

Rashevsky’s approach, nevertheless, was characterized by the attempt to connect the different structural processes and properties observed in living systems. It was an additive procedure which had the purpose to build the relational schemes of biological functions of different organisms. What was Autopoiesis and the Organizational Characterization of the Living 361 conserved in the transformation from one scheme to the other was the hypothetical relational structure of the minimal organization common to all living systems. Rosen started from Rashevsky’s insights but followed a different path. On a higher level of abstraction, he extended the approach of relational biology to the form that the fundamental relations should assume to express the basic autonomy of living beings. He was looking for the mechanism that causes the biological properties to be instantiated. In doing so, he conceptually started from the integrated unity of the living organism, characterised by the capability to self-produce and self-maintain. In the late 50’s Rosen proposed a formal model based on the mathematic of Category Theory, the (M,R)-System, which described the organization of the living as the integration of the functioning and the fabrication of the system (Rosen, 1958a; 1958b; 1959). His early works are quite obscure and he clarified some of their mathematical aspects after many years (Rosen, 1972), but he fully understood the theoretical meaning of his model only later (Rosen, 1991). A recent and clear explanation of the mathematical formulation of (M,R)-Systems together with some attempts to relate it to biochemical processes, was provided by some researchers belonging to the autopoietic school (Letelier et al., 2006). Rosen’s basic assumption in the construction of his model is that every component must be produced, or replaced after its degradation, inside the system. He uses some instrument of Control Theory, considering every process like an input-output one, but in the end he realizes a formal structure very different from those characteristic of machines. The first step is to consider the “metabolic” process that usually takes place in a cell, where some catalysts transform into components the substrates coming from outside the system. It is expressed formally by a set of mappings H(A,B) which transform a set of substrates A into a set of components B. f:A → B f(a) = b with f ∈ H(A, B) The catalyst f, which has a limited lifespan, must also be produced or replaced by some processes inside the system. Living organism in fact, unlike machines, are characterized by a continuous turnover of components. So Rosen introduces another process, called “repair” in his original papers (or “replacement” in Letelier et al., 2006), that produces the metabolic catalyst f out of the components belonging to the set B. φ: B → H(A, B) 362 Leonardo Bich and Luisa Damiano

Fig. 1.

φ(b) = f with φ∈ H(B,H(A,B))

Fig. 2.

Now we have the concatenation of two processes: A → B→ H(A, B) Where: f(a) = b φ(b) = f But again the mapping φ, which plays the role of catalyst in the production of f is not produced inside the system. So another process, another mapping, is needed. It has to be placed inside the system too, but this procedure can lead to an inﬁnite regress. In order to avoid it, Rosen introduces the mapping that makes its model interesting. In fact he looks for it in the components inside the system, in such a way that the models folds onto itself and every mapping becomes the output of another mapping of the system. He calls this third process “replication”, but Letelier et al. prefer to call it “organizational invariance” because it is the process that closes the model onto itself and makes it a self-producing and self-maintaining system under the continuous turnover of components, while replication reminds more of Autopoiesis and the Organizational Characterization of the Living 363 biological reproduction (Letelier et al., 2006). The third mapping is β that produces φ out of f: β: H (A, B) → H (B, H(A, B)) β (f) = φ with β∈ H(H(A,B), H(B, H(A,B))) How can we obtain it from inside the system itself? The answer is to identify β with b ∈ B. The procedure in order to achieve this result is to consider an evaluation map Evb: H(B,H(A,B)) → H(A,B) that evaluates all the possible choices of φ at b. It is the mapping that correspond to an element b ∈ B. Evb: H(B,H(A,B)) → H(A,B) Evb(φ) = φ(b) with Evb ∈ H(B,H(A,B),H(A,B) In order to obtain β from b, the evaluation map must be invertible, that is, it must be injective. with Evb(φ) = φ(b) Evb(φ)= Evb(φ’) implies φ = φ’ By the deﬁnition of evaluation maps Evb(φ) = φ(b) means also that φ(b) = φ’(b) implies φ = φ’ Consequently we can obtain β: H (A, B) → H (B, H(A, B)),as the inverse −1 evaluation map Evb corresponding to b, such that the system closes onto itself. It gives rise to a hierarchical folded chain of processes: A → B→ H(A, B) → H (B, H(A, B)) Where: f(a) = b φ (b) = f β (f) = φ with β corresponding to b.

Fig. 3.

By this procedure Rosen achieved a circularity inside its model, a closure very similar to the organizational one proposed by autopoietic theory. He called it “closure under efficient causation” to mean that all the mappings are entailed by other mappings in the system itself. In fact f entails b, 364 Leonardo Bich and Luisa Damiano that corresponds to β which entails φ which entails f so realizing a causal loop. The only element that comes from the environment is the substrate A, coherently with the physical hypothesis of thermodynamical openness. This model expresses the basic circularity of the organization that characterizes the continuous flux of processes of production of components inside the living systems, together with its maintenance made possible by the organizational invariance embedded by the mapping β. This model has a particular characteristic, in that its formalism allows mappings acting on other mappings, and thus self-referential functions. And also, it assumes the shape of a closed loop. This introduction of circularity into a formal model of a minimal living system is of extreme interest because it has usually been avoided in mathematics and in natural sciences. Instead Rosen, who starts from the biological point of view on the organis- mic unity, puts it at the core of Biology. This kind of model marks a deep differentiation from the study of artificial machines. Even if Rosen starts from mappings that remind of input-output functions, its (M,R)-Systems are characterized by a qualitatively different approach. In fact as already pointed out above, the mappings which represent the active components, can be entailed by other mappings. On the contrary, in artificial systems the mechanic components are not produced inside the system, and consequently their functions are not entailed in the same way as it happens here. In the homeostatic models of the retroaction mechanisms, the loops concern only the variables, which re-enter the functions. In this model instead, the same functions act on each other, so that the system acts not only on some of its own parameters but on its same production rules. The (M/R)-System is also very different from physical models, because of its formal characterization that opens the way to the study of circularity. Also it is placed on a different epistemological level, that of organization. Biology has always lacked an integrated mathematical approach similar to that characteristic of physics (Bailly and Longo, 2006). The reason is in the complexity of its phenomenology, that allows only the construction of structural models with a limited range of application. The purpose of Rela- tional Biology is to find a top-down approach that focuses on the relational character of the organizational order proper of the living, that makes us as observers to identify its unitary identity in spite of its continuous turnover of components and of the variety of its different realizations. Along this line of research, Rosen’s (M,R)-Systems are an interesting attempt to catch formally the order in the nothing, on the abstract level of relations between processes. The purpose in fact is explicitly to answer to Schrödinger’s Autopoiesis and the Organizational Characterization of the Living 365 question on its proper level of analysis. “The graph looks very much like an aperiodic solid, and indeed it possesses many of the properties Schrödinger ascribed to that concept. The novel thing is that it is not a “real” solid. It is, rather, a pattern of causal organization; it is a prototype of a relational model” (Rosen, 2000). The limit of this approach consists in its abstractness, which makes problematic to connect these formal mappings to the biochemical processes observed in the cell, even if it can provide conceptual tools in order to develop a theory of the cell. In fact the mappings, which represent the production rules in the metabolism of the living system, can be conceptually equated with enzymes, that determine the reactions that the metabolic components can undergo. Interpreted along this way, the closure under efficient causation means that all the catalysts necessary for an organism to be alive, are produced by the organisms itself (Cornish-Bowden, 2006). (M,R)-Systems can be considered an explicitation of the concept of organizational closure proposed by the autopoietic theory, especially of the first part of the definition of the living provided by Maturana and Varela (Maturana and Varela, 1973). Thanks to its formal nature it can be very useful in order to understand the epistemological and theoretical consequences of closure. An effort in the direction of an integration of the two theories has already been started (Letelier et al., 2003).

5. Conclusive Remarks Starting from the theoretical perspective we tried to present here in order to deal with the complex domain of biological systems, we can assert that science needs something more than physical structural models. Understanding what makes a living being to be alive and different from physical systems and machines has thus significant consequences for scientific explanation. In fact it can widen the range of the possible descriptions, forcing us to assume different points of view and develop new tools along a path which can lead to the construction of a new class of models. To briefly summarize the conceptual scheme we introduced here, we showed how physical, artificial and living systems can be classified according to three different kinds of order, due to the properties of the different domains where their specific properties are realized. The third kind, the order in the nothing, was shown to require a specific modellistic to be developed, which is the result of a long tradition of transdisciplinary scientific research. The constructivist concept of organization in fact requires a modelization placed on a level of abstraction detached from the properties of 366 Leonardo Bich and Luisa Damiano material components and characterized by the acknowledgment the circularity between the processes that realize the living. The paramount importance of this basic circularity is shown in Rosen’s proto-relational model whose analysis is still only in an initial phase. As a consequence of the remarks expressed in this paper, we can point out that the concept of organization as it has been developed by the line of research that we outlined here, can be put in the middle between two different positions. The first considers organization as an epiphenomenon, as a merely phenomenal consequence of the statistical behaviour of the material component of the system observed. This is the case of the theoretical approach developed by Kupiec and Sonigo presented above under the class of the order from disorder (Kupiec and Sonigo, 2000). The second position considers organization as self-sufficient in itself and puts it on an independent level. This theoretical position gives rise to a strong duality between structure and organization, that could lead to an ontological contraposition. The approach whose most rigorous expression is achieved by the autopoietic theory considers the relation between organization and structure as a descriptive complementarity: “the [organization] of living systems and their actual (material) components are complementary yet distinct aspects of any biological explanation: they complement each other reciprocally” (Varela and Maturana, 1972; see also Varela, 1979). They are mutually defining concepts, because structure is what changes while organization remains invariant. The distinction thus, is an epistemological operation, a useful tool in order to focus on different aspects of the same system. It is also important to underline again that the organizational scheme characterizing a system has important consequences as it allows to make sense of the presence of global processes in the system considered. The basic example is the self-stabilization achieved when a circular relation is realized, like in homeostatic machine. At an higher level of complexity typical of living system the role of organization is crucial in order to understand not only self-stabilization but also self-production. A new step in the development of a Relational Biology that recognizes the mutual interaction between the structural and organizational description can be the attempt to an integration of the thermodynamical and relational description. The line of development that goes from the insight and issues coming from biology to the development of physics is characterized by the attempt to widen thermodynamics in order to make sense of the peculiarity of the living systems (Prigogine and Stengers, 1979; Kauff- mann, 2000). A possible line of research, coherent with the approach that we Autopoiesis and the Organizational Characterization of the Living 367 outlined here, is that of a relational thermodynamics (Mikulecky, 2001b). An interesting result in this direction, if we assume the point of view of autopoietic theory, is Tellegen’s theorem, based on the circular scheme of closure in electronic circuits (Tellegen, 1952; for the possible connections with the autopoietic theory see Letelier et al., 2005). This approach is still related to the description of artificial machines, based on a positional relational order, but it can be a starting point towards the understanding of the consequences of the relational circularity in the thermodynamics of living systems.

References 1. Abraham T. H. (2004) Nicholas Rashevsky’s Mathematical Biophysics, Journal of the History of Biology, v. 37, pp. 333-385. 2. Atlan H. (1972) L’organization biologique et la théoriede l’information, Hermann, Paris. 3. Bailly F. & Longo G. (2006) Mathématiquesat sciences de la nature. La singularitéphysique du vivant, Hermann Editeurs,´ Paris. 4. Bernard C. (1865) Introduction ál’étudede la médecine experiméntale, Baillière,Paris. 5. Bich L. (2005) Autopoiesis and Emergence, in Minati G., Pessa E. and Abram M. (eds.), Systemics of Emergence, Research and Development, Springer, USA, pp. 281–292. 6. Bich L. and Damiano L. (2007) Theoretical and Artificial Construction of the Living: Redefining the Approach from an Autopoietic Point of View Origin of Life and Evolution of the Biosphere (in press). 7. Cannon W. (1932) The Wisdom of the Body, Norton, New York. 8. Ceruti M. (1989) La danza che crea, Feltrinelli, Milano. 9. Cornish-Bowden A. (2006) Putting the Systems back into Systems Biology, Perspectives in Biology and Medicine, vol. 49, pp 475–489. 10. Damiano L. (2007) Unitàin dialogo, Bruno Mondadori, Milano (forthcoming). 11. Dumouchel P. and Dupuy J. P. (eds) (1983) L’auto-organization. De la physique au politique, Seuil Paris. 12. Ganti T. (2003) The Principles of Life, Oxford University Press, Oxford. 13. Jantcsh E. (1980) The Self-organizing Universe, Pergamom, New York. 14. Kauffman S. (2000) Investigations, Oxford University Press, Oxford. 15. Kupiec J. and Sonigo P. (2000) Ni Dieu ni gene. Pour une autre théoriede l’hérédité, Editions´ du Seuil, Paris. 16. Letelier J.C., Mar`ın G. and Mpodozis J. (2003) Autopoietic and (M,R) systems, in Journal of Theoretical Biology, n. 222, 2003, pp 261–272. 368 Leonardo Bich and Luisa Damiano

17. Letelier C., Kuboyama T., Yasuda H., CárdenasM. L. Cornish-Bowden A. (2005) A self-referential equation, f(f) = f, obtained using the theory of (M,R,) systems: overview and applications, in H. Anai and K. Horimoto (eds.) (2005) Algebraic Biology,Universal Academy Press, Tokyo, pp 115– 126. 18. Letelier J-C., Soto-Andrade J., Guinez-Abarzua F., Cardenas M-L., Cornish-Bowden A. (2006) Organizational Invariance and metabolic closure: analisis in terms of (M,R) systems, Journal of Theoretical Biology, v.238, pp 949–961. 19. Maturana, H. (1988) Reality: the search for objectivity or the quest for a compelling argument, Irish Journal of Psychology, 9 (1), pp 25–85. 20. Maturana H. (2000) The Nature of the Laws of Nature, Systems Research, 17, pp 459–468. 21. Maturana H. and Varela F. (1973) De Máquinasy Seres Vivos: Una teor´ıa sobre la organizaciónbiológica, Editorial Universitaria, Santiago. 22. Maturana H. and Varela F. (1987) The Three of Knowledge, Shimbhala, Boston. 23. Mikulecky D. (2001a) Robert Rosen (1934-1998): a snapshot of biology’s Newton, Computers and Chemistry, vol. 25, pp. 317-327. 24. Mikulecky D. (2001b) Network Thermodynamics and Complexity: a transition to relational system theory, Computers and Chemistry, v.25, pp. 369- 391. 25. Morin E. (1973) Le paradigme perdue, Seuil, Paris. 26. Morin E. and Piattelli-Palmarini M.(eds), 1974, L’unitéde l’homme, Seuil, Paris. 27. Nicolis G. and Prigogine I. (1989) Exploring complexity, Freeman, New York. 28. Piaget J. (1967) Biologie et connaissance, Gallimard, Paris. 29. Pichot André(1999) Histoire de la notion de gene, Flammarion, Paris. 30. Prigogine I. and Stengers I. (1979) La Nouvelle Alliance. Métamorphosede la science, Gallimard, Paris. 31. Rashevsky N. (1954) Topology and Life: in Search of General Mathematical Principles in Biology and Sociology, Bulletin of Mathematical Biophysics, v.13, pp. 317-348. 32. Rashevsky N. (1960) Mathematical Biophysics. Physico-Mathematical Foun- dations of Biology, Dover, New York. 33. Rashevsky N. (1972)A Unified Approach to Physics, Biology and Sociology, in Rosen (ed.), Foundations of Mathematical Biology, Academic Press, New York, vol. III, pp 177–190. 34. Rosen Robert (1958a) A relational theory of biological systems, Bulletin of Mathematical Biophysics, v. 20, pp 245–260. 35. Rosen Robert (1958b) The Representation of Biological Systems from the Standpoint of the Category Theory, Bulletin of Mathematical Biophysics, v. 20, pp 317–345. 36. Rosen Robert. (1959) A relational theory of biological systems II, Bulletin of Mathematical Biophysics, v. 21, pp 109–128. Autopoiesis and the Organizational Characterization of the Living 369

37. Rosen R. (1972) Some Relational Cell Models: The Metabolism-Repair Systems, in Rosen (ed.), Foundations of Mathematical Biology, Academic Press, New York, vol. II, pp 217–253. 38. Rosen R. (1991) Life Itself: a Comprensive Inquiry into the Nature, Origin, and Fabrication of Life, Columbia University Press, New York. 39. Rosen R. (2000) Essays on Life Itself, Columbia University Press, New York. 40. SchrödingerE. (1944) What’s Life? The Physical Aspect of the Living Cell, Cambridge University Press, Cambridge. 41. Spencer Brown G. (1969) Laws of Form, George Allen and Unwin Ltd, London. 42. Stengers I (1985) Genealogies de l’auto-organization, Cahiers du CREA, 8. 43. Stengers I. (2003) Cosmopolitiques, 6, La Découverte, Paris. 44. Stewart J. (2002) La modélisationen biologie, in Nouvel P. (a cura di), Enquêtesur le concept de modèle , Press Universitaires de France, Paris, pp 43–66. 45. Tellegen B. (1952) A general network theorem with applications, Phillips Research Report, 7, pp 259–269. 46. Varela F. (1979) Principles of Biological Autonomy, North-Holland, New York. 47. Varela F. and Maturana H. (1972) Mechanism and Biological Explanation, Philosophy of Science, v.39, n.3, pp 378–382. 48. Varela F., Maturana H: and Uribe R. (1974) Autopoiesis: the Organization of Living Systems. Its Characterization in a Model, Biosystems, 5, pp 187– 196. 49. von Bertalanffy, L. (1949) Das Biologische Weltbild: Die Stellung des Lebens in Natur und Wissenshaft, Francke, Bern. 50. von Foerster H. (1980) Observing Systems, Selected Papers of Heinz von Foerster, Intersystems, Seaside. 51. von Foerster H., Zopf G (eds) (1962) Principles of Self-organization, Perg- amon, London. 52. von Neumann J. (1966) Theory of Self-Reproducing Automata, University of Illinois Press, Urbana. 53. Weiss P. (1969) The Living System: Determinism Stratified, in Koestler A. and Smythies J. (eds.), Beyond Reductionism: The Alpbach Symposium, Hutchinson & Co, London, pp. 3-55. 54. Weiss P. (1974) L’archipel scientifique, Maloine, Paris. 55. Wiener N (1948) Cybernetics, or Control and Communication in the Animal and in the Machine, MIT, Cambridge MA. This page intentionally left blank 371

Anticipation in Biological and Cognitive Systems: The Need for a Physical Theory of Biological Organization

Graziano Terenzi Universit`adegli Studi di Pavia, Italy [email protected]

This paper deals with the problem of understanding anticipation in biological and cognitive systems. It is argued that a physical theory can be considered as biologically plausible only if it incorporates the ability to describe systems which exhibit anticipatory behaviours. The paper introduces a cognitive level description of anticipation and provides a simple characterization of anticipatory systems on this level. Speciﬁcally, a simple model of a formal anticipatory neuron and a model of an anticipatory neural network which is based on the former are introduced and discussed. Some learning algorithms are also discussed together with related experimental tasks and possible integrations. The outcome of the paper is a formal characterization of anticipation in cognitive systems which aims at being incorporated in a comprehensive and more general physical theory.

Keywords: Biological Systems; Cognitive Systems; Biophysics; Complex Sys- tems PACS(2006): 87.10.+e; 87.16.Ac; 89.75.Fb

1. Introduction The ability to anticipate specific kinds of events is one of the most amazing features of biological organization. Mainly, it is related to the functioning of complex biological systems which perform cognitive functions. An anticipatory system is a system which decides its behaviour by taking into account a model of itself and/or of its environment; i.e. it determines its current state as a function of the prediction of the state made by an internal model for a future instant of time (Rosen 1985). A system endowed with anticipatory qualities is also named a proactive system. Proactivity and anticipation, thus, are two important features of biological organization. As a matter of fact, anticipatory properties play an important role in the mechanisms which rule learning and control of complex motor behaviours. Indeed, as it stands, motion is the only means biological organism have to 372 Graziano Terenzi interact both with their environment and with other organisms (Wolpert, Gharamani & Flanagan 2001). Recent empirical studies have demonstrated the involvement of the motor system in processes ranging from the observation and anticipation of action, to imitation and social interaction (Gallese, Fadiga, Fogassi & Rizzolatti 1996; Rizzolatti & Arbib 1998; Liberman & Whalen 2000; Wolpert, Doya & Kawato 2003). In this context the problem of anticipation is essentially the problem of understanding how cognitive systems of complex organisms can take into account the future evolution of events which take place in the interaction with their environment in order to take decisions and determine their behaviour (Sutton 1990; Stolzmann 1998; Baldassarre 2002, 2003). The study of anticipation in the context of sensory-motor learning and coordination has been carried out by developing an interesting ensemble of computational models which aim to take into account the anticipatory properties of biological neural networks. Among these models we must mention the Forward Models by Jordan e Rumelhart (1992), the architectures based on Feedback-Error Learning procedure (Kawato 1990) and on the Adap- tive Mixture of Local Experts (Jacobs, Jordan, Nowlan & Hinton 1991; Ja- cobs 1999), such as the MOSAIC model (Wolpert & Kawato 1998; Haruno, Wolpert & Kawato 2001). Essentially, on the basis of computational studies it has been hypothesized that the central nervous system is able to simulate internally many aspects of the sensory-motor loop; specifically, it has been suggested that a dedicated neural circuitry is responsible of such processing and that it implements suitable “internal models” which represent specific aspects of the sensory-motor loop. Internal models predict the sensory consequences of motor commands and, for this reason, are called forward models, as they model forward causal relations (i.e. to a future instant of time) between actions and their sensory consequences. A forward model is employed to predict how the state of the motor system will change in response to a given motor command (motor command ♦ state of the motor system). A forward model is therefore a predictor or a simulator of the consequences of an action. The inverse transformation from the state to the motor command (state ♦ motor command), which is needed to determine the motor command required to reach a given goal, is performed by what is called an inverse model. On the other hand, it is also clear that biological systems are first of all physical systems. For this reason understanding anticipation is a problem for any physical theory which aims at explaining the emergence of biological organization. For this same reason, the quest for such an explanation Anticipation in Biological and Cognitive Systems 373 requires both the clarification of the levels of description of the systems under study, and their integration in a unitary and more comprehensive theoretical framework. The main goal of this paper, then, is to identify a possible characterization of anticipation in biological systems which perform cognitive processing, in such a way as to bring to light some properties that any biologically plausible physical theory must incorporate.

2. Anticipatory Neurons Essentially, a formal neuron is function F which maps an input space I to a corresponding output space A. A non-anticipatory formal neuron is a function which has the form

at = F (it), where it and at are respectively input and activation of the neuron at time t. An anticipatory formal neuron instead can have the following form

at = F (it, it+τ ),

it+τ = G(it), where it+τ is computed by a suitable function G, named a predictor. If the activation at depends both on the input at time t and on activation of the neuron predicted at time t+τ the neuron has the following form

at = F (it, at+τ ),

at+τ = G(at).

Fig. 1. Three diﬀerent models of a neuron: (a) a standard non-anticipatory formal neuron; (b) an anticipatory formal neuron which anticipates over input; (c) an anticipatory formal neuron which anticipates over its own output.

This amounts to saying that the activation of a neuron at time t does not depend only on its input at time t, but also depends on its input (or in case 374 Graziano Terenzi on its output) at time t+τ, as it is computed by a suitable function G. Here, τ represents a “local time” parameter strictly related to predictor G. In the context of this model, we have to distinguish indeed between the global time t and the local time t’= t+τ. Whereas global time t describes the dynamics of function F, local time t0 describes both the activation dynamics and the learning dynamics of the predictor function G. Parameter τ identifies a temporal window over which the dynamics of the anticipatory neuron is defined. It sets up the limit of an internal counter which is used, step by step, to fix values of function G during the learning phase (approximation). In this sense τ represents a measure for the quantity of information that a neuron can store while executing its computation. This fact also means that the neuron is endowed with a kind of internal memory. This memory makes the neuron take into account more information than the information which is already available instantaneously.

3. An Example of Dynamical Evolution of a Single Anticipatory Neuron The following example illustrates the dynamics of a single anticipatory neuron trained to associate input and output by means of the Widrow–Hoff rule (Widrow and Hoff 1960). The predictor also is trained by means of the Widrow–Hoff rule. Let us consider a neuron with the generic form like the one illustrated in Figure 2.

Fig. 2. An anticipatory formal neuron with suitable connections weights.

Let us suppose that F is a sigmoidal function, that is it has the form 1 at = 1 + e−Pi where P is the activation potential of the neuron and it is computed as, X Pi = wij ij j Anticipation in Biological and Cognitive Systems 375 and where wij are connection coeﬃcients of the input links to the i-th unit, and ij are its input values. Let us suppose that G has the same form as F except for computing the quantity, i’t+τ , devoted to approximate its input at time t+τ instead of computing output at that is:

0 1 it+τ = 1 + e−Pi where P, i.e. the potential of G, is computed as the potential of F. As it can be seen, the anticipatory neuron is a system which includes two diﬀerent subsystems, the anticipator F and the predictor G. Whereas F computes its activation on the global time scale t, G computes its activation on the local time scale t’; and the activation is computed by storing in memory the input to the anticipator F both at time t and at time t+τ. Function G, on the other hand, is approximated by resorting to these values. Table 1 illustrates synthetically a generic dynamics for the previously introduced anticipatory formal neuron.

Table 1 - t = 0 - t = 1 - t = 2 - t = . . . - τ =1 - τ =1 - τ =1 - t’ = (t + τ)=1 - t’ = (t + τ)=2 - τ =1 - t’ =. . . . - t’ = (t + τ)=3 - it = i0 - it = i1 - it = . . . - it = i2 -INITIALIZE APPROXIMATE “G” APPROXIMATE “G” ...... VARIABLES - COMPUTE ERROR - COMPUTE ERROR ...... 1 0 1 0 -COMPUTE G EG = 2 (i1 − i1) EG = 2 (i2 − i2) ...... 0 i1 = G(i0) - WEIGHTS UPDATE - WEIGHTS UPDATE ...... ∂EG ∂EG ∆w21 = −η = ∆w21 = −η = -PROPAGATE AC- ∂w21 ∂w21 ...... 0 0 0 0 TIVATION = −η(i − i1) · G (PG) · i0 = −η(i −i2)·G (PG)·i1 0 1 2 ) ) ...... a0 = F (i0, i1) with P = (w i − s with P = (w i − s G 21 0 G G 21 1 G ...... - APPROXIMATE -COMPUTE - COMPUTE ACTIVA- “F” ACTIVATION G TION G ...... 0 0 i2 = G(i1) i3 = G(i2) -COMPUTE ...... ERROR - PROPAGATE ACTI- - PROPAGATE ACTI- ...... 1 T VATION VATION EF = 2 (a0 − a0 ) 0 0 a1 = F (i1, i2) a2 = F (i2, i3) ...... -WEIGHTS APPROXIMATE “F” APPROXIMATE “F” ...... UPDATE ∂EF - COMPUTE ERROR - COMPUTE ERROR ∆w12 = −η ∂w 1 T 1 T ...... 12 EF = 2 (a1 − a1 ) EF = 2 (a2 − a2 ) ∂EF ∆w11 = −η ∂w11 - WEIGHTS UPDATE - WEIGHTS UPDATE ∂EF ∂EF ∆w12 = −η ∆w12 = −η ∂w12 ∂w12 ∂EF ∂EF ∆w11 = −η ∆w11 = −η ∂w11 ∂w11 376 Graziano Terenzi

Essentially, in the course of the dynamical evolution of the neuron, for each couple of patterns presented to the neuron, a propagation cycle and a learning cycle are ﬁrst executed for the predictor G, and then a propagation cycle and a learning cycle are executed for the anticipator F. It must be underlined that the choice of Widrow-Hoﬀ rule for the training cycle of the neuron has been taken only for simplicity and for illustrative purposes; other algorithms could suit well.

4. Networks of Anticipatory Neurons By taking inspiration from the simple model of an anticipatory neuron introduced in the previous section, networks of anticipatory neurons can be designed that carry out complex tasks which involve the employment of suitable “internal models” (be they implicit or explicit) of their external or internal environment in order to take decisions. The scheme in Figure 3 describes a simple example of a neural network constructed by resorting to anticipatory neurons as its building blocks, which calculate their activation as a function of input values predicted at time t+τ. Input units within the scheme are represented by an array i of

Fig. 3. A simple neural network constructed by resorting to anticipatory neurons as its building blocks. input values. Hidden units are represented by a composition of sigmoidal functions, i.e. g and f respectively. Whereas the functions g within the hidden layer give rise to specific activation values, xg, represented here by an array xg, functions fgive rise to another set of activation values. Similarly, for the output layer, the activation of output units is computed by means of Anticipation in Biological and Cognitive Systems 377 composition of two suitable functions g and f, and their activation values are collected respectively in the arrays ug and uf . It must be stressed that each function g is endowed with a stack A, i.e. a memory which stores input vectors spread across a suitable temporal window t+ τ ; it can be represented by a τ x N dimensional matrix M, where N is the number of storable patterns, and τ is the time step when they are given as input to g. For each layer of the network there are two kinds of connections which play different roles: connections to functions f(f-connections) and connections to functions g(g-connections). Indeed, f-connections between input and hidden layers are characterized by the connection weights w (which in the figure are represented without indices); g−connections, on the other hand, are represented by connection weights w0. Similarly, on the one hand, f-connections between hidden and output layers are characterized by connection weights c; on the other hand, g-connections are characterized by connection weights c0. The case reported in figure shows a network in which each anticipatory neuron is endowed with a neuron-specific stack A which lets the neuron store and compare patterns given as input to the neuron in its temporal window (i.e. Input or output patterns, depending on the nature of the device). With regards to the latter, the parameter τ can be approximated by suitable learning algorithms, thus giving rise to networks which are able to perform tasks which require indefinite long-range time correlations for their execution. Training in such a network can be made in many ways. A basic option is to employ a back-propagation algorithm (Rumelhart, Hinton & Williams 1986) in order to learn weights w and c of the functions f within the layers of the network, and to use Widrow-Hoff rule for the weights w0and c0of the corresponding functions g. In the case of an “on-line” learning procedure, a learning step for the whole network can be implemented, for example, through the following succession of phases:

• Initialize Variables • Give a pattern as Input and Propagate Pattern • For each layer,

– Modify weights of functions g according to their temporal window – (Widrow-Hoﬀ) – Compute g – Compute f 378 Graziano Terenzi

– Modify weights of functions f according to the corresponding target pattern (Backpropagation) In table 2, a list of the basic variables of the model, i.e. a set of constructs that can be used to implement and simulate the model under study.

Table 2 Constructs Description i Input vector

xf Vector of the activation of functions f within the hidden layer

xg Vector of the activation of functions g within the hidden layer

uf Vector of the activation of functions f within the output layer

ug Vector of the activation of functions g within the output layer W Weight matrix of the f-connections between input and hidden layers W’ Weight matrix of the g-connections between input and hidden layers C Weight matrix of the f-connections between hidden and output layers C’ Weight matrix of the g-connections between hidden and output layers

Aw τ x N matrix wich stores N input to functions g of input-hidden layer within temporal window τ

Ac τ x N matrix wich stores N input to functions g of hidden-output layer within temporal window τ

5. τ -Mirror Networks: Architectures In the previous sections we have introduced both single anticipatory neurons and a simple network architecture. In the example of the last section each function g was connected to one and only one function f within its layer, and, moreover, it was characterized by a neuron-specific time window. This fact entails the employment of a specific stack for each function g. According to this description, the stack can be represented as a variable- Anticipation in Biological and Cognitive Systems 379 structure quadrimensional matrix which comprises 1) the parameter τ of the temporal window, 2) the number N of storable patterns, 3) a dimension which represents neurons, and 4) a dimension which represents the value of τ for those neurons. Another possibility is to consider network architectures where functions g are connected in output not only to a single corresponding function f but to all of the other functions f belonging to the same layer. Moreover we can consider a generalized window t+ τ for every g of the same layer or of the whole network. The set of all functions g of the network constitutes a real “temporal mirror” for the corresponding functions f. In a sense, we can say that a part of the network learns to represent the behaviour of the other part from the point of view of the specific task at hand, by constructing an implicit model of its own functioning. Briefly, the netwok is capable of self-representation. For this reason the units that approximate functions g of the network within the temporal window t + τ (i.e. the ones we have previously dubbed “predictors”) are called “τ-mirror units” of the network. The schemes presented in Figure 4 and 5 represent this idea synthetically.

Fig. 4. An anticipatory neural network which anticipates over its input space.

Figure 4 illustrates an anticipatory network which predicts over its input spaces. As anticipated, if each function g is connected to every function f within the same layer, and not only to the corresponding ones, then they can be grouped together in a suitable layer, called a “mirror layer”. Moreover, if the same temporal window t+τ is generalized to all of the mirror units within the same layer, then the complexity of both the representation and the implementation of the model can be further reduced. 380 Graziano Terenzi

Fig. 5. An anticipatory neural network which anticipates over its output space.

Figure 5 represents an anticipatory neural network which predict over its output spaces.

6. τ -Mirror Networks: Learning Algorithms We have previously introduced a learning scheme for the training of the neural network model under study. And this scheme is a supervised learning scheme which is based on 1) the employment of the Backpropagation rule for the adjustment of the weights of the functions f(W and C) and 2) the employment of the Widrow-Hoﬀ rule for the adjustment of the weights of the functions g. By the way, there are other possibilities that can be explored.

6.1. Reinforcement Learning A straightforward solution is given by the employment of a reinforcement learning algorithm for approximating the weights of the functions f. Typ- ically, reinforcement learning methods are used to approximate sequential dynamics in the context of the online exploration of the effects of actions performed by an agent within its environment (Sutton & Barto 1998; Bal- dassarre 2003). At each discrete time step t the agent perceives the state of the world st and it selects an action At according to the latter perception. As a consequence of each action the world produces a reward rt+1 and a new state st+1. An “action selection policy” is defined as a mapping from states to action selection probabilities π:S×A → [0, 1]. For each state s in S, a “state-evaluation function” is also defined, Vπ[s], which depends on the Anticipation in Biological and Cognitive Systems 381 policy π and is computed as the expected future discounted reward based on s: X X π 2 a π 0 V [s] = E[rt+1+γrt+2+γ rt+3+..] = [π[a, s] [pss0 (rt+1 + γV [s ])]], a∈A s0 ∈S where π[a, s] is the probability that the policy π selects an action a given the state s, E is the mean operator and γ is a discount coefficient between 0 and 1. The goal of the agent is to find an optimal policy to maximize Vπ[s] for each state sin S. In a reinforcement learning setting, the estimation of expected values of states s, V(s), is computed and updated tipically by resorting to Temporal Differences Methods (TD methods); this amounts to saying that the estimation of V(s) is modified by a quantity that is proportonal to the TD-error, defined as follows

0 δ = r + γ V (s ) − V (s). An eﬀective way to build a neural netowrk which is based on reinforcement learning is to employ a so called Actor-Critic Architecture. Actor-Critic

Fig. 6. An Actor-Critic architecture.

Architectures are based on the employment of distinct data structures for action-selection policy management (“actor”) and for the evaluation function (“critic”). The “actor” selects actions according to its input. In order to select an action, the activation of each output unit of the actor is given 382 Graziano Terenzi as input to a stocastic action selector which implements a winner-take-all selection strategy. The probability for an action ag to become the selected action aw is given by

mg[yt] P [ag = aw] = P , k mk[yt] where mg[yt] is the activation of output units as a function of input yt. On the one hand, the “critic” is constituted by both an Evaluator and a TD-critic. The Evaluator is a network which estimates the mean future rewards which can be obtained in a given perceived state of the world yt = st. The Evaluator can be implemented as a feed-forward two-layer network with a single output unit which estimates V’[s] of Vπ[s], where the latter is deﬁned as above. On the other hand, the TD-critic is a neural implementation of the function that computes the TD-error et as a diﬀerence between the estimations of Vπ[s] at time t + 1 and time t:

π π π π et = ((V [yt]))t+1 − ((V [yt]))t = (rt+1 + γ V [yt+1]) − V [yt] ,

π where rt+1 is the observed reward at time t + 1, V’ [yt+1] is the estimated value of the expected future reward starting from state yt+1 and γ is, as usual, a suitable discount parameter. According to such a TD-error, the “critic” is trained by means of Widrow-Hoff algorithm; specifically its weights wj are updated by means of the following rule π π ∆wj = η etyj = η((rt+1 + γ V [yt+1]) − V [yt])yj. Also the “actor” is trained by resorting to the Widrow–Hoff algorithm and only the weights which correspond the the winning unit are updated by the rule

∆wwj = ζ et(4mj(1 − mj))yj 0π 0π = ζ((rt+1 + γV [yt+1]) − V [yt])(4mj(1 − mj))yj, where ζ is a suitable learning parameter between 0 e 1 and (4 mj (1- mj)) is the derivative of the sigmoid function corresponding to the j-th output unit which wins the competition, multiplied by 4 in order to omogeneize the learning rates of the two networks.

6.2. Reinforcement Learning For Anticipatory Networks Extending a reinforcement learning algorithm to the anticipatory neural networks described above is quite straightforward. Let us consider an anticipatory network like the one described in section 4., but with two-layers Anticipation in Biological and Cognitive Systems 383 only. For this reason it will have a single layer of τ-mirror units. Then, let us consider an “Actor” constructed by resorting to the latter network; in this case, input units code for the state of the world at time t, whereas output units code for possible actions that can be executed at time t. The “Critic” can be considered as analogous to the one just discussed in the previous section. In this context, the approximation of weights for the functions f within the output layer of the network goes exactly as described in the previous section, that is by modifying the weights of the f-connections according to the TD-error computed by the Critic. But how can the weight over the g−connections be updated? Also in this case, the extension is straightforward if we consider the fact that the adjustment of the weights of these units is based on the Widrow–Hoﬀ rule. The idea is to employ the TD-errors corresponding to the reference temporal window rather than the quadratic error corresponding to input (or output) patterns. This essentially means that the stack of the g units has to store, beside the input patterns to g within the temporal window τ, also the corresponding TD-errors. Therefore, the error signal is computed not by considering the quadratic error with respect to suitable targets, but by resorting to the TD-error corresponding to the reference time step.

6.3. Other Learning Algorithms Beside the ones discussed in the preceding sections, there is a wider set of learning methods that could be employed for the training of this kind of networks. However, their application must be suitably assessed and analyzed ﬁrstly by discussing their theoretical implications. Even if it is not the goal of this paper to discuss this topic, we recognize that it is a very interesting possibility to extend the application of non-supervised learning algorithms (such as Kohonen-like-methods) to this training setting. Another possibility is to employ genetic algorithms.

7. Experimental Tasks for Simulation Assessment How can the viability of such an approach to the study and design of neural architectures be assessed? If, on the one hand, the assessment of the viability of a theoretical model essentially amounts to test its predictive and/or explicative power with respect to natural phenomena, on the other hand, the assessment of its utility from a practical point of view requires it to satisfy suitable constraints and goals within a design setting; in this sense 384 Graziano Terenzi it must solve specific scientific and design problems. Scientific evaluation, on its side, calls for the construction of a model whose behaviour can be confronted with real neuroscientific data. In this sense, the τ-mirror model can be employed in the context of problems which entail time-series processing. Problems of this kind are fundamental in the Neurosciences, but the domain which can mostly provide the basis for assessing the anticipatory architecture introduced above is the one that relates to the understanding and reconstruction of the neural mechanisms which perform sensory-motor learning and control. For example, a possible experimental task is based on an simulation set up which includes a simulated artificial environment interacting with an artificial organism endowed with suitable sensors and effectors; specifically it is characterized by a robotic arm. In this context the agent learns to perform a so called “pointing” task which consist of following a trajectory of an object which moves in the environment. Moreover it must learn to anticipate the trajectory of the object. In order to do so, the artificial organism must be endowed with suitable sensors and actuators; specifically it must be endowed with an artificial retina, a two-joint arm (whose position can be represented in a polar coordinate reference system ) and an anticipatory cognitive system.

Fig. 7. A possible setting regarding the simulation of a pointing task by a robotic arm.

A simpler simulation task, which is similar to the one just described, is to follow and anticipate a trajectory of an object in a more abstract manner, that is by employing the whole body of the simulated agent and not the simulated arm. Anticipation in Biological and Cognitive Systems 385

Other experimental settings can be conceived that involve the execution of both planning and navigation tasks in simulated environments. More abstract simulation tasks can involve learning and anticipation of things such as numeric series, grammars and automata, which do not possess necessarily a material counterpart.

8. Conclusions and Future Work The construction of neural network architectures endowed with self- representation abilities as well as with the proactive ability to show behaviours which do not depend only on their state at time t but also on their predicted state at a future time step is a strategic lever for understanding phenomena ranging from sensory-motor control to thought and consciousness. In this explorative paper we have introduced a possible form of both anticipatory neurons and networks which have been named “τ- mirror networks”. We have proposed some learning algorithms for this kind of networks and have put forward the possibility to extend other algorithms in order to train them. Among the future developments, a possibility (which has to be evaluated in the detail) is to change the form the anticipatory networks in order to make them more biologically and physically plausible. For example, Dubois (1998) has introduced an anticipatory form of the neuron of McCulloch and Pitts which is based on the notion of hyperincursion and that could be employed in the construction of another kind of anticipatory network. The hyperincursive computing systems are demonstrated to show stability properties much relevant if compared to the non-anticipatory counterparts. Therefore, a future development could be the integration of hyperincursivity within the theroetical framework presented in this paper. Notwithstanding the aforementioned theoretical developments, the most important and critical step is to develop a more comprehensive physical theory which will be able to describe anticipatory systems including those of the kind introduced in this paper. We strongly believe that such an integration is necessary to understand this very important aspect of biological organization. Without such a step, indeed, it would be impossibile to account for a wide range of phenomena occurring in both the biological and cognitive realms.

References 1. Baldassarre (2002) Planning with Neural Networks and Reinforcement Learning. PhD Thesis. Colchester-UK: Computer Science Departmente, 386 Graziano Terenzi

University of Essex. 2. Baldassarre (2003) “Forward and bi-directional planning based on reinforcement learning and neural networks in a simulated robot” in Butz, Sigaud & G`erard(Eds), Adaptive Behaviour in Anticipatory Learning Systems, Springer, Berlin, pp 179–200. 3. Dubois (1998) “Computing anticipatory systems with incursion and hyperincursion”. AIP Conference Proceedings 437, pp 3–29. 4. Gallese, Fadiga, Fogassi & Rizzolatti (1996) “Action recognition in the pre- motor cortex”. Brain 119, pp 593–609. 5. Haruno, Wolpert & Kawato (2001) “Mosaic model for sensorimotor learning and control”. Neural Computation, vol. 13, 2201-2220. 6. Jacobs, Jordan, Nowlan & Hinton (1991) “Adaptive mixtures of local experts”. Neural Computation, Vol. 3, 79-87. 7. Jacobs (1999) “Computational studies of the development of functionally specialized neural modules”. TRENDS in Cognitive Sciences, Vol. 3, No. 1, pp 31–38. 8. Jordan & Rumelhart (1992) “Forward models: supervised learning with a distal teacher”. Cognitive Science, Vol. 16, pp 307–354. 9. Kawato (1990) “Computational schemes and neural networks for formation and control of multijoint arm trajectory”, in Miller III, Sutton & Werbos (Eds) Neural Networks for Control, MIT Press, Cambridge, MA. 10. Liberman & Whalen (2000) “On the relation of speech to language”. TRENDS in Cognitive Science, Vol. 4, pp 187–196. 11. Luppino & Rizzolatti (2000) “The organization of the frontal motor cortex”. News in Physiological Science, Vol 15, pp 219–224. 12. McClelland & Rumelhart (Eds) (1986) Parallel Distributed Processing. Ex- ploration in the Microstructure of Cognition, MIT Press, Cambridge, MA. 13. Miller III, Sutton & Werbos (Eds) (1990) Neural Networks for Control, MIT Press, Cambridge, MA. 14. Rizzolatti & Arbib (1998) “Language within our grasp”. Trends in Neuro- sciences, Vol. 21, pp 188–194. 15. Rizzolatti, Fogassi & Gallese (2001) “Nerophysiological mechanisms underlying the understanding and imitation of action”. Nature Reviews Neuro- science, Vol 2, pp 661–670. 16. Rizzolatti & Craighero (2004) “The mirror-neuron system”. Annual Review of Neuroscience, Vol 27, pp 169–192. 17. Rosen (1985) Anticipatory Systems, Pergamon Press. 18. Rumelhart, Hinton & Williams (1986) “Learning Internal Representations by Error Propagation”, in McClelland & Rumelhart (1986). 19. Stolzmann (1998) “Anticipatory classiﬁer systems”. Genetic Programming 1998: Proceedings of theThird Annual Conference, pp 658–664. Anticipation in Biological and Cognitive Systems 387

20. Sutton (1990) “Integrated architectures for learning, planning and reacting based on approximating dynamic programmino”. Proceedings of the Seventh International Conference on Machine Learning, San Mateo, CA: Morgan Kaufman, pp 216–224. 21. Sutton & Barto (1998) Reinforcement Learning: An Introduction. MIT Press, Cambridge, MA. 22. Widrow & Hoﬀ (1960) “Adaptive Switching Circuits”, IRE WESCON Con- vent. Rec., pt. 4, pp 96–104. 23. Wolpert & Kawato (1998) “Multiple paired forward and inverse models for motor control”. Neural Networks, Vol. 11, 1317–1329. 24. Wolpert, Gharamani & Flanagan (2001) “Perspectives and problems in motor learning”. TRENDS in Cognitive Science, Vol. 5, No. 11. 25. Wolpert, Doya & Kawato (2003) “A uniﬁed computational framework for motor control and social interaction”. Philosophical Transactions of the Royal Society, 358, pp 593–602. This page intentionally left blank 389

How Uncertain is Uncertainty?

Tibor V´amos Computer and Automation Research Institute, Hungarian Academy of Sciences, Hungary [email protected]

The gist of the paper is the fundamental uncertain nature of all kinds of uncertainties and consequently a critical epistemic review of historical and recent approaches, computational methods, algorithms. The review follows the development of the notion from the beginnings of thinking, via the Aristotelian and Skeptic view, the medieval nominalism and the inﬂuential pioneering metaphors of ancient India and Persia to the birth of modern mathematical disciplinary reasoning. Discussing the models of uncertainty, e.g. the statistical, other physical and psychological background we reach a pragmatic model related estimation perspective, a balanced application orientation for diﬀerent problem areas. Data mining, game theories and recent advances in approximation algorithms are discussed in this spirit of modest reasoning.

Keywords: Uncertainty; Probability; Randomness; Epistemics; History of Sci- ence; Computational Pragmatics PACS(2006): 01.90.+g; 02.50.Cw; 01.65.+g; 02.50.r; 01.70.+w

1. They have No Science? We are unable to get deep into the minds of our ancestors. Nevertheless we can draw some hypotheses on their beliefs, on their view about the world order mirrored in their fantasy. With some certainty, we can state that most of the events, circumstances were uncertain for them, especially from our point of view. The difference in definition is relevant for the whole essay. What I take as certain, can be uncertain for any contemporary and even more for somebody living in the future and possessing a much deeper knowledge about the phenomena. The world was nearly totally uncertain for them who had to transfer all causes to imagined beings. Even the most certain event, death was fuzzified by several rites into a life transition and continuation in somehow similar manner. That, for us, primitive looking manner of burial cults survives still in more sublime forms of current high brow civilizations. 390 Tibor Vámos

Tales, sculptural figures, preserved folklores testify the hypothesis that due to a general feeling of uncertainty, being exposed to epidemics, dis- asters of natural environment, famine, the ancient man was much more hysteric than the everyman of today. That was true as long as the Middle Ages, Huisinga [21], a great scholar of the period, described that with lively documentation. The divorce of uncertainty and certainty started with science. Science interpreted in our sense, that is with the Greek enlightenment. Generally speaking, until the height of the Greek philosophy, people of the Mediter- ranean and Eastern civilizations collected plenty of remarkable phenomena, and some can be still in our eyes considered to remain of scientific character; but they looked at these phenomenological facts as naturally given realities (all kinds of spirits, gods included) and had no curiosity feeling to search for further causes, interactions, generalizations, especially not in the frames of an organized, culturally adopted search discipline. This means that, for the advanced Greeks, what could be included into the frame of their contemporary science was structured and disciplined in a new edifice of science; and all that remained, formed the uncertain realm of non-science, beliefs, religions, curiosities. In that evolving rationalism, combined with the mythical foggy environment of the Pythagorean age, even science and arts were not separated. According to a witty remark of Bertrand Russel [35], Pythagoras, could have been some kind of a mixture of Einstein and Mary Eddy Baker, the 19th century founder of the American Church, Christ Scientist. May we add Schönberg? Aristotle, in his unique effort in creating a rational, causal conceptual world, adhering to the sensible and conceivable world of the observable, separated the basic, causal and eternal looking primitives from the secondary, accidental features of those. He tried to do this separation with a reference of the accidental co-occurrence of the causal. That view is still surviving in the philosophy of strong causality, though what we consider to be causal now, after experimental demonstration of quantum physical entanglement is even more open than for the geniuses of the last two and a half millennia. The discussion of the current state of causal, accidental and other intermediate philosophical considerations, lay much beyond of the extent of this paper. We can only admire those pioneers of scientific thinking and thinking in general who tried to understand the world, first without prejudices and without any of the fantastic experimental observation devices of modern ages. The Aristotelian analysis of the accidental secondary looking features How Uncertain is Uncertainty? 391

[3] conclude in a definition very close to our information theory view about signal and noise. And, a wonderful scientific manifesto added to all those: I do not dwell with these secondary (συµβεβηκòς accidental) phenomena because they have no science The inclusion of these, mostly affections related phenomena, into the scope of science, as we understand now, is a rather recent advance as modeling of human behavior in economy and social sciences became possible subjects of computational efforts. Aristotle was “only” a beginning of modern scientific thinking but a long row of Greek philosophers added still remarkable thoughts to that beginning, some contradicting to the rather fundamentalist looking causal- logical teaching of Aristotle. The dogma of the excluded middle was and remained the cornerstone of that way of thinking. Yes or not, true or false is always a pragmatic, sometimes necessary simplification of problems, though the reality covers the whole space between the two.

2. The Skeptics, Forerunners of Modern Science Philosophy From the few but in remains admirable and admirably rich treasury of Greek philosophy we give a prominence to the Skeptics. The unclear origins go back to the fourth-third century BC, to Pürrhón(Πυρρω0ν), and continued under heavy criticism and several later additions to the texts of Sextus Empiricus [37] and Diogenes Laertios [13] in the second and third century AD reporting especially the modalities of Aneisidémos(Aνεισιδηµóς), who lived in the first century BC. In these modalities, the tropos (τρóπoς), are collected ten various arts of relativities in the perception of the reality. Most of these relativity modalities are parts of our commonsense thinking that, per se, are strikingly modern in our eyes, and provoked well understandable resistance on behalf of the most lucid contemporary Greek minds. We can recognize rather precise definitions of modern ideas about the relativity of scientific ideas due to social conditions, i.e. the ideas of Bloor [6] and the school of the social-philosophical “strong program”. Even more interesting is the tropos about the fuzzy nature or estimations, not different from the current fuzzy concept of Lotfi Zadeh. The world of uncertainty was alien for the rational, scientific oriented philosophy that tried to include all phenomena into well understandable and sensible general frames without the instruments of further periods that could differentiate much more, based on more sophisticated theories about causal relations. 392 Tibor Vámos

3. Nominalists of Middle Ages: de dicto - de re Uncertainty The philosophy of the Middle Ages, referring to several ideas of the Greek philosophy, concentrated on the divine organized, overall regulated world view and, on the other hand the same divine, not understandable creation of mysticism, surprising wonders of the everyday life, i.e. on those, not comprehended by the contemporary discourse of science. For us, the most relevant stream of thinking is the growing deepness of nominalism, the separation of the name from the named, de dicto – de re connection and antithesis. In the course of developments in thinking about logic Kneale, W. and Kneale, M. [24] proved an excellent orientation. That line of thinking is one of the most important research areas of communication, the problem of adequate information emission and its perception. The novelty is the inclusion of the third constituent, the vehicle of information. Nevertheless, we can trace some past characteristics of the vehicle of information history by concerning the destiny of scripts and books. Pro captu lectoris habent sua fata libelli, the booklets have their own fate, how they are accepted by the reader, wrote Terentianus Maurus, a gram- matologist in the second century. By the ubiquity and instant nature of electronic information world, these questions received a highly practical relevance. One of these aspects is the indirect, electronic representation of the individual, of the Self, used for all kinds of substitution of the direct presence. Personal identification is a crucial subject of individual freedom, privacy and security. The same, relativistic nominalist problem arises in the effects of various mass media, unprecedented manipulation technologies by electronic communication. The problem of legal regulation is now an everyday and ubiquitous scruple, depending on the supposed weights of communication perception of the population concerned. We can return to the Aristotelian dictum: has no science with the relevant complement: has no unified science and a disputable, not provable remark: will not have a unified science. Unified science is addressed, in spite of their theoretically provable incompleteness and continuous developments, moreover reinterpretations, not similar to other branches of mathematics.

4. Mathematics as the Last and Computable Form of any Science Here we have to make a small detour defending that rather awkward definition about science, mathematics and computability. We speak about How Uncertain is Uncertainty? 393 science, in the Anglo-Saxon sense, i.e. science in the sense of reproducible, experimentally strongly provable phenomena, those having firm definitions for models of reproduction under defined conditions; and these model definitions are firm enough to be translated into disciplinary formulae and mathematics. The last step is more obvious for the present everyday view: based on the mathematical formulae the model can be programmed and the outcome, the computational reproduction computable. The last statement clarifies the way of reasoning. First in physics, but now every practically manageable problem of biology related medicine and agriculture, statistics and finance related economy followed the overall dom- inating modern production processes of technology: computer aided design, computer aided scheduling and manufacturing. All additional problems (remember the Aristotelian συµβεβηκóς, i.e. additional, secondary, accidental) are related to some kinds of uncertainty, not well computable incidents.

5. Ars Coniectandi, Statistics Ian Hacking0s excellent book, The Emergence of Probability [15] ries to analyze the other reasons of the late emergence and the relevance of Pascal’s [32] and Jakob Bernoulli’s [5] contribution to the ars coniectandi, the art of dicing (and we add, the same Latin word is for reasoning based on a guess) and soon later to the fast growing interest on problems, related to statistics in economy, social conditions, meteorology and measurement in experiments on physical, chemical and biological phenomena. Observation of returning frequencies and estimation of future events run connected and independent; the second more on the Bayesian conditional probability lines. The detailed history of mathematical ideas about probability and the roles of great mathematicians are well detailed in that cited book of Hacking [15] and many others e.g. recently by Halpern [17]. Models of uncertainty defined the progress of calculations. For a rather long time dicing and similar games served as models, the way of observation developed statistics and the epistemic curiosity was the contradiction between the rather good approximations given for large numbers of experiments and the complete uncertainty of a single roll. Statistical models followed the hypotheses about the nature of the investigation. Typical are distributions around a certain mean value, variations like the biological data of height, weight, life expectancy, etc. symmetric and not symmetric. E.g. life expectancy at a certain advanced age is typically asymmetric. Another model, used later for many purposes, is the reliability of 394 Tibor Vámos content estimation in beer bottles, the Student distribution, measured on a limited number samples. The problem is similar to the Nala – Kali story of mythology from India, that will be cited a little later, but now it received a firm scientific analysis and a conjecture, where to use, how to use? This remark put light to the question, where does science start and where sophisticated observation ends.

6. The Models of Random Motion A series of application rich models started with the collision problem of rigid and elastic objects, first, the Brownian motion but expanded towards the more and more sophisticated variants of different border conditions, objects, elasticity, not only in physics but in move of human and other hysteric masses. For masses, events where the motion is not primarily defined by collisions the Monte Carlo random walk model is applied best, as one of the main patterns of data mining or any other search in a completely unknown distribution field. A further constraint of independence from previous events, or remembering only a fixed certain number of preliminaries, is the Markovian model, applied for instance in the great search machines. These methods work mostly in search on the internet, i.e. on huge amounts of information.

7. Models as Hypotheses or Prejudices The models serve as hypotheses and prejudices, consciously or hidden. The best example of an ideal model is the duly celebrated Kolmogorov [26] model of probability, with the condition of the closed world assumption based on the excluded third logic and the stationary temporal behavior, ergodic, i.e. equal chance for experimentation with one sample many times or many samples. The classic probability hypothesis works well in those worlds, ensembles of events that can be approximately modeled by these constraints, on the other hand not appropriate applications can be totally misleading, especially those related to relatively fast changing conditions, not with constant frequency changing events. Typical examples are the stock exchange prices in hectic, crisis periods or application of experience with mental and physical capabilities of significantly different ages, environments. The circle looks to be really vicious: having a number of observations we start to calculate evidence on the basis of a certain theoretical, or experience suggested model and get results that should be checked on the same How Uncertain is Uncertainty? 395 hypothetical model or supposed similar circumstances. The whole history of science can be recounted on that scheme and the only consolation what we can attribute as virtue, practicality and power of science is the continuous mass refinement of model approximations and getting more and more proofs for better forecasts. The story of all more or less sophisticated technologies, mass production and reproduction is one of that proofs, the same is more or less valid for progress in medicine, much less in pedagogy and economy. Science can be considered as an infinite story of model approximations with check of experimental improvement, receiving more justifications, more standing forecasts.

8. Mystic of Uncertainty, Mystic of Life: The Early Message from India Uncertainty starts to be nonexisting if it can be modeled in the way and reliability of scientific certainty. That is the reason why we cannot speak about uncertainty in general, as about a subject of science and not as a concept of philosophy and general discourse. This mystic of uncertainty could be one of the reasons why European science alienated itself from these irrational looking problems. Indian mathematics was put in a different framework, may be due to the more natural attitude to nature. That was first admired by Europeans in respect of love, after this continent started to liberate itself from the ideological shackles by the fresh air of the Renaissance and of the Age of Reason. One of the most famous love story about a success, related to statistical estimation, was the tale of Nala and Kali, who latter was also a demigod of dicing! Nala could give a perfect estimation about the number of leaves and fruits on a big tree by calculating the relation between two, typical singular branches and of the whole. The original story was more ancient than the last referred version from about 400 A.D.

9. Where does Science Start and where Sophisticated Observation Ends, a Message from Persia: Serendipity The other story from a rich civilization is the Persian fairy tale about the three princes of Serendip. The narrative is a metaphor of making fortunate and unexpected discoveries by accident. The story was made popular by Horace Walpole but the beautiful origins and witty metaphors are best in the original tales, and the reader is advised to look after the excellent reviews to be found in details on the internet. 396 Tibor V´amos

The answer from the European scientist after the blossoming Age of Reason, Pasteur: In the ﬁeld of observation chance favors the prepared mind. The question is open for several interpretations as any, related to the general subject of uncertainty.

10. An Uncertain Classification of Uncertainty Returning to the different arts of uncertainty, a rough and somehow arbitrary classification can be given:

• objective, physical uncertainty of the quantum world; • uncertainty due to some existing but not yet discovered phenomena, e.g. causes and medicament of some diseases; • uncertainty due to non computable complexity, e.g. long range, exact meteorological forecast; • uncertainty due to lack of knowledge, non availability of data, especially of historical events; • uncertainty due to unknown, unexpected interactions, especially human behavior.

These different kinds of uncertainties can reach a more definite classification after a clarification of the problem, i.e. after scattering the fog of uncertainty. The explanation of human behavior can be ordered into discovery of a neural disorder, due to anticipation of a meteorological change, uncertainty about a not yet disclosed verdict and of the not yet discovered interactions in the brain. Looking at this enumeration it is clear that in spite of the uncertainty of typology the different cases require different approximation models. The model matching is sometimes more an art of serendipity than science of Nala and Kali, an art of sensitivity for the opportunities and pitfalls of analogous thinking. All these are good reasons why scientists hated or abhorred uncertainty and why it is the great challenge of science now. The epistemic loop of observation, creating hypotheses based on observations analogies, affirmation these hypotheses by new observations and consolidating that knowledge by theories, is the progress rout of current science. The difference to the past is the availability of immense observation data, possibility of review their features by renewed, moreover methodically revised observations. That loop is strengthened by the critical norms of unbiased science, its possible independence from ideological biases. How Uncertain is Uncertainty? 397

11. The Pragmatic View of Methodologies From our point of view we emphasize two relevant features of the process: the critical and pragmatic view of the methodologies and the computer science support of data management, especially data mining. Methodologies are enumerated here only. They are well discussed in excellent textbooks and university courses. The enumeration serves the overview of practical but not general devices as a set of tools for manufacturing diﬀerent shaped surfaces:

• statistics, with its temporal, conditional, dependence constraints and evaluation methods; • classic probability, with its definitional constraints and developments for calculation the effects of disturbing these constraints; • Bayesian methods for conditional probabilities, estimations with combination of statistical, probabilistic data; • possibility methods (like Dempster-Shafer, certainty factor, etc.) with attention to combinations of statistics, probability concepts and anomalies related to cumulating of marginal estimations; • fuzzy methods for estimation of memberships within hypothetical or real groups if instances, major method for using human estimations and neglecting uncertain relations of distributions. In its non probabilistic view the method has a different course in acceptance and applications.

All these methods have some natural roots in set theory as dealing with hypothetical sets of data, instances, qualifications; in algebra and logic due to the tasks of combination uncertainty data. All these methods, sooner or later, have less or more attention to the problem of evidence based on measurement data and on human estimation. Even the theoretically strongest classic probability theory had a rather early branch called subjective probability. We [38] and the literature use the words probability, vagueness, possibility, evidence, estimation in different contexts and textbooks in similar manner and in strictly defined separate meaning with reference of the origin of data, the calculation method, the state of reliability, measure of confidence. This variety of definitions’ naming reflects the intrinsic nature of uncertainty. 398 Tibor Vámos

12. Where are we Now? The greatest diﬀerence to earlier situation of science was underlined: • the availability of never before reliable, wide range instrumentation increasing the possibilities of human observation by about thirty magnitudes; • the possession of never before imaginable amount of observation data and computational power, mathematical methods for management of those; • a continuously developing epistemic discipline for critical revisions of hypotheses and theories, freedom of since achieved in permanent struggle against interests, alien to science.

13. Data Mining Data mining is a collective designation of all modern methods for discovery connections, structures among seemingly unstructured data. The first answer was the application of classic methods in statistics: significance, factors, etc. Soon turned out that in case huge amounts of data that is mostly not enough, the relations are more hidden than according to the popular saying, needle in a haystack. The escape route returned to the model hypotheses but in a conscious way of hypothesis control. The challenge mobilized a wide variety of mathematical disciplines. The most frequently applied methods are the hidden markovian procedures, search for returning phrases, expressions, contexts. From this short reference it is visible that sophisticated connection searches should apply several tricks, hypothetical schemes and a possibly comprehensive expertise within the special subject concerned. Data mining is in most cases equivalent with the search for hidden structures. Mathematics in some sense is the theory of structures, search for structures and building structures. All disciplines of mathematics can be interpreted in that way and this generalization helps the continuous interplay within the realm of mathematics, especially between geometry, its further abstractions, like topology and all others, algebra, algebraic foundations of logic, analysis, etc. The search for structures in data is a search within the certain branch of science and practice, related to technology or any phenomena of nature, the human being included. If we accept this interpretation, the search can be understood as application of knowledge about abstracted structures to the investigated discipline and in between the method and the special How Uncertain is Uncertainty? 399 discipline, the hypothetical model. These selected models are the guesses of the solution hypotheses and further some guiding hints to the possibly independent methods of hypothesis control. Possibly independent refers to the model itself and the computational procedure, too. The search applies the old usual methods, too i.e. analysis, matching data to some basic functions of temporal behavior and frequency analysis based on the classical methods and on different waveforms, like wavelets, fractals. These methods are in some sense closest to the model view, to the evolutionary exponential and logarithmic decay, to powers of energy relations and further, well known actions and interactions of complicated motion or frequencies of all kinds of oscillations, returning phenomena. Functional analysis is in some sense a search for a model, for the best looking class of functions and by that a reverse reasoning to the nature of the data scheme, the model. The application of functional analysis approximation search is a recent attempt in mining economy and psychology data [1].

14. Spaces of Events and Search The first model and even linguistic metaphor is geometry. The fantasy rich development of Riemann-surfaces, non-Euclidean geometry, topology [14], the new world of related metric is the most promising present and future field of model building and search procedures on these strange, multidimensional spaces. These spaces are strange for a traditional view of a human culture, first attached only to the two-dimensional surface of a flat looking earth and slowly accustomed to a similar view in the third dimension. Modern science has shown that nature is not as unimaginative as we were. The strange multidimensional, topological space is really the dynamic representation of nature, not only of the far cosmological one but representing all kinds of internal and external phenomena, those not understood until now. Not understood is a highly relative attribute. The models and so the spaces of present science are rather strictly bound to our early visual im- pressions; and though they are beautiful and admirable further abstractions, but “only” expressions of relations in our language. Language means also abstractions of natural language towards any means of representation. The vicious circle has pragmatic solutions. The models are understood in that working sense, they are accepted as long as they are in conformance with the data, the observations. Data mining is the search for a good model, good in adjustment to the data, in coherence with other related knowledge 400 Tibor Vámos and with possibility of computational application. Model means in this respect the relative understanding, the place of the model within the general world of current knowledge. Nothing more in the view of final totality and nothing less than an admirable instrument of everyday life, of orientation in the complexity of the practical world.

15. Restitution of Approximations, the Sorcerer0s Hulls The next remark concerns the approximation itself. The origins of mathematics were for a long time the art of approximations, approximations in the sense of calculation everyday phenomena. Especially after the rev- olutionary achievements of science in the 19th century, approximation for mathematics was a kind of dirty, fuzzy practice, not the world of elegant mathematical theory. That view changed once more in the past very few decades. The change is due to the dramatic development of computer power and in connection with that a new horizon of the earlier only theoretically considered limits of computability. As stated before, one of the arts of uncertainty is the computability limit, because of the exponential growth of complexities. That art of uncertainty promoted to be a major obstacle, possible, the major obstacle. All others, related to the unknown, not discovered, not understood phenomena are or can be the subjects of further research, new experimental and theoretical ideas, they are open challenges for research. Even the most quoted example of objective uncertainty, the Heisenberg state-measurement puzzle is or can be an emerging problem from the realm of mystery. High complexity, problems beyond the current physical computability limits have a different nature. The mathematical background, the nature of problems where the models are unavoidably beyond the limits of polynomial time solutions, the classes of these hard algorithmic problems is now the subject of excellent university textbooks, not to be detailed here [20]. The efforts for model transformations are until now partly limited for several important real cases, how to interpret some non polynomial models as polynomials related to the algorithmic operation requirements. The celebrated traveling salesman problem0s broad analogy class belongs to those. A great many of environmental and mental subjects lie certainly beyond those, in the infinities of computational steps. If we accept the theses about the modeling role of mathematics for any scientifically treatable problem and the conclusion of manageability of those all, by computation, than we find the new limit of knowledge in the How Uncertain is Uncertainty? 401 computability problem. Short: if not computable, not manageable by the firm reproducible methods of rational science. Not too much satisfactory for any kind of rational, goal oriented human intellect. That is the main reason why approximation reached a new status in rigorous science and by that and as a natural feedback, great new theories, mathematical instruments of approximation calculi and their of the confidence of those. The best example is the development of topology related methods, creation of possibly convex hulls around the feasible solutions, reduction, transformation of these hulls into manageable dimensions and patterns. The manageable mean realistic conditions of computation and of pattern matching. The last refers to the analogy thinking, finding similarities, measures of similarity and dissimilarity. All these are, on one hand by highly sophisticated and until now only perceivable for very distinct mathematical brains, and on the other hand, in fast progress, practical tools for routine engineering design. All problems of nonlinear and stochastic nature, and that is the practical case for most engineering tasks, are somehow covered by these Sorcerer0s hulls. Similar to that trend was accepted the idea of demonstrating mathematical proofs by computational experimentation. That has also a very relevant practical meaning, not only for the development of mathematics, but in the mirror of the above applications, a new instrument for validity, confidence estimations, a key problem for engineering design. Not completely independent is the renaissance of number theory as a new, stimulating subject in modern cryptography and in general search procedures after structures in huge data masses. An apparent connection point is the ancient problems of prime numbers, as somehow mystic elementary building blocks of all numbers, and by that for hidden keys and hidden structures. The secrecy of the keys is a routine requirement of the whole society, the discovery of hidden structures is an objective of most research problems, e.g. pharmaceutical chemistry and genetics.

16. Compromise: Inclusion of Human Estimates and Conclusion about the Ubiquity of Research The third novelty is also an unorthodox development: for calculation of human influenced processes, that is economy, other types of negotiations, planning strategies in complex action tasks the guess of subjectivities is unavoidable. Analyzing the human decision process two hypotheses and some other won Nobel- prizes. The first group is characterized by the modern continuation off classic, Adam Smith founded rational choice school, later 402 Tibor Vámos enriched by Max Weber, Talcott Parsons, Buchanan and Tullock and further, the other by the psychology based groups, characterized by the works of Kahnemann, Slovic, Tversky and their followers. All these is flavored by he Condorcet-Arrow line of proof for the not solvability of optimal choice between more different, well founded interests and not even the possibility of a definitely acceptable voting system. These developments were applied in the frames of the Neumann initialized game theories and the 18th century historical beginnings of optimization continued now by theories of reasonable compromises, balances, started by Pareto and in our age, Neumann, Nash, Harsányi, Black and Scholes, Aumann and Schelling, only to mention the Nobel-class of economy. The main gist was and is the balance among different attitudes, objec- tives, environments by combinations of rational strategies and motivations of individuals and various masses, explainable by present circumstances and by the several hundred million years evolutionary heritage of the human animal. That immense task of present human existence and coexistence demonstrate the ubiquity of uncertainty, of its research and the intrinsic complexity, moreover coherence of all related scientific disciplines.

References 1. Aczél,J., Falmagne, J. C., Luce, D. (2000) Functional equations in the behavioral sciences, Math. Japonica 52 3, pp 469-512. 2. Andréka, H., Madarász,J., Németi,I. (2006) Logic of space-time and relativity theory, Book manuscript. 3. Aristotle, Metaphysics, Dir William David Ross ed., Oxford, Clarendon Press 1924, repr. 1981, Book. K8, 1065a line 38–39. 4. Bayes, T.: An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53, pp 37–418, 1763 5. Jacob Bernoulli (1744) Opera Omnia 2. Vol. Geneva, repr. Bruxelles, 1975. 6. Bloor, D. (1976) Knowledge and Social Imaginary, Routledge and Kevin Paul, London. 7. Bollobás,B., Varopoulos, N. T. (1975) Representation of Systems of Mea- surable Sets, Math. Proc. Cambridge Philos. Soc. 78, 2, 323-325 8. McCarthy, J.:Epistemological problems of artificial intelligence, Proc. IJ- CAI’pp 0-1038-1044,Cambridge, MA, MIT Press 9. Chazell, B. (Dec 2006) Proof at a roll of the dice, Nature, 28, pp. 1018-1019 10. Chomsky, N. (1955) Logical Structure of Linguistic Theory. MIT Humanities Library. Microfilm. 11. Darwin, C. (1979) The Origin of Species by Means of Natural Selection, 1859-1872 Random House Value Publishing, New York. 12. Dennett, D. (1995) Darwin’s Dangerous Idea, Simon & Schuster. How Uncertain is Uncertainty? 403

13. Diogenes, L. (1925) Lives of Eminent Philosophers, tr. by Hicks, R.D., Lon- don, 2 Vols., Loeb Classical Library, Harvard University Press, Cambridge, MA. 14. Gromov, M. (1998) Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser,Boston. 15. Hacking, I. (1975) The Emergence of Probability. A philosophical study of early ideas about probability, induction and statistical inference, Cambridge University Press, Cambridge. 16. Haenni, R., Hatmann, S. (2006) Special Issue of Minds and Machines on Causality, Uncertainty and Ignorance, Mind ann Machines 16, 3. 17. Halpern, J. (2003)Reasoning About Uncertainty, MIT Press. 18. Halpern, J., Pearl, J. (2001) ”Causes and explanations: A structural-model approach. In Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence, San Francisco, CA: Morgan Kaufmann, pp 194–202. 19. Hardy, G. H. (1940) A Mathematician’s Apology, Cambridge University Press. 20. Hromkovic, J. (2002) Algorithmics for Hard Problems, Springer. 21. Huisinga, J. (1937) The Waning of the Middle Ages, London. 22. Jaeger, G. (2006) Quantum Information: An Overview, Springer, Berlin. 23. Jones, S. (1999) Almost Like a Whale: The Origin of Species Updated, Dou- bleday. 24. Kneale, W., Kneale, M. (1962) The Development of Logic, Oxford University Press, Oxford. 25. Kornai, A. (2006) Mathematical Linguistics, Book manuscript. 26. Kolmogorov, A. (1933) Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer, Berlin. 27. Kuhn, T. S. (1970) The Structure of Scientific Revolutions, International Encyclopedia of Unified Science, University of Chicago Press, Chicago. 28. Lakatos, I.: Falsification and the Methodology of Scientific Research Pro- grammes, in I. Lakatos and A. Musgrave (eds) Criticism and the Growth of Knowledge, Cambridge, Cambridge University Press, 970, pp 91–196 29. Minsky, M. (1985) The Society of Mind, Simon and Schuster, N.Y. 30. Minsky, M. (2006) The Emotion Machine 9/6/2006, draft book. 31. Popper, K. (1972) Conjectures and Refutations, Routledge and Kevin Paul, London. 32. Pascal, B. (1654) his first ideas about the new science of probability in: Oeuvres de FermatT, Vol. 2, (Eds.: P. Tannery and C. Henry) Gauthier- Villars, Paris 1894. 33. Pearl, Judea (July 2004) ”Robustness of Causal Claims” In Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence, AUAI Press: Arlington, VA, pp 446–453. 34. Ramsey, F. P. (1931) Truth and Probability, The Foundations of Mathemat- ics and other Logical Essays, ed. R.B. Braithwaite, London. 35. Russel, B. (1946) A history of western philosophy, George Allen and Unwin, London. 404 Tibor Vámos

36. Savage, L. J. (1954)The Foundations of Statistics, John Wiley and Sons, New York. 37. Sextus Empiricus, 4 Vols., tr. by Bury, R. G., Loeb Classical Library, Har- vard University Press, Cambridge, MA, pp 1933–1949 38. V´amos,T. (1991) Computer Epistemology, World Scientiﬁc, Singapore. 39. Venn, J. (2006) The Logic of Chance, London, 1866, Dover Publications. 40. Weinberg, S. (2003) Facing up and Its Cultural Adversaries, Harvard Uni- versity Press, Cambridge, MA. 41. Wittgenstein, L. (1956) Tractatus logico-philosophicus, Routledge and Kevin Paul, London. 405

Archetypes, Causal Description and Creativity in Natural World

Leonardo Chiatti Laboratorio di Fisica ASL VT Via S. Lorenzo 101 01100 Viterbo, Italy ﬁ[email protected]

The idea, formulated for the first time by Pauli, of a “creativity” of natural processes on a quantum scale is briefly investigated, with particular reference to the phenomena, common throughout the biological world, involved in the amplification of microscopic “creative” events at macroscopic level. The involvement of non-locality is also discussed with reference to the synordering of events, a concept introduced for the first time by Bohm. Some convergences are proposed between the metamorphic process envisaged by Bohm and that envisaged by Goethe, and some possible applications concerning known biological phenomena are briefly discussed.

Keywords: Metamorphosis; Synordering; Quantum Leap; Archetype PACS(2006): 91.65.Pj; 03.65.Fd; 87.10.+e; 77.84.Dy; 77.80.-e

1. Introduction Not many researchers today would be convinced that the Naturphilosophie approach to the knowledge of nature would return. Even though excellent arguments have been provided to support this return (Card [1996]), there is no doubt that it meets a great deal of resistance. In fact it is vital that the assumptions of the Natural Philosophy are reformulated in order to har- monize them with contemporary natural sciences. It is necessary to recover the contrast between the causal, local and mechanistic descriptions which these sciences have chosen and the holistic, morphological and qualitative inclination that has always characterised the philosophy of nature. Is this possible? Does an authentic form of “creativity” in natural world exist, one that is not a mere epiphenomenon of basic causal and deterministic processes? And how should these unpredictable “creative” phenomena be described, since they cannot be represented by the predictive models formulated in the context of the mechanical causality? Could these descriptions 406 Leonardo Chiatti utilise the same concepts familiar to the once professed Naturphilosophie such as “archetype” (Goethe’s Urbild) or “metamorphosis”? In the attempt to address such diﬃcult questions, and without any pretence of resolving them, we will take into consideration not the speciﬁc aspects of the naturalistic realm, but those of microphysics. If we wish to reformulate these questions in really innovative terms we must begin our investigation at molecule level, from the atoms and from the subatomic particles. Subsequently, we will be in a better position to discuss the macroscopic world of daily experience and its environment, the biosphere.

2. Quantum Level: The Physical World as a Network of Events One of the fundamental aspects of the structure of matter is atomism: each physical object is an aggregate of homogenous entities subdivided in classes of elements identical to each other. Among these we can enumerate the atoms and molecules: all the atoms of iron in a determined quantum state are identical, and the same applies to water molecules. The set of the internal states of each of these micro-entities is generally discrete, not continuous; this characteristic is not common in macroscopic objects, whose configurations generally can be varied continuously. Molecules and atoms are not elementary: clashing against each other at sufficiently high energy, these entities are broken into smaller entities, such as electrons and nuclei, thus they are no longer atoms nor molecules. Enti- ties are considered elementary when, however high the energy involved in the collision process, the products obtained still belong to the same class of reagents. This occurs to those micro-entities called “elementary particles”, such as electrons, protons, neutrons, etc. The term particle is derived from the Latin word particula, i.e. small part. It reminds us of that particular entity that in classical physics is called “material point”: a point provided with mass that moves describing a trajectory in space and persisting for a determined time. This image has generated an infinite number of controversies over the real nature of elementary particles, which are not the material points of classical physics. In fact the material point is an abstraction that does not have a genuine counterpart in nature other than as an idealization that can be mapped on certain classical physical systems in some circumstances (i.e. “small” object of negligible size), and dynamics of micro-entities such as atoms, subatomic particles, nuclei is in effect completely different to that of the classical macroscopic objects. Micro-dynamics is described, at a level Archetypes, Causal Description and Creativity in Natural World 407 today considered fundamental, by the quantum field theory, while a more approximate description is given by quantum mechanics. Since the aspects of micro-dynamics which we wish to focus on briefly are qualitatively the same in both descriptions, we will limit our discussion to quantum mechanics (Caldirola [1974]). In order to adequately understand the significance of what we intend to demonstrate, it is necessary first of all to eliminate some classical prejudices, whose persistence has contributed in creating the myth of incomprehensi- bility of the quantum level of reality; a myth which is deeply rooted in many physicists. In the macroscopic world of daily life the distinction of an entity and its properties is obvious: the entity is, in a certain sense, the support of its properties. For example, a leaf is a different thing compared to its colour, its form, its dimensions, etc. The group of properties of the leaf is not the leaf itself, intended as a material substance endowed with those properties. Therefore an object is considered to be a material support persistent in time and to which given characteristics can be attributed, independent of possible interactions with other objects. The situation of a micro-entity such as an elementary particle, an atom or a molecule, is completely different: a micro-entity is nothing other than the connection of two particular events, one following the other in time. The first event is the creation (that is the physical manifestation) of a determined set of characteristics (A), while the second event—chronologically successive to the first—is the destruction (physical de-manifestation) of a second set of characteristics (B). Set A is the “initial state” of the micro- entity, while set B is its ”final state”. These events are the effect of the interaction of the micro-entity with other micro-entities; for example but not necessarily, with those macroscopic aggregates of atoms which are measurement apparatuses (Fock [1978]). One should pay particular attention to the fact that during the interval between the first and the second event the micro-entity does not exist at all; in particular, it cannot be conceived as a corpuscle that follows a trajectory or as a wave that is propagated in space, the two modes of matter and energy transfer known in classical physics. Nor can it be conceived as a combination of these two modes, as various theorists of the 20th century (De Broglie, Vigier and many others) hoped, elaborating many different (and often internally inconsistent) models that have not been supported by experiment. Reducing the existence of the micro-entity to only two separate instants 408 Leonardo Chiatti means that the micro-entity is not a persistent material object. Such condition is only apparently paradoxical: an elementary entity is by definition without an internal structure, therefore its “structure” can be defined only by its behaviour during the interaction with other similar entities. But the events “creation of A” and “destruction of B” are precisely interactions of the micro-entity with other micro-entities or aggregates of micro-entities, therefore it is only in these events that the “structure” of the micro-entity can be and is manifested: it is manifested as the initial or final state of that micro-entity. To ask ourselves what is the state of the micro-entity in an intermediate instant has no sense, unless an interaction of the micro- entity with other micro-entities occurs at that instant; but in this case a new and completely different interaction scheme is defined, in which the micro-entity created in state A during the first event is destroyed in state C at the intermediate instant, with the creation of a new micro-entity in the same state C; a micro-entity which is subsequently destroyed in the final state B during the last event. In other words, the fact that the structure of the micro-entity is definable only during external interactions limits the possibility of defining its physical state only at those instants in correspondence to events of interaction; in all other instants the physical state of the micro-entity is not definable. For example, the position of the micro-entity is not defined, thus the micro-entity cannot be localised in space. An important consequence is thus derived: the micro-entity cannot be viewed as a material substance that persists in time and is extended in space, support of its own characteristics. Therefore, the micro-entity is nothing other than the combination of characteristics A and B: it is not different from its initial and final properties evaluated during the only instants of its existence. And what types of properties are the elements of A and B? They are physical quantities such as energy, angular momentum and linear momentum, etc. that describe the behaviour of the micro-entity during the interactions when it is created or destroyed. In this way, the properties created or destroyed during the interactions are in effect the labels of the interactions themselves. These ideas can be illustrated considering the particular formulation of quantum mechanics known as wave mechanics. In such formulation, properties A and B are represented by the initial and final wavefunctions of the micro-entity, indicated by ψA and ψB respectively. Two distinct deterministic and causal laws of the evolution of these wavefunctions are postulated, one towards the future or “forward” and the other towards the past Archetypes, Causal Description and Creativity in Natural World 409 or “backward”. In the most simple case of nonrelativistic systems without spin, these two laws are represented by the forward and the backward Schrödingerequations. These laws define respectively the value of the function ψforward(t) for each instant t following instant tA when properties A are created, and the value of the function ψbackward(t) for each instant t before instant tB when the properties B are destroyed. We can imply that:

ψforward(tA) = ψA, ψbackward(tB) = ψB;

but generally there is no direct relation between ψforward(tB) and ψB, nor between ψbackward(tA) and ψA . Consequently, we are not dealing with the time evolution of a single entity from the past towards the future, but with two distinct time evolutions of two distinct wavefunctions, one from the past towards the future, and the other from the future towards the past. Let’s investigate now the physical significance of the functions ψforward(t) and ψbackward(t). First of all, we must say that from their values at times tA, tB respectively [that is from the functions ψA, ψB respectively] it is possible to mathematically derive the sets of observables A and B. Such possibility has lead physicists to incorrectly conclude that these functions represent the state of the micro-entity at time t intermediate between tA, tB, while no micro-entity exists at that instant at all. In effect, the conditions represented by the two relations illustrated above are the two initial conditions under which the two distinct Schrödingerequations can be solved, therefore they cannot be derived starting from such equations. If we admit to specify as “casual” everything that is not predictable through causal laws, these two initial conditions would therefore in effect be considered casual events. More precisely, we should be aware that the second condition, relative to the evolution backward in time, is in effect a final condition. If the genuinely initial condition (i.e. the first one) is determined, for example as an effect of the preparation of the state of the micro-entity in a given experiment, the second remains undetermined. Experience has taught us that this condition is manifested, at an arbitrary time t B, with a probability (understood as the frequency of its recurrence on an ensemble of trials with identical initial preparation) equal to: Probability (result of B at time tB given the preparation A at time tA) ∗ 2 = | ∫ ψBψforward(tB)| , [where, for simplicity, the variables of integration and the constant of nor- malisation have been omitted]. This is the well known Born’s postulate, that affirms the casual and probabilistic character of quantum events such 410 Leonardo Chiatti as the creation of A or the destruction of B. The existence of a conditioned probability expressed in Born’s postulate should not surprise us: result B appears in an event of interaction between a determined micro-entity and other micro-entities and is therefore conditioned by the structure of the interaction and the initial preparation of that micro-entity. Therefore, the appearance of this result can not be a completely random event : it will be manifested with a definite and computable conditioned probability. How- ever, it is not a deterministic event. Function ψB becomes the new function ψforward(t) for t = tB, a fact known as “quantum jump” or “quantum leap”. In quantum mechanics we have therefore the association of two distinct evolutions: the deterministic and causal evolution of wavefunctions, and the probabilistic and acausal evolution of real quantum events. An entire period of research was aimed at identifying a supposed “sub-quantum” level of reality, governed by causal, deterministic and local laws; the hope was to derive the Born’s postulate from these laws as a statistical approximation. This period ended with the EPR experiments of 1980-90, which finally demonstrated the impossibility of the project (D’ Espagnat [1980], Licata [2003]). A final note. The description of the micro-entity and the temporal evolution of wavefunctions mentioned above are completely symmetric respect to the direction of time: they do not distinguish the past-future direction from the opposite direction. However the physical quantities such as energy can be demonstrated to propagate from the past to the future, reproducing the usual chronological order of physical phenomena (Cramer [1980],[1986],[1988]; Chiatti [1994],[1995],[2005]). In any case, the existence of the final conditions on quantum processes introduces a sort of finality, if not teleology, in these processes. Thus, we ask: what influence does all this have on macroscopic aggregates composed of an enormous number of micro-entities like, for example, living systems? Such aggregates are constituted of volumes in space where billions of billions of billions of quantum creation-destruction events occur every nanosec- ond. In each event in which certain micro-entities are destroyed other micro- entities are created and vice versa; therefore we are dealing with a very dense network of interactions in time and in space. Any “object” is really an aggregate of relatively constant aspects in this never-ending flow of events. Therefore the “essence” of the object, which supports these aspects, is nothing else but this very flow of events. Only in the so called classical limit, that is in the case of an enormous amount Archetypes, Causal Description and Creativity in Natural World 411 of mutually interacting micro-entities, can the distinction between the material support and its properties be observed. It is at this level that the objects appear, and when we can talk about the “form” of the object and the “morphogenesis” in naturalistic terms. In the classical limit, the probabilistic laws of quantum mechanics become the usual deterministic laws of classical physics. The acausal evolution disappears and the causal evolution of the state of the system, described by the laws of Newton, Lagrange or Hamilton, remains. We thus ask ourselves: does the acausal evolution really disappear or is it merely concealed by the language of the classical description ?

3. Acausality and the “Creativity of Nature” According to Pauli The laws of classical physics determine the temporal evolution of the state of an object. In the definition of this state, all references to “flow of events” (i.e. the essence of the object) is eliminated: references are made only to the invariable, or slowly variable, properties of this macroscopic flow. Having adopted this level of description, the acausal component of events of the flow is concealed, in that it is hidden in the language used for the description. At this point let us make some considerations, before proceeding further. In all evidence it appears that the search of an authentically creative∗ or innovative activity of Nature cannot be performed at the purely macroscopic level of natural phenomena, because this is governed by the inflexible necessity expressed in the local, deterministic and causal structure of classical laws. For example it is useless to search for a creativity in the hypothetical violation of the Newton laws in a phenomenon such as the mechanics of the opening of a flower: such phenomenon is correctly and rigorously described by these laws. The attempt made by many, and in which many research workers still insist on, to identify “biological laws” that would describe the biological phenomena, giving predictions different from those obtained from the application of known physical laws to the same phenomena, has ever been inconclusive and seems destined to total failure if aimed at introducing a sort of creativity of Nature. Creativity is something that is not a product of the past, and in the

∗Obviously, when we refer to creative activity of Nature we should not think of an intentional activity associated with a self-reflexive consciousness overseeing the phenomena, but of a sort of free self-determination of the phenomena themselves. 412 Leonardo Chiatti language of physical laws this means essential unpredictability. Only at the level of single quantum events constituting a macroscopic process can we find it, in the terms briefly described above. This was one of the last major ideas of Wolfgang Pauli; he believed (Peat [2000]) that Nature exercised a faculty of choice through the discontinuous variation of the quantum state, † i.e. exactly the acausal process ψforward(t) → ψB described previously. Among the natural, and in particular biological, phenomena there are many in which a discontinuity of this kind (commonly known as, even though inappropriately, “quantum jump”) produces effects that can be amplified—by classical mechanisms within the system or organism—until they appear at a macroscopic level. These amplification mechanisms, therefore, connect the micro-world of quantum jumps (that have an acausal component) with the macroscopic world governed by classical laws, thus introducing in it a real novelty. Among the biological cases, we are reminded of the genetic point-like mutation, that can produce a mutant phenotype with macroscopically different characteristics from the wild phenotype; the transduction of a single optical photon in an electrochemical signal propagated along the neurons (culminating in a sensation), after its interaction with the rods and cones of the retina; in the embryogenesis, the modified growth of tissue produced from the division of a cell in which the DNA is mutated by a single direct interaction with ionising radiation. It is necessary to keep in mind that these phenomena do not violate the classical laws. Simply, the primary events of the causal chains (i.e. quantum jumps) are considered, in the classical language, only in their aspect as cause of the following events, such as “random noise” or “external signals”, and not in their essence that includes an acausal aspect. Paying attention exclusively to the causal relations between the consequences of the initial event, the simultaneously causal and acausal nature of the event itself (and then of the entire chain of consequences) is concealed. The concealment of acausality in turn prepares the concealment of synordering patterns between events, as we discuss in the following section.

4. Levels of the Implicate Order If we look at the quantum level of reality, we understand that two distinct ontological levels are present: one evident and manifest, the other hidden. The manifest level is constituted by quantum jumps (interactions); they are

†Other authors, such as Wheeler and Just, have expressed analogous interpretations; see also Zeilinger [1990]. Archetypes, Causal Description and Creativity in Natural World 413 manifest because they form the real substance of the physical world, and as such every possible entity “observer” or “observed”, consists of them. Subsequently there is the level of the connection between quantum jumps, formally described by the twofold temporal evolution of the wavefunctions that we previously mentioned. We are reminded that during the interval between the event in which the micro-entity is created and the event in which it is destroyed, the micro-entity does not exist as a localised object in space in any form. The link between these two events seems to claim something magical, since it remains hidden. The point is that the two events are connected by a “nucleus” of physical reality which is beyond space-time. For example if the ﬁrst event consists of the creation of properties A and the second event of the destruction of properties B, we will have the scheme:

creation of A ← “nucleus” ← destruction of B.

It seems justifiable to presume that properties B are “reabsorbed” by the nucleus de-manifesting itself from physical reality, while properties A are “expressed” by the nucleus manifesting itself in physical reality. Since every possible observer-observed system is constituted by expressions-absorptions of this kind, the “nucleus” remains forever unobservable. It is the primum mobile, the immobile cause of the physical realm; but it is, in itself, physically unobservable. It is unobservable since each conceivable physical observation consists of events projected from it and reabsorbed in it. Peat [2000] therefore proposed to term it as “vacuum”.‡ If we assume the vacuum is unique and universal, all elementary physical events (i.e. quantum jumps) will be connected through it. Therefore, two of these could be generally connected by a couple of wavefunctions in the previously explained way, but could also be connected in a different way. In the first case they would be considered causally connected events, in the second case simply synordered. It is obvious that the causal order is only a particular case of synorder. The term “synordering” was firstly introduced by Bohm [1980], who denominated the two ontological levels described at the beginning of this section respectively as “explicate order” and “implicate order”. At the level of implicate order, all the quantum jumps that form the entire history of

‡However, the meaning of this term should not be confused with that intended by physicists as “vacuum” in field theory. This latter case is simply concerned with the configuration of the minimal energy of the field, and therefore with a certain physical property, instead of the source of all the physical properties. 414 Leonardo Chiatti the Universe (past, present and future) are potentially connected,§ resulting in the entire physical Universe being formed of potentially synordered events. The potential connection of each single elementary physical event with all other past, present and future events, form a sort of significance of the event itself. We cannot in this paper go into the bohmian analysis of the relationship between implicate and explicate orders; we refer to the original texts (Bohm [1980]), which remain unsurpassed in clearness and depth.¶ We limit ourselves to remember that such relationship is expressed generally by non local transforms, representing the whole in one of its parts; just as in a hologram, each fragment of which contains phase-coded information of the entire represented object. Similarly, the vacuum must correspond to an unobservable level in which all the information concerning the physical Universe in its entire past, present and future history is codified in compact form. An essential difference with respect to the hologram is that, while it is static, in the present case the entire temporal domain is codified and therefore it is more correct to speak of holomovement. Bohm found that an appropriate mathematical description of holomovement could be provided by a certain algebra, which he called holoalgebra. The process of asymptotic convergence towards the complete identification of this holoalgebra is one of the characteristics of scientific research.k

§For example, modifying the boundary conditions in a so called “delayed choice” experiment of quantum optics, the behaviour of a single photon present in the apparatus complies with this change, without any signal informing it of the event (Wheeler [1980]). ¶See also the interesting interview of Bohm by Peat and Briggs [1987]. Bohm died in 1992. kOn the possible identiﬁcation between the “vacuum” as interpreted in this paper, the “implicate order” and the Jung’s “unus mundus” see in particular Peat [2000]. Bohm himself saw in the implicate order the common root of physical matter and psychical phenomena (Bohm [1980]). This issue is relevant in order to extend the notion of archetype, that we are introducing here for the physical-biological domain, at the psychological domain, and to compare it with the homonymous Jung’s concept. See also Card [1996] and Mc Farlane [2000]. This fairly controversial argument (Fournier [1997]) will not be approached in this paper. Archetypes, Causal Description and Creativity in Natural World 415

5. The Concept of Metamorphosis According to Bohm and Goethe Even though the bohmian analysis has had a great impact on the metathe- oretical interpretation of quantum formalism (Licata [2003]), its idea of metamorphosis (Bohm [1980]) has not been object of much attention by later academics. The idea is the following. A given type of explicate order (one that is manifested, for example, in form or in function of an organ or organism) is generally defined by a certain aggregate E of geometric transforms such as spatial and temporal translations, axial rotations, etc., that change the configuration of the process while remaining within the same explicate order. A metamorphosis M is a transform mapping the considered explicate order into a more implicate level of order; un optical analogy may be offered by the holographic imaging of a given object, which we have already seen to be non-local. Let us indicate with M−1 the inverse operation of unfolding the implicate order into a new explicate order; in the optical analogy, it consists of the projection of a three-dimensional image of the object through irradiation of its hologram with coherent light. Assuming this, the transform E’ = MEM−1then transports explicate order E into a new explicate order E’, and can be called a similarity metamorphosis. The reason for this name, instead of the more conventional “similarity transform” indicated in the standard textbooks of abstract algebra, is the desire to underline the difference between transform that conserve the explicate order and metamorphosis that converts the explicate order into a certain implicate order and then eventually reconverts it to a new explicate order. Transform maintains the causal and local structure of the dynamics of explicate order, while similarity metamorphosis codifies, in a given explicate order, information regarding the ”rest of the Universe” in a non-local way. And it is in such codification that we can find the synordering of physical events. It is well known that the structure that Bohm proposes for the holomovement is the similarity metamorphic flow that alternates the enfolding of the explicate order into implicate order and unfolding of the implicate order into the explicate order. The creation of the electronic state ψA at time tA, and the subsequent destruction of the electronic state ψB at time tB, causally connected by the dynamics described in the Schrödingerequa- tions, are in effect interpretable as particular metamorphosis that connect, in a generally non-local way, events which are separated in space-time. The 416 Leonardo Chiatti movement of the electron is not the movement of a persistent material point in an external space environment, but the alternation of its enfolding (M) into the implicate order and its unfolding (M−1) into the explicate order, within a metamorphic process (Peat e Briggs [1987]). Therefore one can see that the causal connection is a particular case of similarity metamorphosis or, in other terms, a particular case of synordering. However, the concept of metamorphosis can be applied, without any substantial changes, also to the macroscopic structures of classical objects. It is possible, for example, that in a biological system specific combinations of quantum events (for example an extended group of coordinated mutations) could occur as the unfolding phase of a similarity metamorphosis which, during the enfolding phase, could have codified the state (non only in the present, but also in the past and in the future) of the entire system, of the species or of the environment. Such events could then, via the causal amplification processes discussed above, lead to macroscopic modifications within the system, thus to metamorphosis in the biological usual sense of the term (in which we include the mechanisms of differentiation during ontogenesis and speciation during phylogenesis). Nothing of this will ever enter into conflict with the causal nature of the sequence of classical states of the system. Even though the comparison of different concepts matured in different historical periods and in different cultural contexts constitutes an un- doubtedly hazardous operation, we believe it worth noting the convergences between the concept of metamorphosis developed by Bohm and that elaborated by Goethe. Regarding this subject the introduction by Stefano Zec- chi to a collection of Goethe’s works, focused on the Metamorphosis of Plants, (Goethe [2005]) is remarkable. Goethe was aware that an object, bearer of its own form, is in itself flow and movement. As Zecchi eluci- dates: “The metamorphosis of plants emphasizes that the form is immersed in the becoming and we can perceive it during its metamorphic process. In the Steigerung (the ascending process of the composition of parts) the form is not determined through an abstraction, or a progressive rarefaction of its material nature; the mere existence of form requests a continuous re- newal of this nature. It is this metamorphic re-materialisation that keeps the breath of life within the form. This does not occur in a linear time, but in a cyclic time with a beginning and an ending. The classical concept of time can be seen in the Goethe’s idea of metamorphosis: the law governing the development of phenomena has nothing to do with history; its time is a constant present that includes both past and future. The study of Archetypes, Causal Description and Creativity in Natural World 417 metamorphosis and its rules aims to make explicit what is stable and eternal within the contingency. The Steigerung of forms is not therefore linear movement in time, but circular movement around its own starting point. Birth and death represent the metamorphosis of what is self-identical, stable and eternal, but that also has life”. And again “ . . . Arber has also suggested the most correct way of scientifically interpreting the phenomenon of metamorphosis: it must be intended not as a visible phenomenon, but as a process of transformation which operates within the same transformative force. Metamorphosis is therefore not a material phenomenon, but an immaterial process in which its effects are perceived in certain particular cases . . .” The idea of metamorphosis presents itself spontaneously in two important research areas in biology, that is the ontogenesis and the phylogenesis. And it is well known that Goethe applied it in these areas. In the last section we attempt to re-propose this approach, that was dismissed in the 19th Century.

6. Archetypes and Causal Description An important point to emphasise is that metamorphic transformations are superposed on the ordinary causal temporal evolution of living systems, which is in reality none other than a particular aspect of metamorphic flow: the causal relation is only a particular case of synordering. However, in general perspective the synordering is expressed as a sort of “chance structure”. Naturally, while all events can be synordered because of their connection through the vacuum, not necessarily is each event synordered with each other in reality. Actual synordering occurs by means of the (extra-temporal) metamorphic process. The structure of metamorphosis is the pattern of the synordering to which it is associated. The actual patterns of synordering of events constitute what we will call archetypes. Archetypes can be represented mathematically, but such representation is not generally causal, because they are patterns in the implicate order. As we can see, such concept is significantly different from the platonic concept of “idea”, intended as a sort of “cast” or “blueprint” of the manifestation process.∗∗ In particular, it should not be confused with

∗∗In this conception a logical contradiction arises, concerning the relation of inherence. Let’s consider for example an atom of iron of the railway. To which archetype it would be associated as its partial manifestation? The archetype of “atom of iron”? Of “railway”? 418 Leonardo Chiatti

Goethe’s Urbild, of what remains as an important example the Urpflanze (archetypal or model plant), that inspired Goethe during his travel in Italy. A more complete description of the natural world would attribute the same importance to the dynamics of the causal-temporal evolution of the single systems (typically described by means of differential equations of motion or variational principles) and the extra-temporal patterns of synordering – i.e. the archetypes – often represented as graphs and taxonomies. As one can easily argue, this involves a new conception of the natural world in which contingencies are no longer mere casual accidents, not subsequently examinable and therefore of minor importance compared to the laws. As a consequence, the usual distinction between prescriptive (such as physics) and descriptive sciences (such as botany) disappears, remaining relatively valid only in the operational sense. In other words, we can claim that the existence of poppies is no longer an incident of evolution. In each instant of its growth, the quantum jumps constituting the “poppy” system can express information relative to the outline of the entire life cycle of the plant, of the outline of the entire life cycle of the poppy species, or to aspects of the environment in which it lives. The amplification phenomena capable of reverberating these events at the macroscopic level, lead to the ontogenesis of the poppy that will no longer be an exclusively local, deterministic and causal process: it will admit non-local, acausal and authentically creative aspects. An important part of scientific inquiry must then aim to isolate these aspects from the other merely mechanical ones: exactly the programme of Goethe. It is also evident that through the described processes, the environment can be codified in mutations, therefore the basic “casuality” of the background mutations could be in effect oriented from the environment; this could be the basis of an additional adaptation process other than the process of natural selection. It may be that the environmental processes that exercises a selective pressure on the organism and the background mutations occurring in it are synordered. If this is the case, we should look at the phylogenetic tree no longer as a map of evolutionary incidents, but as a partial mathematical representation (in fact it is a graph) of the archetype involved with the evolution of the biosphere on this planet. It is worth noting that such approach is not necessarily in conflict with

Of “planet Earth” (from which the atom was extracted)? Any object or entity can be seen as an element of distinct aggregates, each one, in platonic terms, associated to an “archetype”. That means the exclusivity of the relation of inherence to only one aggregate [the one which should correspond to the generative process of that entity] fails. Archetypes, Causal Description and Creativity in Natural World 419 the neo-Darwinian theories: it can represent a completion of them. In fact the basic scheme of the casual mutation subsequently selected by the environment is not denied, but we simply affirm the possibility of an acausal connection between these two factors. Naturally this does not exclude the fact that the neo-Darwinian model could be extended along multiple directions as, for example, the Barbieri’s semantic theory (Barbieri [1985]). It would be interesting to consider biological phenomena, fundamentally inexplicable from a causal point of view, in the terms proposed in this paper. An example can be seen in the migratory phenomena of birds. The fact is that the knowledge of the migratory destination cannot be genetically transmitted (the information is too detailed!) and that in some cases it is not transmitted by learning. It could be that instead of “knowledge” to be “utilised” (possibly an anthropocentric preconception of the biologist) one should talk about the expression of a complex of reactions, codified at the archetypical level as part of the process consisting of that specific species of birds, aimed at determining a flight plan under certain external circumstances. These phenomena would therefore be associated to the spontaneous canalization of the manifestation on selected lines, a sort of “impersonal intentionality” that reminds us of the Chinese doctrine of Chi. In conclusion, we must mention an issue that is too complex and delicate to be rigorously analysed in this article: what instruments are available to distinguish the causal aspects from acausal ones in a determined dynamical system? There is no simple answer, but the question is unavoidable if one wishes to make scientific and then testable hypotheses. It is evident that synordering can induce constellations of coordinated events which, in its absence, would be extremely improbable. Therefore, wherever it is possible to identify such constellations and statistically estimate their improbability, the quantification of the level of confidence to find oneself in front of synordering phenomena is possible. Of course, this is a rather indirect and a posteriori procedure, but we must keep in mind that the synordering connects the implicate order to the explicate order, and that our capability of effective control and observation is limited to the latter. For instance, it is possible to make conjectures that well known sud- den jumps of biological evolution, involving the simultaneous appearance of numerous distinct and independent but coordinated mutations, capable of expression in a functional and favourably adapted phenotype, are phenomenon of this kind. 420 Leonardo Chiatti

Naturally, much theoretical and experimental work is necessary to arrive at secure conclusions. But the stakes are a deeper understanding of the natural world going beyond the supposed absolute character of causality (recently restated through the myth of “intelligent designs”).

References 1. Barbieri, M. (1985) La teoria semantica dell’ evoluzione. Boringhieri, Torino. In italian. 2. Bohm, D. (1980) Wholeness and the implicate order. Routledge & Kegan Paul, London. 3. Caldirola, P. (1974) Dalla microfisica alla macrofisica. EST Mondadori, Mi- lano. In italian. 4. Card, C.R. (1996) The Emergence of Archetypes in Present-Day Sci- ence And Its Significance for a Contemporary Philosophy of Nature, in Philosophia Naturalis. Beitrage zu einer zeitgemassen Naturphilosophie, Arzt T., Hippius- Grafin Durckheim M., Dollinger R. (Eds.). Konigshausen & Neumann, Wurzburg, Germany. Reprinted in Dynamical Psychology. The work is available on the web. 5. Chiatti, L. (1994) Wave function structure and transactional interpretation, in Waves and particles in light and matter. Van der Merwe A. and Garuccio A. (Eds.), pp 181–187. Plenum Press, New York. 6. Chiatti, L. (1995) Path integral and transactional interpretation. Found. of Phys. 25, pp 481-490. 7. Chiatti, L. (2005) Le strutture archetipali del mondo fisico. Di Renzo, Roma. In italian. 8. Cramer, J.G. (1980) Generalized absorber theory and the Einstein- Podolsky-Rosen paradox. Phys. Rev. D 22, pp 362—376. 9. Cramer, J.G. (1986) The transactional interpretation of quantum mechanics. Rev. Mod. Phys. 58 (3), pp 647–687. 10. Cramer, J.G. (1988) An overview of the transactional interpretation of quantum mechanics. Int. Journ. Theor. Phys. 27 (2), pp 227–236. 11. D’ Espagnat, B. (1980) La teoria dei quanti e la realtà. Le Scienze (it.transl. of Scientific American) XXIV, 137, pp 52–69. 12. Fock, V.A. (1978) Fundamentals of quantum mechanics. Chapter I. MIR Publisher, Moscow. 13. Fournier, J. (1997) Synchronicity and acausal connectedness. The work is available on the web. 14. Goethe, J.W. (2005) La metamorfosi delle piante. Zecchi, S. (Ed.). Intro- duction. Guanda, Parma. In italian. 15. Licata, I. (2003) Osservando la sfinge; la realtàvirtuale della fisica quantistica. Di Renzo, Roma. In italian. 16. McFarlane, T.J. (2000) Quantum Physics, Depth Psychology and beyond. California Institute of Integral Studies. – The work is available on the web. 17. Peat, F.D. and Briggs, J. (1987) Interview with David Bohm. Omni, January 1987. Archetypes, Causal Description and Creativity in Natural World 421

18. Peat, F.D. (2000) The blackwinged night, creativity in nature and mind. Perseus, Cambridge MA. 19. Wheeler, J.A. (1980) Delayed choice experiments and the Bohr-Einstein di- alog. Common Conference of American Philosophical Society and Royal Society. London, 5 June. 20. Zeilinger, A. (1990) Problemi di interpretazione e ricerca di paradigmi in meccanica quantistica, in Che cos’`ela realt`a?. Selleri F. (Ed.). Jaca Book, Milano. In italian.