A SIGN LANGUAGE for QUANTUM MECHANICS Prashant Abstract Semiotics Is the Language of Signs Which Has Been U

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QUANTUM SEMIOTICS: A SIGN LANGUAGE FOR QUANTUM MECHANICS

Prashant

Department d’Informatique et de recherché operationnelle, Universite de Montreal, Montreal. Canada. {prashant}@iro.umontreal.ca http://www-etud.iro.umontreal.ca/~prashant/

Abstract

Semiotics is the language of signs which has been used effectively in various disciplines of human scientific endeavor. It gives a beautiful and rich structure of language to express the basic tenets of any scientific discipline. In this article we attempt to develop from first principles such an axiomatic structure of semiotics for Quantum Mechanics. This would be a further enrichment to the already existing well understood mathematical structure of Quantum Mechanics but may give new insights and understanding to the theory and may help understand more lucidly the fundamentality of Nature which Quantum Theory attempts to explain.

Keywords: Semiotics, non locality, non contextuality, Quantum Mechanics, axioms.

1. INTRODUCTION TO SEMIOTICS:

The field of Semiotics is a rich field of sign language study. It has fundamental ramifications in many areas of science and technology because of its being a general sign language. Language has always played a vital role in the communication of scientific facts from laboratories to the scientific community. In fact language is the most vital link between the propagation of scientific fact to gain legitimacy in the scientific world. A fact which can’t be communicated but is true to a researcher may not subject itself to scientific definition because of the reason that it can’t be reproduced or communicated to others. Semiotics is being practiced from time immemorial by living organisms including human beings to express feelings, knowledge, and wisdom in daily life. Many famous

1 proponents of the stature of Leibniz, Pierce and others have tried to formalize this field and have given their own philosophies about it.

In the field of Physics emphasis in the Peircean semiotic categories has been attempted in different ways. There are three modes of being, the three phenomenological categories of C. S. Pierce:

1. Firstness = the potential. 2. Secondness = the actual. 3. Thirdness = the general.

In Peirce's philosophy these categories are very broad concepts with applications in metaphysics, cosmology, psychology, and general semiotic. In Classical Mechanics only Secondness occurs: There is no spontaneity (Firstness) and no irreversible tendencies to seek equilibrium in various types of attractors (Thirdness), only specific states leading to specific trajectories through the state space. In Thermodynamics both other categories enter the scene: Thirdness by the irreversible tendency of the systems to end in an equilibrium state, determined by the boundary conditions, where all features of the initial state have been wiped out by internal friction. Firstness is reflected in thermodynamics by the spontaneous random fluctuations around the mean behavior, conditioned by the temperature and the frictional forces. The Firstness category is the most difficult to grasp, because when we try to exemplify it by specific examples and general types we are already introducing Secondness and Thirdness. However, Firstness has made a remarkable entry into Quantum Mechanics through the concept of the wave function as describing the state of a system. The properties of a system that are inherent in its wave function are only potential, not actual. An electron has no definite position or momentum; these properties only become actualized in the context of specific types of apparatus and acts of measurement.

2. SEMIOTICS IN QUANTUM MECHANICS

The quantum formalism, although well- established and confirmed, still seems counter-intuitive in many ways. How do we develop our mental images of reality and an epistemological framework such that the whole thing will seem intuitively clear and

2 sensible? The mechanical part of Quantum Theory, i.e. equations of motion and the construction of operators for observables, seems quite uncontroversial; there is no need for different "schools" disputing, e.g. the proper form of the Schrödinger- or Heisenberg- equations, because this part of the theory is so well integrated with the canonical methods of classical mechanics. However, Quantum Semantics, the way to define the meaning of Quantum-symbols in terms of ordinary language and classical concepts is still a matter of dispute between different schools of interpretation, and recent experiments, e.g. Aspect's (1982) [1] and those of the Innsbruck-group (1998) [2], that have validated the non-local, or "entanglement" properties of quantum states seem to have accentuated, rather than settled, the semantical problems. According to Niels Bohr (1935) and the "Copenhagen interpretation" [3]:

"There can be no question of any unambiguous interpretation of the symbols of quantum mechanics other than that embodied in the well-known rules which allow to predict the results to be obtained by a given experimental arrangement described in a totally classical way."

This statement of Quantum Semantics leads into a vicious circle for a theory of measurements: How can we know that a given piece of measurement equipment "described in a totally classical way" will behave according to the "well-known rules"? A quantum mechanical account of the action of the apparatus is out of the question because the meaning of the quantum symbols is undefined, unless the classical description alone provides a sufficient guarantee for its proper functioning. The Copenhagen interpretation is thus unable to tackle the measurement problem, except for some vague hints to the Correspondence Principle and an anti-realistic conviction that the reduction or collapse of the wave function is a purely conceptual tool with no physical background.

In a delightful little book entitled “Quantum Theory: A Very Short Introduction”, the author, John Polkinghorne, contrasts his philosophy of quantum mechanics with that of Heisenberg’s as follows:

“In experiments about atomic events we have to do with things that are facts, with phenomena that are just as real as any phenomena in daily life. But the atoms or elementary particles are not as real; they form a world of potentialities rather than of things or facts.”

“An electron does not all the time possess a definite position or a definite momentum, but rather it possesses the potentialities for exhibiting one or the other of these if a measurement turns the potentiality into an actuality. I would disagree with Heisenberg in thinking that this fact makes an electron ‘not as real’ as a table or a chair. The electron simply enjoys a different kind of reality, appropriate to its nature. If we are to know things as they are, we must be prepared to know them as they actually are, on their own terms, so to speak.”

Here we apply to the above problem the principle of Peircean semiotics that all phenomena exhibit three basic elements – Firstness (quality, potentiality), Secondness (reaction, actuality, fact), and Thirdness (law, habit, representation, etc.). The result shown below not only accommodates both Heisenberg and Polkinghorne but also adds something new, thirdness:

“Electrons and atoms (or any quantum objects referred to by Herbert as ‘quons’) have three irreducible aspects: Quons as they really are (Firstness); quons as measured (Secondness); and quons as represented in signs or theorized (Thirdness)”

We can depict the content of this statement as shown in Figure 1.

__ __

| Firstness |

| Potentiality |

| (Wave function, ) |

| /\ |

Quons = | / \ |

| /____\ |

| Secondness Thirdness |

| Measurement Quantum Theory |

| (Operator, ) (Eigen values, λ) |

|__ __|

Figure 1. A semiotics-based metaphysics of quantum theory.

The essence of Figure 1 is that quons are real entities whose complete description requires elucidating their three irreducible ontological aspects of Firstness, Secondness, and Thirdness as shown. The parentheses contain the relevant mathematical concepts. The large bracket symbolizes the irreducibility. Quons are real (i.e., they are as they are regardless of what we think of them), because all of the three vertices are real.

If the above analysis is right, it may be concluded that discussions on the metaphysics of quantum theory can benefit enormously from utilizing the semiotic framework enunciated by C. S. Peirce (1839-1914) over a century ago.

3. A BRIEF REVIEW OF PEIRCEAN SEMIOTICS:

The definition of a sign, according to Peirce, ought to be free from reference to human consciousness, or language, although this may seem awkward. The "purest" formulation in this sense makes it clear, that self-reference must be inherent in the definition of the sign. Below, we analyze this definition, which for the sake of clarity is here divided into smaller sections:

1. A Sign, or Representamen, is a First 2. Which stands in such a genuine triadic relation to a Second, called its Object, 3. As to be capable of determining a Third, called its Interpretant, to assume the same triadic relation to its Object in which it stands itself to the same object.

The three sections of the definition are seen to reflect the three phenomenological categories. The first section is self-contained: the sign as a First refers to nothing else; it is an icon of itself. Reference to an object is introduced as a dyadic relation in the second section, for although the text speaks about a genuine (i.e. irreducible) triadic relation, the third factor has not been introduced yet. The Secondness of the sign is an index of the object.

Figure 2: The sign as a triadic relation. R = Representamen, O = Object, I = Interpretant

6 At the bottom of the figure is shown a linear representation of the sign relation, I-R-O with two sign links (-) connecting the three factors, R, O, and I. The sign links represent physical processes and are interaction bonds, i.e. each link contain two oppositely directed causal relations, as can be seen by comparison with the triangular causal diagram above. The linear diagram has the advantage of showing that the causal relation between O and I can only exist as mediated by R. Also, it makes it easier to depict a chain of signs, where the interpretant of the first relation becomes the representamen of the second relation, and the representamen of the first relation becomes the object of the second relation.

The reflexivity of the sign relation is the feature that makes this particular way of chaining signs possible. This is the idea of unlimited semiosis, the potential of creating new meaning that is inherent in Peirce's conception of meaning and in the Law of Mind. In continuation of the sign definition quoted above he says:

"The Third must indeed stand in such a relation, and thus must be capable of determining a Third of its own; but besides that, it must have a second triadic relation in which the Representamen, or rather the relation thereof to its Object, shall be its own (the Third's) Object, and must be capable of determining a Third to this relation. And this must equally be true of the Third's Thirds and so on endlessly; - - "

The diagram below shows how the new interpretant (J, the "Third's Third") can be chained to the first relation I-R-O, so we get J-I-R-O where the second relation J-I-R has I as the representamen and R as the object.

Figure 3: A more detailed diagram of the sign relation showing its potential for creating a new sign with I as the representamen and R as the object.

7 The causal relations in figure 3 are labeled with letters, f, g, and h which have the additional meaning of category numbers assessing the relations as either 1: potential, 2: actual, or 3: general. The following selection rule must be valid in order to ensure that the chain is unbroken:

h g f.

The links can then be similarly categorized: the R-O link by f, the I-O link by g, because the connection between I and O is established by the I-R link. The g-relations (I-O and I- R) represent the ground of the sign relation (this is not very clearly stated in Peirce's verbal formulations, and there is no general consensus about how "the ground" ought to be defined). The ground (as defined here) and its category number g gives rise to the basic sign classification: 1: icon, 2: index, and 3: symbol.

3.1 INTERACTION BONDS AS SIGN LINKS:

Sign relations can be synthesized by means of the interaction-bond-graph-technique, developed by H.M.Paynter [11]. This framework for the building of dynamic models has a natural affinity to the principles of thermodynamics, relativity, and quantum mechanics. Although all interactions are local, the system also contains non-locality in the form of causal constraints that account for the non-local entanglement of quantum states. Such constraints enter the definition of some of the icons, like the two "junctions" that represent Kirchhoff's node- and mesh- equations for networks. The iconical (pictorial) representation of dynamic relations in the formalism makes it possible to circumvent the prohibition against ontology and the use of mental images of the quantum world. An implementation of the bond-graph icons as computing elements may provide a crucial step in the development of quantum computers. In the most general formulation of the technique every bond is associated with a metric tensor that defines the scalar product of the two complex vectors (effort and flow) that constitute the interaction. The technique thus incorporates the general principle of relativity, including Einstein- locality. The Reticulation (network-structuring by bond- graph-icons) of the sign relation leads for the representamen R to the partial differential wave- equations described as sections repeated in space like a three dimensional crystal. The processes of Preparation and Detection, associated with the R-O, and the I-R bonds, respectively, are naturally

8 Quantized as causal shifts that are distributed through the network from the connecting junctions. In this way the quantization of action and the projection postulate for measurements are given a natural and realistic explanation.

4. AXIOMS OF QUANTUM SEMIOTICS:

1. The quantum mechanical state vector is a sign. 2. A sign or representamen (R), according to Peirce, is a first standing in such a genuine triadic relation to a second, called its object (O), as to be capable of determining a third, called its interpretant (I), to assume the same triadic relation to its object in which it stands itself to the same object [5]. 3. The representamen R in a quantum semiotic sign relation mediates between the quantum mechanical object O and the interpretant I: I-R-O 4. The interpretant I is a potential, actual, or general purely physical result of measurement. 5. The sign links ( -- ), in the dyadic parts R-O and I-R of the sign relation are interaction bonds corresponding to the physical processes of preparation (the R- O link) and registration (the I-R link). 6. Each sign link is characterized by the Peircean categories as either 1: potential, 2: actual, or 3: general. 7. The category numbers, f and g, of the R-O link and the I-R link are restricted by the selection rule: g f.

8. The qualisign 11 (g = f = 1) is the continuum of the Hilbert space H. The symbol (g = f = 3) is synthesized from the lower signs by successive actualizations of potential links (1 2) and generalizations of actual links (2 3). 9. The six classes of signs (gf) are connected with Peirce's semiotic definitions and Dirac's bra-ket notation in the following way

(33) symbol

q p

(13) (23) iconic legisign indexical legisign

p p

(11) (12) (22) qualisign iconic sign indexical sinsign

10. A measurement is a permanent registration. The physical setting of an interpretant (the I-R link) preceding the registration is an irreversible process. 11. Registration is a dissipative and noisy process.

12. For a dissipative admittance the quantum noise on the current, whose spectrum is given by the fluctuation- dissipation (FD) theorem[6] corresponds to a time-series of discrete events

13. The collapse or reduction of the state vector requires the setting of a dissipative

sign link corresponding to the appropriate ray of H before the measurement. The

10 projection on the ray is the first of the quantum events predicted by the FD theorem.[7] 14. The collapse of a state vector for more than one particle requires prospective coincidence counting. 15. The violation of Bell's inequalities and other superclassical correlations is due to a common context of detection of several particles represented by preset coincidence counters. 16. Quantum Mechanics is strictly local and all the so called "non-local" effects can be simulated in a purely classical and local scenario provided there is a common context for the registration of individuals. [8].

5. NON-LOCALITY OR CONTEXTUALITY:

Classical, non-contextual logic implies Bell's inequalities, which are clearly violated by the formalism and experiments. To explain this violation by non-local action-at-a- distance is unsatisfactory, because there is no such thing in the fundamental principles of relativistic physics and no practical ways of using it (e.g. for super-luminal communication). A common context for the detection of the two separate (but entangled) particles might explain the violation of Bell's inequalities even without action-at-a- distance [8]. Such a common context could be provided by the coincidence counters in Aspect's experiments [1], but the experiments in Innsbruck [2] have shown that the inequalities are violated and the predictions of Quantum Mechanics are satisfied under circumstances where all coincidences are found retrospectively by comparison of the arrival times of the individual particles. It seems, thus, that contextuality alone is not a sufficient explanation. One has to accept some sort of non-locality, or "passion-at- a- distance" as it has been called by Shimony(1983).[9]

Thus we can study the effects of non locality and non-contextuality using the semiotic language which has been proposed and work out the appropriate interpretation which it admits to. P.V. Christiansen has attempted one such explanation of Bell Inequalities from the semiotic point of view.[8]

6. ENTROPY, INFORMATION AND MEANING:

An alternative approach to the discussion of meaningful structures on a physical basis is the attempt to discuss "information" as something related to physical entropy. There have been some attempts to reformulate quantum mechanics on information theoretic primitives and some remarks here may be useful.

"Information" may be without any meaning and still exist. "Meaning" is something general, Thirdness, but a specific string of letters, like "sjxipesdqo" that contains information, does not necessarily contain meaning. From the viewpoint of the telecommunication engineer it is essential that the information transmitted through a telephone line is actual, but meaningless, i.e. it exists on a level of Peircean Secondness. Entropy in a physical system is not "lack of information" i.e. non-existing information, but potential information about the exact microscopic state of the system. In principle this information could be obtained or actualized whereby the microstate would be known and the entropy reduced to zero. In this case the existing information would change its mode of being from firstness to secondness, but thirdness is out of the question, because it is specific information that cannot be generalized. So, entropy is a potential, meaningless information. If we try to build a theory of meaning by saying that "entropy is lack of information, so information must be negative entropy, and information has meaning", then we are in fact saying that meaning can be understood as "lack of meaninglessness".

There is nothing wrong with Shannon's "Theory of Information" when we remember that the potential meaning of the Shannon-information is outside the scope of the theory and is left to the human listener on the telephone. Also, there is nothing wrong with the information theoretical description of physical entropy as potential information about the microstate of a physical system, so it is not just a funny coincidence that Shannon's information and Gibbs' entropy are given by the same probabilistic formula. The physical information theory, as worked out by Szilard and Brillouin [10] in the discussion of Maxwell's demon has something important to say on the physical limits of how much structure a system can exhibit, but it can say nothing about whether these structures are meaningful or not.

12 A theory of meaning must take departure in the concept of a sign, and here Peirce's semiotic philosophy, physical theories of spontaneous order formation, and Ren‚ Thom's catastrophe theory are applicable, because they build on the notion of an underlying continuum from which discrete categories may emerge by continuous growth and habit formation. In this way we may be able to bridge the gap between naturalistic and humanistic studies.

7. CONCLUSION:

Thus the field of semiotics can be effectively applied on quantum mechanics to develop a new language which may give new fundamental insights to many controversial aspects of interpretations of quantum theory in general. Axiomatization of quantum semiotics is a fundamental step forward to understand and consistently develop and give a new direction to the meaningful expression of quantum mechanical primitives. It can also be suggested that semiotics can be applied on similar lines to the newly developing field of Quantum Information Theory and its ramifications can be checked and analyzed. I sincerely hope that semiotics of quantum mechanics can establish itself as a missing link between the well understood area of mathematical structure of quantum mechanics with its philosophical structure of interpretations to give us a new insight to the correctness of its interpretations as they stand today.

REFERENCES 1. A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, 1804 (1982). 2. G. Weihs, A. Zeilinger, et al. "The EPR Gedankenexperiment in Real World", Preprint, Institut fur Experimentalphysik, Universite Innsbruck, (june 24,1998). 3. N. Bohr "Can Quantum Mechanical Description of Physical Reality Be Considered Complete", Phys. Rev. 48, 696 (1935). 4. Peirce's letter to Lady Welby, december 23., 1908. See: I. C. Lieb, Charles S. Peirce's Letters to Lady Welby, 1953, New Haven. 5. Collected Papers, ed. Hartshorne & Weiss, CP 2.274. 6. H. B. Callen and T. A. Welton, Phys. Rev., 83, 34 (1951). 7. P. V. Christiansen, The Semiotics of Quantum-Non- Locality, IMFUFA text no. 93 (1985). 8. P.V. Christiansen, "Peircean Local Realism Does Not Imply Bell's Inequalities", Preprint, Symp. the Foundations of Modern Physics, Joensuu, Finland, 1990. 9. A. Shimony."Controllable and uncontrollable Non- Locality", Proc. Int. Symp. Foundations of Quantum Mechanics, Tokyo. (1983). 13 10. L. Brillouin, Science and Information Theory, 1962, Academic Press, N.Y. 11. H. M. Paynter, Analysis and Design of Engineering Systems, MIT, Cambridge, (1960).