Quantum Decoherence and Interlevel Relations A

QUANTUM DECOHERENCE AND INTERLEVEL RELATIONS

A Dissertation

Submitted to the Graduate School of the University of Notre Dame in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy

Elise M. Crull

Don Howard, Director

Abstract by Elise M. Crull

Quantum decoherence is a dynamical process whereby a system’s phase relations become delocalized due to interaction and subsequent entanglement with its environment. This delocalization, or decoherence, forces the quantum system into a state that is apparently classical (or apparently an eigenstate) by prodigiously suppressing features that typically give rise to so-called quantum behavior. Thus it has been frequently proposed by physicists and philosophers alike that decoherence explains the dynamical transition from quantum behavior to classical behavior. Statements like this assume the existence of distinct realms, however, and the present thesis is an exploration of the metaphysical consequences of quantum decoherence motivated by the question of the quantum-to-classical transition and interlevel relations: if there are in-principle “classical” and “quantum” levels, what are the relations between them? And if there are no such levels, what follows? Importantly, the following philosophical investigations are carried out by intentionally leaving aside the measurement problem and concerns about particular interpretations of quantum mechanics. Good philosophical work, it is argued, Elise M. Crull can be done without adopting a speciﬁc interpretational framework and without recourse to the measurement problem. After introducing the physics of decoherence and exploring the four canonical models applied to system-environment interactions, it is argued that, ontologically speaking, there exist no levels. This claim—called the “nontological thesis”—exposes as ill-posed questions regarding the transition from the quantum regime to the classical regime and reveals the inappropriateness of interlevel relations (like reduction, supervenience and emergence) operating within metaphysical frameworks. The nontological thesis has further important consequences regarding intralevel relations: not only are there no meaningful ways to carve the world into levels, but there are no meaningful ways to carve the world into parts and wholes either. These conclusions, supported by quantum decoherence and the empirical success of its models, drastically alter the philosophical terrain—not just in physics or in the philosophy of physics, but in traditional metaphysics as well. To Mom and Dad ad astra per aspera

ii CONTENTS

TABLES ...... vi

ACKNOWLEDGMENTS ...... vii

CHAPTER 1: INTRODUCTION ...... 1 1.1 Background ...... 1 1.2 What this dissertation is about, and why ...... 9 1.3 What this dissertation is not about, and why ...... 15 1.3.1 The measurement problem problem ...... 15 1.3.2 The interpretation problem ...... 30 1.4 Chapter synopsis ...... 33 1.5 Conclusion ...... 39

CHAPTER 2: DECOHERENCE FOR NONSPECIALISTS ...... 41 2.1 Introduction ...... 41 2.2 Preliminaries: Understanding quantum coherence ...... 42 2.3 The ubiquity of decoherence ...... 51 2.3.1 Assumptions ...... 52 2.3.2 Robustness, interaction Hamiltonians and the pointer basis 57 2.3.3 Minimal environments: Scattering-induced decoherence . . 64 2.3.3.1 Takes little prodding ...... 65 2.3.3.2 Takes little time ...... 67 2.3.3.3 Wavelength limits of decoherence ...... 69 2.4 The density matrix formalism ...... 73 2.4.1 Density matrices ...... 76 2.4.2 Reduced density matrices ...... 83 2.4.3 Master equations ...... 87

iii 2.4.3.1 Born-Markov master equation ...... 91 2.4.3.2 Non-Markovian dynamics ...... 95 2.5 An alternate perspective: The Wigner representation ...... 96

CHAPTER 3: MODELING DECOHERENCE ...... 99 3.1 The general practice of modeling ...... 99 3.1.1 Cartwright on models ...... 101 3.1.2 Responding to Cartwright ...... 103 3.2 An overview of the four canonical models of decoherence . . . . . 106 3.3 Master equations: From general to speciﬁc ...... 113 3.4 Oscillator environments ...... 120 3.4.1 Osc-osc models: Quantum Brownian motion ...... 123 3.4.2 The Wigner approach to osc-osc models ...... 132 3.4.3 Assumptions in the osc-osc model ...... 135 3.4.4 Spin-osc models ...... 139 3.5 Spin-1/2 environments ...... 144 3.5.1 Spin-spin models ...... 148 3.5.2 Osc-spin models ...... 150 3.6 Decoherence models in biological & chemical systems ...... 152 3.7 Conclusion ...... 154

CHAPTER 4: WHENCE “CLASSICALITY”? ...... 157 4.1 Introduction to Part II of this thesis ...... 157 4.2 Deﬁning classicality ...... 160 4.3 Denials of quantum fundamentalism ...... 164 4.3.1 Cartwright once again ...... 164 4.4 Attempts at a middle ground ...... 168 4.4.1 Wiebe, Ballentine and Emerson ...... 170 4.4.1.1 The Hyperion dispute ...... 171 4.4.1.2 Emerson’s thesis ...... 178 4.4.2 Other attempts at a middle ground that won’t do . . . . . 183 4.4.3 Caveat: Classical spacetime ...... 193 4.5 Batterman, Bokulich & the obstinance of classical structures . . . 196 4.6 Closing the (non-vicious) epistemic circle ...... 201 4.7 Conclusion ...... 205

iv CHAPTER 5: NONREDUCTIVE QUANTUM MONISM ...... 208 5.1 Introduction ...... 208 5.2 Reduction ...... 214 5.2.1 Kim’s functional reduction ...... 219 5.3 Supervenience ...... 224 5.4 Emergence ...... 226 5.5 Conclusion ...... 230

CHAPTER 6: WHY WE CAN’T HAVE NICE THINGS ...... 232 6.1 Introduction ...... 232 6.1.1 A note about terminology ...... 235 6.1.2 A note about method ...... 237 6.2 Why we can’t have nice things ...... 239 6.2.1 A thought experiment to illustrate this point ...... 243 6.2.2 A real experiment to illustrate this point ...... 245 6.3 Consequences for quantitative mereology ...... 250 6.3.1 Compositional mereology ...... 253 6.3.2 Traditional bundle theory ...... 257 6.3.3 Mereological nihilism ...... 258 6.4 Consequences for qualitative mereology? ...... 260 6.4.1 One more thought experiment ...... 260 6.5 Conclusion ...... 264

CHAPTER 7: CONCLUDING REMARKS ...... 266

BIBLIOGRAPHY ...... 271

v TABLES

2.1 Λ (in cm−2s−1)...... 66

vi ACKNOWLEDGMENTS

I must begin by acknowledging my great debt to Don Howard. An advisor who believes in you, tirelessly advocates on your behalf and whose sole fault in musical taste is Patsy Cline (though perhaps the fault lies with me) is a gift of immeasurable worth. The enthusiasm with which he shepherded me through difficult stages in my graduate career—the present work not withstanding—was such that I can only hope to impart some portion of the same to my own students some day. Abundant thanks of a similar nature are owed to Katherine Brading and Guido Bacciagaluppi, for seeing a colleague in me long before I did. In a way, I suspect their confidence propelled me to that very status. I am also thankful to Michael Rea for inviting a philosopher of physics into a metaphysician’s world, and for the ways he has encouraged me (not least by example) to engage in metaphysics with clarity, confidence and a good measure of charity. In addition to those already mentioned, I have benefited frequently from count- less discussions (along with the occasional impassioned debate), helpful suggestions and insightful commentary provided by John LoSecco, David Wallace, Jos Uffink, Amit Hagar, Michel Janssen, and my good friend and peer, Tom Pashby. I also wish to thank the committee and participants of Seven Pines XIV: I learned a great deal from a great many people that week, evidence of which is surely to

vii be found in these pages. Thanks also to the organizers and participants of the 2009 Geneva Summer School in the Philosophy of Physics, for exploring with me certain issues pertinent to my thesis as well as certain regions of the Pennine Alps. Special thanks are long overdue to Max Schlosshauer and Kristian Camilleri, both of whom showed deep interest and unbridled support regarding this project from its very inception. Since those first conversations in Australia, I have con- tinued to profit from their council and friendship. I hope the future brings us to the same part of the world as one another as often as possible. Andrew White deserves abundant thanks for his Herculean editing skills on earlier drafts. Thanks also to Chicory Caféand Martha Precup for many Saturday mornings spent together. Lauren Whitnah, I am blessed through your friendship and will think of you often from across the Pond. Also, thanks for letting me ride your horse. I would be remiss not to mention the understanding and accountability (except, perhaps, with respect to the daily crossword) provided by my law school study companions, Laura Porter and Kate Kennedy. My love always to Brooke Peterson, who brought me yellow tulips during a particularly dark winter, and to Greta & Will-bo, who might never understand just how therapeutic their dance parties, train sets and giggle-fits were for “Eeese.” To my academic big brothers, Greg Macklem and Erik Peterson: we laughed, we cried, we became legend. I will keep near to my heart always the folks of SBCRC and the home group: but for the plenteous love you’ve shown me, graduate school would have been impossible indeed. To Giovanni, Pierre Louis, Seamus, Dean and Little Burl I offer hearty thanks for sorely needed extra-curricular amusement. Karen Vegter (“Mom Too”),

viii you are a font of wisdom and good humor, and your friendship in these years has meant a great deal to your “Daughter Also.” Gram and Gramps, you’ve been a continual source of support and advice all through my long career as a student. I cannot tell you how much it has meant to have grandparents who understand life in academia. The same can be said to Sarah Van Dyke. You are, and always have been, more a sister than a cousin to me. Finally, I wish to thank my family. The weight of my love for all of you is too much to bear most days. To a most excellent brother trio—Daynan, Brett and Justin (and Julie, too!): my debt to you each is beyond counting. Mom and Dad, I have dedicated this work to you. What have I ever done to deserve the kind of love you continually give? I am here because of you.

ix CHAPTER 1

INTRODUCTION

1.1 Background

One usually assumes the existence of an ideal closed system when testing or deriving physical principles. Of course experimental physicists in particular have long been aware that the physical principles, in order to apply generally and therefore prove useful, have demanded this sort of idealization, among others (e.g., the introduction of ceteris paribus clauses), that allow one to neglect or correct for certain environmental effects throughout the measurement process. Typical lab procedure demands shielding the system of interest from the perturbative effects of the environment and controlling environmental coupling—whether it be mechanical, thermal or otherwise; physicists have been extremely successful in performing these isolations, and thus the assumption of a closed system remained unchallenged beyond applications of perturbation theory for quite some time. Already by the mid-1930s Erwin Schrödingerhad realized that quantum interactions would inevitably give rise to nonlocal states, or so-called entangled states (Schrödinger,1935, 1936). The consequence of this was to realize (and the physics community has been slow to do so) that assumptions about the isolation

1 of quantum systems needed to take entanglement into consideration in addition to dissipative effects. How to isolate a quantum system from such nonclassical interactions was a question whose import for theory and practice took several decades to realize. With the advent of quantum mechanics, a different yet related issue came to the fore: it was understood that while any system might be treated entirely quantum mechanically in principle (which would include assuming the existence of certain nonclassical consequences for said system), larger systems were nevertheless more accurately handled with classical mechanical calculations, and none of the strange physical consequences of quantum mechanics seemed to manifest themselves at the macroscale. This seemed to indicate the existence of different domains for the various mechanical laws—a “quantum” world exhibiting strange and unintuitive quantum behavior, and the usual “classical” one, behaving just as it had appeared to for millennia. Was there, then, a way to demarcate between the quantum and classical domains? And if such a borderline existed, in what sense: was there a true divide to be discovered empirically that explained our perception of classical objects despite their underlying quantum composition? How can we understand fully the dynamics of these systems, which are themselves quantum mechanical yet must be observed (relatively) macroscopically? What are the conditions under which one can say that the quantum state has transitioned into a classical or quasi-classical state? Some decades after the development of quantum mechanics, the two issues

2 presented above—the idealization of systems as closed and having local state descriptions, and the question of the transition from the quantum to the classical domains—were seen to be intricately and interestingly related. This realization came via discussion of the decoherence of quantum systems upon interaction with their environment. Feynman, Vernon and others had already realized in the 1960s that quantum nonlocality would introduce perturbative effects upon an environment in interaction with a quantum system that were dynamically independent from dissipative effects; such realizations were theoretically fleshed out by the works of Leggett and Caldeira, but not until the 1980s. Even then, the term “decoherence” was not applied to these dynamical processes.1 Independently of the theoretical developments of Feynman and Vernon, physicist H. Dieter Zeh published in 1970 a concise paper (often considered the origin of decoherence studies) in which he argued that if one assumes universal validity of the Schrödingerequation, then,

Since the [quantum] interactions between macroscopic systems are effective even at astronomical distances, the only “closed system” is the universe as a whole. The assumption of a closed system... is hence unrealistic on a microscopic scale. (Zeh 1970, p. 73)

In the same paper, Zeh goes on to state that in taking the dynamics of the interaction between environment and system more seriously (i.e., taking the interaction to subsist in more than mere mechanical coupling), one must realize that nonlocal states necessarily exist after interaction. Why, then, are the phenomenal

1Cf. Feynman and Vernon (1963), Caldeira and Leggett (1983), Caldeira and Leggett (1985) and Leggett et al. (1987). For a nice overview of this early history, see Stamp (2006).

3 consequences of the “quantumness” of these interactions never (or hardly ever) observed? Part of Zeh’s insight in his 1970 paper (expressed in more detail in his 1985 paper with his student E. Joos) was to realize that the vast number of degrees of freedom in a typical environment act to delocalize the state of the quantum system. In other words, the coherence, or degree of locality originally existing in the system’s state, upon quantum interaction with its environment is rapidly leaked into the many more degrees of freedom composing the environment. As a result, the quantum system is delocalized, or decohered. Hence the term “decoherence.” Though the next two chapters will explain in greater detail the physics of quantum decoherence, a ﬁrst-pass picture of this highly unintuitive phenomenon (one that nevertheless derives straightforwardly, and with no additional mechanics, from the quantum theory of 1927) can be given using an example involving the chirality (handedness) of molecules. Zeh’s graduate work in the late 1960s had been done in low-energy physics, where he investigated degrees of freedom in various interacting nuclei (Camilleri 2009, p. 292). It was through these pursuits that Zeh began to wonder whether interactions between systems and environments might explain certain puzzles concerning superselection rules—why some molecules always appear one way and not another—including what chemists call “the paradox of optical isomers.” Let us consider Joos and Zeh’s explanation of this paradox in light of decoherence, as described in their 1985 paper.2 Chemists have long known that most molecules of signiﬁcant size are always

2The example of optical isomers given below is also presented in Joos’s contribution to Giulini et al. (1996) on pp. 89–95.

4 observed to be in well defined spatial positions—what they call “configurations.” Certain molecules form what are called “optically active” configurations, and it is here that the paradox of optical isomers occur. For optically active molecules, the two important configurations are mirror images; this is called a parity symmetry in physics. However, this parity symmetry is in certain molecules only spatially symmetric and not properly rotationally symmetric due to the complicated relationship between the various elements composing the molecule. Such molecules, like sugar and ammonia, are thought of as “handed” molecules, and are therefore called chiral. Instead of identically symmetric parity states, molecules like sugar and ammonia have left-handed or right-handed states. Owing to the superposition principle (an axiom of quantum mechanics to be discussed in more detail in chapter 2), the set of all possible spatial states a chiral molecule might occupy upon measurement includes not just left-handedness and right-handedness, but also all linear combinations of these two states (with various coefficients). Thus it must be true that the likelihood of measuring a spatial state for chiral molecules as some superposition of left- and right-handed states is nontrivial. Why, then, do we always observe sugar to be in one of the two non-superposed states—right-handed or left-handed? To compound the issue, the relatively structurally similar ammonia molecule is often observed to occupy superpositions of chiral states and rarely (if ever) is seen occupying a definite handed state, in accordance with our premeasurement expectations. Thus, a problem: why does nature act according to our expectations with regard to the ammonia molecule’s spatial position but not with the structurally similar sugar molecule?

5 Joos and Zeh (1985) consider a parity eigenstate (i.e., a purely left- or right- handed state) of these molecules in interaction with a single unpolarized photon (a photon without its own intrinsic dynamics, which simpliﬁes the interaction between the molecule and the photon). The photon will be our means of measuring which chiral state the given molecule is in. The insight of Joos and Zeh was to recognize that the interaction dynamics of the new system must include not just the molecule and the photon measuring device but also the environment. Joos and Zeh demonstrate that in the case of the sugar molecule, the photon-plus- molecule system will become strongly correlated (that is, quantumly coupled) to environmental photons. This strong quantum coupling—or entanglement— with the environment destabilizes the phase relations that exist among the sugar molecule’s possible chiral superposition states. These superposition states become unstable, while the non-superposed states (left- or right-handed), which lack phase relations, are not aﬀected so strongly by this entanglement with the environment and remain dynamically stable as the total system evolves. Joos and Zeh calculated rudimentary values for the rate at which this destabi- lization (or decoherence) process occurs, and they realized that it would happen on a timescale many orders of magnitude faster than the measurement process being carried out in this experiment. In other words, by the time the measurement event can tell us which state the sugar molecule is in, decoherence has already desta- bilized to prodigious degree all of the possible superposition states and more or less let the right-handed and left-handed eigenstates alone. This explains why we have always (thus far) measured one of these two states instead of a superposition

6 of them, though the latter is more likely, statistically. Nature does not “favor” or “select” such states—they are just those states most robust under environmental decoherence. Decoherence also satisfactorily explains the behavior of the ammonia molecule: the dynamics of its spatial states under environmental interaction are such that superpositions are not susceptible to correlation with a typical photon environment (you might say colloquially that the ammonia molecule’s spatial states don’t “see” the environment of photons, and vice versa). As such, no entanglement happens in the position basis, and therefore no decoherence occurs among phase relations of chiral states. Thus superpositions of handed spatial states remain stable for ammonia and constitute the usual observed position states. Thus Joos and Zeh argue that decoherence processes can account for the chemists’ paradox: it is nothing above and beyond quantum dynamics of a total system—that is, system plus environment—that explains our measurement results in the case of both the ammonia and sugar molecules. This result might be stated more generally as follows: the inevitable interaction between a system and its environment gives rise to the process of decoherence. And it is precisely the dynamics of decoherence that account for the existence of macroscopic states of aﬀairs (typically considered “classical” states) that appear to lack quantum characteristics.3 The problem Joos and Zeh took themselves to be explaining in their early papers was precisely this general issue of the quantum-to-classical transition: if

3Though what exactly counts as “classical” shall be discussed rigorously in chapter 4.

7 all domains of physical objects can be described quantum mechanically in principle, then why don’t we see the strange consequences of quantum interactions at the macroscale? Indeed, why do we consider such situations (like entanglement) strange in the first place when they ought to be ubiquitous? The consequences of these early findings are still being realized and articulated today (especially in the burgeoning fields of quantum computing and quantum information theory), though as mentioned, decoherence itself brings with it no supplements to the mechanics of quantum theory, which have been known in full for the better part of a century. Decoherence describes no new physical principles, nor is it an interpretation of quantum mechanics in the same way that collapse models or many-worlds interpretations are (though there are links between these questions; I will touch upon this below). Instead, decoherence theory presents a fascinating lens through which we might understand with greater precision the dynamics giving rise to quantum phenomena, allowing us to study the strange effects of entanglement from a broader perspective. Some of the findings of earlier work on decoherence include the universality of decoherence in a stunning majority of realistic cases and the extraordinary rapidity of decoherence compared to other physical processes, again in a majority of cases. These results have led some to claim that decoherence provides the key for understanding why there exist in the pantheon of everyday objects things that can practically be labeled “classical objects,” despite the nearly century-old dictum that the underlying processes of such “classical objects” are fundamentally quantum mechanical.

8 This particular mismatch between what is observed and what is known about the quantum versus the classical “worlds” is commonly referred to as the problem of the quantum-to-classical transition. In his 2007 entry on decoherence for the Stanford Encyclopedia of Philosophy, Bacciagaluppi explains the question of the relationship between classical and quantum concepts as follows:

The question of explaining the classicality of the everyday world becomes the question of whether one can derive from within quantum mechanics the conditions necessary to discover and practise quantum mechanics itself. (Section 3.3, emphases original)

Decoherence theory is thus considered by some to close an explanatory loop: it was in virtue of analogies with classical mechanics that we were able to develop quantum mechanics, and now we ﬁnd that it is in virtue of quantum processes that objects appear classical to begin with.

1.2 What this dissertation is about, and why

The following research on decoherence was done with the larger aim in mind of examining with philosophical scrutiny the widely shared, broadly construed con- ception of this dynamical process as somehow explaining the quantum-to-classical transition; in other words, to examine the claim that the classical world emerges or arises from the quantum world via decoherence. More speciﬁcally, the aim was to discover the nature of interlevel relations as illuminated by a study of the process of decoherence (e.g., can we use a term like emergence at all? If so, in what sense?). Such an investigation is in accordance with the standard questions being posed in the philosophy of science regarding decoherence and other microphysical

9 phenomena. But what the physics reveals instead is the impossibility of carrying out any such inquiry. As one becomes more deeply acquainted with the science, one becomes increasingly aware of the inability of the physics itself to ever define, in principle, levels or domains that might serve as the relata for an ontological investigation of interlevel relations. It is clear from the viewpoint of decoherence, understood with minimal physical and philosophical assumptions (the latter including assumptions about the proper interpretation of the mechanics), that explanations of physical processes at a quantum mechanical level cover not only phenomena considered properly quantum mechanical, but those phenomena considered “classical” as well. The world, contra appearance, is not sliced into categories—into micro, meso or macro domains—nor does it give rise to physical properties that fit neatly within those domains, if those domains are defined in principle, no matter how one chooses to define them. One of the first lessons of quantum mechanics plus decoherence seems to be there are no levels. Investigation of decoherence exposes the ad hoc nature of attributing phenomena to one level or another and demonstrates that talk of levels, realms or categories of phenomena cannot be ontologically grounded.4 If there are no ontic levels derivable from our best microphysical picture of the world, then it follows that there is no ontologically deep meaning attributable to talk of the “classical world” as distinct from the quantum world or of transitioning (emerging, reducing,

4That is, at least if one wished to do ontology with an eye toward the ﬁndings of real, experimental physics. If not, then this isn’t a problem, though one can no longer claim to be doing proper ontology. More will be said along these lines in chapter 6.

10 supervening) one to the other. Such considerations have a consequence that may sound radical to nonphysicists (and, indeed, even some physicists and philosophers of physics themselves seem loathe to accept it): there is no classical world. I shall refer to this claim (along with the qualifiers “according to decoherence” and “ontologically speaking”) as the nontological thesis. The nontological thesis will be part, but not all, of the philosophical consequences one derives from a careful study of decoherence. Classicality (where this term, whatever it may stand for, is a property attributed to any of the usual ontological suspects: objects, processes, fields, etc.) is merely an illusion—or more appropriately, the claim that there exists a distinction between classicality and quantum-ness is an illusion that decoherence theory dismantles by providing a universal physical description of a world wherein all entities (whether they be objects, fields, processes, etc.) fit into a single category. Regardless of what one chooses to name that category, it cannot be labeled “classical” in any sense by which one commonly understand the term, and it is one in number. Thus I will argue that decoherence leads us to a quantum-monistic metaphysics. In order to argue for the nontological thesis, several topics must be explored, among them establishing that decoherence (when properly and philosophically understood) does indeed provide us a picture of the world that contradicts our intuitions regarding phenomena heretofore considered classical. Before doing this, however, I must present clearly the scope of decoherence theory, as the nontological thesis is predicated not only upon the universality of decoherence but also on

11 crucial physical assumptions necessary for its application. I must argue that these assumptions are well founded. A second discussion that must take place involves the question of classicality: What is meant by it? What are the necessary and/or sufficient conditions for something’s counting as “classical”? I argue that so-called classical states of affairs are empirically indistinguishable from quantum mechanical states of affairs, and decoherence provides the dynamics explaining this indistinguishability.5 Lastly, since my thesis is an ontological (rather, a nontological) one, I must answer for the ontological misconceptions that currently exist regarding levels or worlds. For example, I must demonstrate that explanations of the appearance of classicality are fundamentally quantum-mechanical, and show that widespread tropes claiming the supposed emergence of classical phenomena from quantum phenomena are simply wrong. I do not by these claims mean to deny the obvious perceptual facts that speak constantly and consistently for the existence of a Newtonian world—civil engi- neers have succeeded at their tasks for many a generation, on the whole without negative consequence, by assuming the world behaves purely classically within certain domains. But these are facts of perception, and if continuing study of microphysics teaches us one thing, it is that our intuitions about the behavior of the world often fall a great distance from the mark when applied beyond the limits of immediate perception, or even in the limits of applied mechanics. But I am not primarily interested in questions of perception, nor am I concerned with the limits

5By empirical indistinguishability I simply mean the following: if we lived in a world without decoherence, phenomena would appear to us the same as they do in a world with it.

12 that arise in practice, warranting the application of different sorts of mechanics at different size or energy scales. I am engaged in a metaphysical inquiry. As such, I have written this thesis for an audience of reasonably scientifically literate philosophers. I have risked being too slow and/or obvious to the philosopher of quantum mechanics or the physicist in the hope that metaphysicians might join the discussion. Despite the metaphysical nature of my inquiry, the strong urge we have to explain the world as it appears to us (in other words, to pose the traditional quantum-to-classical transition question and subsequently attempt to solve it) may itself cry out for explanation. I propose that decoherence theory can explain why there appears to be a distinct classical regime and in doing so explain away the question of the transition from a world in which quantum properties manifest themselves to the world of everyday experience. The way decoherence answers such questions involves no small amount of irony: by pursuing the consequences of quantum principles, we learn that it is a quantum feature of the world—namely, the universality of entanglement, which is a consequence of decoherence—that gives rise to states of affairs that are empirically indistinguishable from “classical” states of affairs. In other words, one’s observation of apparently usual states of affairs is a trivial consequence of the world’s nonclassical constitution. This is, once again, the closing of the explanatory loop alluded to above. I should think that most metaphysicians would at least grant my nontological thesis as feasible if all it amounted to was a claim that quantum theory is uni-

13 versal.6 It should not be terribly cavalier, so say the physicists (who have long understood the ubiquity of quantum theory), to state that all things considered classical or even semiclassical could be described in principle in terms of quantum phenomena alone, and that we merely slice the world as is most convenient for creatures of our size and with our particular faculties of perception. But I wish to go further and propose that even at the quantum scale of description one cannot provide deﬁnite or meaningful demarcations between the domain of one object and that of another. Even more radical metaphysical theses follow from this. For instance, if there are no objects available even at the quantum level of description, how does one assign properties to them? These sorts of questions will be investigated in the second half of this work. In the end, the hope is not to conclude with a devastating nontological thesis as such but to suggest in a more positive light that perhaps the lessons taken from this investigation of decoherence can serve to enrich general philosophical discussion concerning, for example, the ontic status of properties by oﬀering ways of thinking about such relations that perhaps we’d have never guessed were it not for the example of nature. Indeed, the novel aspects of the present inquiry primarily consist not in dismantling but in expanding and developing the space in which metaphysicians play by drawing their attention to the possibilities demonstrated for us by physical phenomena. For if decoherence teaches anything, it is that the daily operations of the world are much stranger and more wondrous than dreamt

6Though, interestingly enough, there is notable dissent from some philosophers of science on this point. Cf. chapter 4 for a discussion of dissenters regarding the universality of quantum theory and/or the ubiquity of decoherence.

14 of in any of our philosophies.

1.3 What this dissertation is not about, and why

Most of the philosophical literature on decoherence deals directly with the measurement problem and various interpretations of quantum mechanics. I do not wish to focus on either of these topics. I believe my question to be prior to them in the sense that, before one can properly evaluate arguments regarding the measurement problem or interpretations of quantum mechanics, substantial progress must occur regarding clariﬁcation and terminological housekeeping: What can we really say based on this new aspect of quantum theory? What conclusions are premature, overgeneralized or underdetermined regarding it? Not only do I believe my question to be antecedent to such considerations, but I further justify my neglecting these two canonical topics by arguing below that questions framed primarily in terms of the measurement problem or interpretation issues either dissolve upon careful investigation of decoherence, or they confuse the things that function as central components to my work—namely, questions of the universality and scope of decoherence, various philosophical and physical assumptions made in constructing and applying decoherence models, and so forth.

1.3.1 The measurement problem problem

The ﬁrst topic this dissertation will not deal with is the measurement problem. My reason for purposefully avoiding substantial discussion of the measurement problem in the body of my argument is in part inspired by the rampant ambiguity

15 with which a whole family of mysteries related to quantum measurement have come to be viewed as a single problem. More importantly, I have doubts about the unchallenged tradition of beginning any philosophical discussion of quantum mechanics with the measurement problem, given that, in even cursory glances at decoherence and its consequences, one ought to realize that the role of the observer and what counts as measurement are changed substantially in light of these dynamics. For example, most will agree that for a measuring entity to count as such requires only that some information about the system to be gained or somehow transfered to the entity by an event—the measurement. But if the only necessary conditions are that (i) there be an interaction even called a measurement and (ii) the measuring entity gains information about the system via this interaction, then even our ﬁrst-pass glance at decoherence ought to be enough to convince the reader that, due to the nonlocal nature of post-interaction quantum states, a single interaction between the quantum system and any environmental particle satisﬁes these criteria. On this understanding, uncontrollable measurements are taking place constantly in natural situations—interactions between systems and environments that could yield information about the interaction at later times. Obviously, the role of measurement in philosophical discussions regarding quantum theory will not look the same with the addition of decoherence and the lesson it teaches about the importance of taking all environmental-system interactions to bear in our explanations of the underlying dynamics. Indeed, a new approach to such discussions in which the measurement problem is less central

16 seems appropriate.7 Before describing in more detail the way in which decoherence sheds new light on the measurement problem, I want to be clear about what I take to be the problem. As mentioned above, the frequency with which multiple issues are made to correspond to the measurement problem is itself a problem. Take, for example, just such a confusion from the opening comments in Joos and Zeh (1985)—a foundational paper on decoherence, written by physicists, who are nonetheless driven by questions of a philosophical nature. They begin the paper as follows:

The relation between classical and quantum mechanics is at the heart of the interpretation problem of quantum theory. Outcomes of measurements are usually expressed in classical terms at a certain level of description: the pointer position is assumed to be definite like the position of a classical point mass in space. On the other hand, the general applicability of quantum theory—that is, essentially, the superposition principle—is important for many phenomena of macroscopic objects, for example, in solid state physics. However, if applied rigorously, this principle would lead to possible states never observed in nature, like superpositions of macroscopic objects in very different positions or of other “macroscopically different” states. One may also wonder why microscopic objects are usually found in energy eigenstates, whereas macroscopic objects occur in time-dependent states. (Ibid., p. 223)

Within this brief paragraph one can see traces of the following issues: the quantum-to-classical transition, the problem of why macroscopic superpositions are never observed, why certain eigenstates seem to dominate at diﬀerent scales,

7Even if, after reading this entire work, one is unconvinced of my argument here, I refer to a thorough treatment of decoherence and the measurement problem made in Janssen (2008). Though many of her conclusions align with those I give below, I still maintain that, on the whole (and taking the lessons of Janssen’s 2008 to heart), the measurement problem can be laid to rest as much ado about (almost) nothing.

17 and the assumption that measurement outcomes are deﬁnite. Furthermore, it has been (and continues to be) claimed that decoherence solves the measurement problem.8 But which measurement problem? While a study of decoherence may have something to say as per certain of the questions grouped under the measurement problem rubric, the theory does not, I argue, solve all the questions sometimes attributed to it. Max Schlosshauer (Schlosshauer 2007, pp. 49–50) has devised a neat catalogue of the diﬀerent problems sometimes called “the measurement problem”; I borrow his breakdown of the problem into three distinct questions, arguing for each that decoherence theory either obviates the question or cannot be expected to answer it.9

1. The Problem of the Preferred Basis

The superposition principle, which lies at the heart of quantum theory and will be described in more detail in the next chapter, has the logical consequence of allowing a vast number of basis choices in which to carry out measurements on a given quantum system.10 For example, when considering

8A representative sample of this sort of claim can be found in Anderson (2001b) and Anderson (2001a), in which Anderson argues that decoherence eliminates the need to invoke a collapse of the wave function. See also the response to Anderson by Adler (Adler 2003); more glimpses of the question of decoherence solving the measurement problem can be found in Albert and Loewer (1990), Zeh (2002), Bacciagaluppi (2007) and Wallace (2007). 9Though I follow Schlosshauer’s cataloguing of the measurement problem, my arguments for what decoherence has to say about each one are not necessarily Schlosshauer’s. 10All measurements must be carried out with respect to some coordinate frame; in quantum mechanics, since we are dealing with systems that occupy nonphysical Hilbert spaces (more will be said along these lines in the following chapter), we describe the structure of our coordinate system in terms of bases in this mathematical space. Whatever basis one chooses for a particular measurement of a quantum system will dictate the form of the quantities being measured; the goal in measurement is therefore to choose to measure the system of interest in a basis that will

18 a quantum system with spin, one might ﬁnd (upon measurement) that the system is in one spin state along one axis, even though the many superpositions composed of combinations of spin states and axes are also possible bases one might ﬁnd the system in. The question then arises, why do we consistently observe systems in only a miniscule subset of all the allowed bases? This subset of bases corresponds to classical observables in an overwhelming majority of cases (i.e., to diagonalized bases, or bases without (so it appears) interference terms). For example, we usually observe larger systems in position bases but not superposition of position bases. Given the statistical improbability of always observing bases that are apparently classical, why should such preferences for them appear in nature?

Zurek (1982) and Schlosshauer (2007), among others, argue that decoherence theory provides something by way of an explanation to the puzzle of why certain bases (and bases that typically correspond to classical variables) appear to be favored. Zurek explains that decoherence brings about a process he has named “environment-induced superselection”—often abbreviated “einselection” in the literature—and it is this latter process that accounts for the prevalence of certain bases in nature. Einselection refers to a dynamical process that arises as a system continues to interact with its environment: during interaction, the space of possible superpositions is dynamically nar- give us the results of our measurement in simplest form, and we do so by picking a basis that corresponds in particular linear-algebraic ways to the variable we are concerned with measuring.

19 rowed to a subset consisting of those states of the system considered robust with respect to the environment’s eﬀects. In other words, strongly favored bases are precisely those in which the quantum system is resilient to the quantum-coupling inﬂuence of the environment. The bases composed of such robust states are stable during environmental interaction; it comes as no surprise, then, that stable bases are the ones typically observed and are those we have come to designate as corresponding to “classical” observables (like position).

A provocative example of decoherence dynamically selecting a preferred basis involves a study done by Cucchietti, Paz and Zurek in 2005. This study considers the evolution of a spin quantum system in a spin bath (that is, an environment at thermal equilibrium) and demonstrates that under certain limiting conditions regarding system-environment interactions certain polarization components of the system stabilize very rapidly.

The master equation (that is, the equation used to describe the evolution of a system and environment interaction—the derivation of master equations for decoherence will be discussed in the next chapter) for the sort of model considered by Cucchietti et al. contains a polarization vector with components for each of the three possible spin axes. It is then simply a theoretical exercise of taking the equation as input and mapping its evolution through time, then observing which component of the polarization vector stabilizes

20 in various limiting cases. The component that is most robust under environmental interaction will determine into which eigenstates the system may fall upon measurement.

What the authors found mapped exceedingly well onto empirical data. They considered two limiting cases: the first interaction they modeled involved assuming weak intrinsic dynamics of the system, and the second interaction involved the limit of strong intrinsic dynamics of the system. Intrinsic dynamics correspond to the degree of quantum coupling that occurs between different aspects of the system itself. For example, strong intrinsic dynamics might be generated in a quantum system by that system’s polarization (its spin degrees of freedom) interacting with the system’s translational degrees of freedom. When the system has strong dynamics internally, its dynamics tend to dominate the overall dynamics of the system-environment interaction. Thus in the second limiting case (with strong internal dynamics), one would expect the system to evolve in such a way that the environment has negligible effects on the system, and vice versa. In the first limiting case, however (weak intrinsic dynamics), the dynamics of the system-environment interaction will be far stronger than that of the system itself, and so we expect to see the system responding keenly to the influence of the environment.

This is in fact what Cucchietti et al. found. In modeling the weak limiting case, the system evolved such that the x and y components of the polar-

21 ization vector decayed extraordinarily rapidly to approximately zero and maintained values approaching that limit. The z component, however, remains more or less stable at a value substantially greater than the others at the relevant scale. This means that in the case of weak intrinsic dynamics, the eﬀect of the system-environment interaction (which, you’ll recall, was the dominant aspect of all the dynamics in this case) is such that it quickly and eﬀectively suppresses all spin-basis options except for one—the z basis, which corresponds to possible measurements of the quantum system in either a spin up or spin down state.

In the case of strong intrinsic dynamics, the system’s evolution results in the steady decay of oscillating values for the y and z components of polarization but a stable value for the x component. This indicates the dynamical emergence of a stable basis that will yield energy states upon measurement; again, this is what we expect and in fact measure in cases where the system’s own dynamics are dominant: such systems are stable in an energy basis and thus occupy energy eigenstates of the system.

Thus decoherence theory gives us a dynamical story explaining why certain bases seem to be preferred in diﬀerent situations. No spooky, biased world consciousness is at work: it is only the dynamics of entanglement manifesting itself according to the nature of the given system-environment interaction.

22 2. The Problem of the Non-observability of Interference

As discussed above and in the example of the chiral molecules, in the space of possible quantum states for a system to occupy, superpositions of states (which are themselves a state) statistically ought to be dominant. Above I have given an argument for why so-called preferred bases tend to be classical ones—that is, state spaces spanned not by superpositions but by component states. A separate question involves the eﬀects of these supposedly ubiquitous superpositions, namely, interference phenomena. If superpositions ought to be the usual measurement outcomes, then interference eﬀects ought to be much more prevalent in nature, yet they are observed much less frequently than one would expect.

It took some years for physicists to realize that the interference terms missing from observation were not necessarily absent and perhaps merely decay extremely rapidly and are therefore seldom observed. However, this merely puts a diﬀerent puzzle on the table, namely, why do interference eﬀects decay so quickly? It cannot be due to the dissipation of the system’s energy into the environment (or vice versa), for decoherence theory provides us with the means to calculate a rate of decoherence (given some basic assumptions about the behavior of the system and the environment), and it is extremely fast—much faster than the rate of dissipation in a vast majority of realistic

23 cases.

The usual textbook explanation for the non-observability of interference in matter, as reported in Schlosshauer (2007, pp. 55–56), goes something like this: consider the analogy of a classical light wave and the interference that results from its passing through a suitable diﬀraction apparatus—e.g, the traditional double-slit arrangement. In this case, students are told that the distance between the slits must be comparable to the de Broglie wavelength of the light in order for interference phenomena to appear, owing to the resolution power of a double-slit apparatus so constructed. It follows that in the case of matter waves, interference phenomena are practically unobserved because we lack the technological ability to manufacture a double-slit apparatus with adequate resolution at that scale.

There is no denying that this technological limitation prevents our observation of interference between matter in many laboratory setups. However, this does not explain instances of clever engineering wherein experimenters have been able to observe interference effects at a scale well within the op- erationally defined mesoscopic (sometimes referred to as quasi-classical or semiclassical) domain. Such is the case, for example, in experiments performed with massive C70 molecules (cf. Brezger et al. 2002) and the even more massive fluorinated fullerene molecules (C60F48) studied by the same

24 research group (Hackerm¨ulleret al. 2003).11 Yet there must be some other explanation for the pedestrian fact that in usual, uncontrolled states of affairs, interference is not observed as often as one should expect.

The further explanation is once again provided by decoherence. Though the full details of the process and its effect on interference terms in a system’s state description will be given in the following chapter, suffice it to say that it is the interference terms—the off-diagonal elements of the state in a given basis that most keenly “feel” the environment and become suppressed with great rapidity upon interaction with its many degrees of freedom. Put simply, we do not observe interference because without utmost precision and care, we are always dealing with a system whose interaction with a huge number of external degrees of freedom prevents our ability to observe (but note well, not destroy) interference terms in the einselected basis.

3. The Problem of Outcomes

This problem is the one most frequently taken to be “the measurement problem” in the literature and is perhaps the question most directly related to the old problem of explaining the quantum-to-classical transition. Schlosshauer

11For more experiments in the mesoscopic regime, see Arndt et al. (1999), Arndt et al. (2002) or the numerous experiments discussing the production of stable mesoscopic superpositions, or so-called Schr¨odingerkittens. Many such experiments have been done and are underway by research groups in Paris, among others. These will be discussed in more detail in chapter 3.

25 (2007, p. 57) breaks the problem of outcomes into two questions: the generic problem of why we get deﬁnite outcomes from measurements on quantum systems with probabilistic state distributions, and the speciﬁc question of why we get the particular outcome we do.

According to Schlosshauer, decoherence makes sense of the first question but cannot answer the second; consequently, it is the specific problem of outcomes that he takes to be all that remains of the measurement problem so construed. I depart from Schlosshauer with respect to his discussion of the generic problem of outcomes. I consider this problem to be obviated by the lessons of decoherence, not solved. I believe decoherence irrefutably demonstrates that the generic question is ill posed to begin with, but nevertheless explains why the question of outcomes has appeared to be a problem until now. However, I agree with Schlosshauer’s assessment of the specific problem, more on which will be said presently.

(a) The generic problem: why do we get deﬁnite measurement outcomes?

At first blush, decoherence does not seem to explain why measurements on indeterminate systems yield definite results. But this is no matter; I argue that what decoherence does is not explain but rather explain away the question of definite outcomes. Because decoherence—which is effectively universal and exceedingly rapid in most realistic cases does not destroy but only suppresses interference terms, what we observe to

26 be a definite outcome is not. The nonlocal aspects of the system still exist, but they are damped to such an extent that their consequences would be unobservable in any realistic timeframe.12 Thus the answer to the question, “Why do we get definite outcomes for quantum measurements?” is simply that we don’t get definite outcomes, except in the smallest percentage of situations. What we usually get is a distinctly quantum (nonlocal) state of affairs that is empirically indistinguishable from a definite outcome.13

The role of decoherence regarding the generic problem of outcomes has also been discussed by Bacciagaluppi.14 However, he responds to this problem slightly diﬀerently: instead of saying that the generic problem of outcomes is obviated by decoherence as I claim, Bacciagaluppi sees the problem as being exacerbated by decoherence theory. His thinking is that with the knowledge of the underlying dynamics given us by decoherence not only are we confronted with the puzzle of deﬁnite outcomes for quantum measurements, but we must further puzzle at the attainment of determinate outcomes in all domains where entanglement

12Although for certain interactions there is a time of recurrence, or a time after which nonlocal aspects of the system will re-achieve their initial values, this length of time is so great in most situations involving uncontrolled decoherence as to be ignored. 13Again, whether or not deﬁniteness is a suﬃcient condition for an outcome’s appearing “classical” is a separate issue, to be addressed in chapter 4. 14See Bacciagaluppi (2007), though Bacciagaluppi’s explanation of decoherence’s role regarding this particular problem has been elucidated further in private conversation and correspondence with the author.

27 is present. Though I disagree with this way of approaching the problem, in either case there is little investigation left to be made as per the question of obtaining deﬁnite outcomes in quantum measurements, and that was our original focus.

(b) The speciﬁc problem: why do we get the particular outcome we do?

While the generic question asks why we measure a point on the screen instead of an indeﬁnite splotch, the speciﬁc questions asks why we measure that point at, say, location (x0, y0) instead of (x00, y00).

First we must reframe the specific question of outcomes in a way that eliminates the expectation of a definite measurement result, as I have just argued from decoherence against this possibility in an overwhelming majority of situations. The question must be rephrased in terms of appearances, then: why does a measurement outcome appear to be that quantity and not this one? In other words, why does Schrödinger’s cat appear to be approximately dead instead of approximately alive, or vice versa?

Though decoherence obviates the general question, I agree with the assessment of Schlosshauer and others that the question of speciﬁc outcomes is not, and perhaps cannot, be answered by decoherence. Thus, this problem alone remains outstanding of all those routinely

28 categorized as “the measurement problem.” Decoherence explains why it is that measurement outcomes assume the appearance of eigenstates despite the far greater statistical likelihood of measuring superposed states. But once the process of decoherence has dampened interference terms between components of possible superposed states, the component states themselves remain robust possible states. How it is that one among these comes to be measured remains an open question.

It is my suspicion that at the root of this desire to render quantum mechanics complete is a deep discomfort or failure to internalize what it means to say that quantum mechanics reveals a truly indeterministic world—a world that does not contain a causal (or any other) story about the “choice” of one approximately non-superposed outcome over other approximately non-superposed yet equiprobable states. If physics has taught us that the world is indeterminate, then an answer to the speciﬁc problem of outcomes might well lie outside the scope of what is accessible or demonstrable.

In sum, when the measurement problem is broken down into separate questions as above, we see that decoherence has something to say about three of the four: the problem of preferred basis is solved by considering the dynamical consequences of system-environment interactions, the problem of lack of observable interference is explained by the suppression of interference terms resulting from system-environment interactions, and the general problem of outcomes is seen

29 to be ill posed, in that we are rarely obtaining definite results but are instead obtaining results that have the appearance of definiteness due to decoherence.15 Decoherence itself cannot answer the question of specific outcomes. I have sug- gested that to hope for this sort of answer from physics may be a result of our misunderstanding or hesitating to accept thorough-going indeterminacy in our world. Having dispensed with the measurement problem in large part—or at least loosening historic moors to this question—I now turn to the other topic to be methodically avoided in this thesis: the debate over the correct interpretation of quantum mechanics.

1.3.2 The interpretation problem

Of the extant philosophical (and even scientiﬁc) literature dealing with decoherence, one would be hard pressed to ﬁnd many articles attempting to treat the issue irrespective of or prior to a particular interpretation of quantum mechanics. Even Zeh’s foundational 1970 paper—a mere seven pages in length—spends nearly a full page advancing an Everettian approach.16 Personally, I remain unconvinced by any one interpretation, and for that reason I attempt to present the following work in an interpretation-neutral manner. Even were I a dues-paying member of a particular interpretational program, I

15Though, of course, the strength of my arguments as presented here rest crucially on the investigation to be carried out in forthcoming chapters. 16And, perhaps more interestingly, a recent contribution of Zeh’s—(Zeh, 2002), “Basic Con- cepts and their Interpretation”—concludes with a rousing (though not entirely substantial) commendation for the many-minds interpretation.

30 would still choose to present an interpretation-neutral thesis, for two reasons. The first reason explains why a vast majority of writing on decoherence deals entirely or essentially with the question of interpretation, and the second reason explains why argument for the nontological thesis will not bother with this question. First: the simple truth, direct from the physics itself, is that the process of decoherence itself does not supplement quantum mechanics with an unambiguous or univocal interpretation of the relevant empirical data. Though most of those investigating decoherence have realized this very fact and ceased making arguments that decoherence unambiguously solves the question of interpretation, many authors persist in arguing that their favored interpretation is significantly helped by the findings of decoherence research.17 As is hopefully obvious from this mere gesture toward the rumbling debate over interpretations, much is yet to be decided, and there is no space for a com- prehensive argument of this sort in the present work (nor could I supply such an addendum in good conscience, given my deep ambivalence on this point). Second: a fruitful, interpretation-neutral investigation of decoherence can be carried out and ought to be. If my primary aim is to investigate carefully the consequences of the physics itself, it behooves me to avoid alignment with one interpretational camp or another. Without importing major assumptions, we can get most of the relevant decoherence material on the table, so to speak—

17For a nice overview of what the proponents of various interpretations had hoped (or continue to hope) decoherence contributes to their positions, see Schlosshauer (2007, chapter 8). For a few particular examples, I refer the reader to Allori and Zanghi (2001) for a Bohmian approach, Bacciagaluppi and Hemmo (1996), Bene and Dieks (2002) and Lombardi et al. (2010) on modal interpretations, Halliwell (1995) on the decoherent histories and, in defense of Everettian views, Wallace (2007) and Wallace (2010).

31 especially since I am also bypassing the measurement problem, whose weeds are often tangled with those of the interpretation question. In fact, many who have written on interpretations of quantum mechanics are motivated by something like the question of the quantum-to-classical transition or the definite outcomes aspect of the measurement problem. In the case of the former, the present work is precisely an attempt to explore the question of what decoherence can add to our understanding without assuming that any substantial metaphysical baggage is necessary to explain the transition from one “world” (with all its associated properties) to another. Relatedly, the motivation for seeking an interpretation stemming from the problem of definite outcomes becomes no motivation at all if one agrees with the argument given in the previous subsection—to wit, we need no metaphysical interpretation to explain the transition from a probabilistic distribution of possible states to a single, definite state because we are almost never dealing with truly definite states. In my demonstration of the nontological thesis, I take pains to be minimalist with respect to the assumptions that must be made both physically and philosophically. Indeed, it is part of the larger project to evaluate these very assumptions and their ontological consequences. If decoherence—stripped of supererogatory assumptions like those of interpretation and those arising in connection with the measurement problem—manages to say some important things to metaphysicians and philosophers of science, then we’re all the better for it.

32 1.4 Chapter synopsis

The narrative of the project as a whole can be divided into two parts: the ﬁrst part establishes the nonexistence of the classical world, and more generally the idea that there are no real “levels.” The support for this conclusion is drawn from a careful investigation of the basic concepts involved in decoherence, exploration of the four canonical decoherence models, and the failure of various strategies for deﬁning classicality in a metaphysically robust sense. Throughout this introduction to decoherence and the dynamics involved, I establish premises crucial to my nontological thesis: (i) that decoherence is fundamental and universal, and (ii) that decoherence not only demonstrates that the world contains no ontological levels or domains, but it furthermore explains why this is so, despite the apparent contraction with everyday experience. The second part of the dissertation then explores the various consequences of the nontological thesis. I focus on consequences regarding the character of interlevel relations and what follows for those who wish to make relational claims involving terms like emergence, supervenience, reduction, and so forth: if all the world is one category, how useful can two-category relations such as these ultimately be? Similar commentary can be made with respect to more general metaphysical projects, whose fundamental tenets cannot be sustained given the physics of decoherence and what it reveals about the way the world is (rather, how the world is not). However, I do not wish to end with a demonstration of the nontological thesis and the negative consequences that follow. I conclude on a constructive note,

33 suggesting that a positive consequence of the project might be to provide a way of describing physical relations so unintuitive (yet nevertheless sustained by our best theories) as to have escaped the purview of contemporary metaphysicians. Meta-ontologists in particular have been concerned with the (arguable) deﬁciency of metaphysical language to capture the world as it is described in our physical theories. Perhaps a deeper investigation of our best conﬁrmed theory—quantum theory—as seen through the lens of decoherence can contribute fresh perspectives to weary debates. Below I provide a brief overview of the main objectives of each chapter.

Part I: The physics

Chapter 2 will serve as an introduction to the basic concepts and minimal formal apparatus necessary for a careful philosophical analysis of decoherence theory. I introduce the mechanics in a way suitable to the lay person with some quantum mechanics background, stressing the universal scope of entanglement (and therefore the process of decoherence). I will focus speciﬁcally on clarifying the connec- tions among three concepts pivotal for an in-depth investigation of decoherence: entanglement, interference and superpositions. I describe the construction of the basic equations used for modeling decoherence: the total Hamiltonian and its application to the density matrix formalism used to generate master equations for decoherence processes. In this description of the basic conceptual components of decoherence and

34 quantum mechanics, the particular arguments given above regarding the dissolu- tion of various components to the measurement problem will become more clear. We will see how the pointer basis can be said to arise from the dominant terms in a system-environment Hamiltonian, how superpositions and associated interference phenomena are suppressed (though a quantitative analysis of this point will be reserved for the next chapter, treating the various models), and moreover, we will see that much of the physics involved in decoherence can be explored from just the axioms of the theory itself, without assuming an interpretation of quantum mechanics. After a few words on the role of modeling in physics and the nature of the link between metaphysics and physical models, chapter 3 goes on to introduce the reader to the canonical decoherence models, of which there are four. Treatment of the physical and philosophical assumptions employed by the various models of decoherence has been scant in the literature. For this reason, in addition to the qualiﬁcations a study of these models will inevitably bring to the ﬁnal strength of the nontological thesis, this sort of investigation is critical for all who undertake a study of decoherence theory or make claims based upon it. The overwhelming lack of discussion regarding models in the philosophical literature strikes me as particularly disturbing, as does the lack of candor in admitting that many of the deepest arguments from decoherence theory to some philosophical conclusion depend essentially on an idealized version or interpretation of the theory, rather than being predicated upon the theory as given to us by importantly limited physical models. This cannot be neglected; to what extent decoherence and sub-

35 sequent philosophical implications can be trusted is importantly dependent upon the empirical strength of models as applied to physically disparate situations.

Part II: Philosophical consequences of the physics

Chapter 4 asks, Whence classicality? Where might one claim classicality, classical behavior, classical objects, the classical world, etc., come from, and how are the properties of a classical object distinct from those of its quantum counterparts? I begin by sketching the space of answers to this question currently entertained in literature, and find that trouble arises when classicality is defined in terms of what it is not, i.e., quantum mechanical. Such a definition contradicts the nontological thesis, and in this chapter I investigate the range of positions on offer that embrace (to varying degree) just such a definition of classicality. A large part of the body of this chapter will focus on dissenters to the claim that decoherence is ubiquitous and explains the classical appearance of the world— claims that I have explicitly argued for in previous chapters. I look in some depth at one area of research that purportedly provides the most support for anti-universality claims regarding quantum mechanics and decoherence—that is philosophical work done on semiclassical mechanics and quantum chaos. I argue that notions of classicality that seem important for the explanation and understanding of such borderland cases between classical and quantum mechanics can be given a deeper explanation in terms of only quantum mechanics; decoherence plays a crucial role in generating such explanations.

36 This chapter concludes with a brief discussion of the way decoherence is meant to close the epistemic loop between quantum mechanical observables (or variables, or degrees of freedom) and classical observables (variables, degrees of freedom). Though our knowledge of quantum mechanics was at first dependent upon classical notions, I argue that this epistemic circularity is not vicious and is natural given the history. The physics, it appears, does not have strange preferences for certain observables, bases, or states (these preferences being encoded in our division as “classical” variables or “classical” behavior versus those that are not). The world is impervious to divisions among its degrees of freedom: how one defines a subsystem (or any system) is arbitrary, and decoherence illuminates precisely this feature of the world. In sum, whether one attempts to stake out metaphysical ground for classicality by looking at various limiting cases, by turning to semiclassical or chaotic systems, or by arguing against the epistemic loop, none of these strategies suffice at the metaphysical level, and the nontological thesis stands. Chapter 5 is concerned with a survey of the damage the nontological thesis inflicts upon philosophical positions (typically philosophy of physics) whose premises depend upon the existence of levels or domains. For example, those arguing for interlevel relations of the emergence or reduction or even pluralist type will encounter difficulty, given what we’ve seen in previous chapters. Metaphysi- cal interlevel relations that depend upon fundamental divisi cannot succeed if the nontological thesis stands. Though I maintain that our metaphysics is necessarily underdetermined by

37 even our best physical theories, I argue that a few conclusions regarding ontology can nevertheless be made. In particular, if there are no levels, then it follows that the world is entirely quantum mechanical. I call this position nonreductive quantum monism, and this chapter will begin exploring, via discussion of emergence, reduction and supervenience, the meaning of this view. Chapter 6 continues the project of the previous chapter by discussing the implications of the physics with respect to intralevel relations. If there is no metaphysically meaningful way to divide the world into levels, it is not hard to argue there is also no metaphysically meaningful way to divide the world within this resultant quantum monism. In other words, the physics of decoherence demonstrates the inappropriateness of parts and wholes talk in metaphysics. In this chapter, I look at studies typically endorsed by metaphysicians as viable modes of inquiry—e.g., compositional mereology—and see what can be said in the wake of decoherence regarding the ability to construct a successful ontology in terms of parts and wholes. This includes discussion of properties and what our investigations of decoherence entail regarding the relationship between properties and the things that hold them. Chapter 7 will provide brief concluding remarks concerning both parts of the dissertation and suggest future investigations that might bear fruit in diﬀerent areas of philosophy.

38 1.5 Conclusion

A criticism likely to be made against the philosophical conclusions of the present thesis might be along the lines of the following: “What you say is interesting within the realm of quantum mechanics. But we know that quantum mechanics isn’t the whole story, as it is nonrelativistic. What about field theories?” In response, the reader will see that none of my conclusions are dependent on the fact that quantum mechanics is nonrelativistic; in fact, the arguments herein would run equally well, I should hope, within a relativistic framework like quantum field theory or quantum electrodynamics. Entanglement is a feature of the world no matter how fast things are going. By extension, decoherence will be (and has already been) considered a crucial feature of dynamical explanations in any physical theory that deals with interacting systems.18 In closing, I stress once again that this project is intended to be a metaphysical one. That is, I am asking ontological questions, questions concerning what there is—or, more appropriately given the nontological thesis, what there is not—from the perspective of quantum mechanics with decoherence. I am not, therefore, asking about the relationship between quantum theory and classical theories like Newtonian mechanics. Neither am I asking (at first) about the relationship between quantum properties and classical properties, although the properties question (or at least clarification regarding what we mean when we say that something

18Work has been done already since the mid-1990s on decoherence and cosmology or quantum ﬁeld theory: cf. Giulini et al. (1995), Habib et al. (1996), Kiefer (1992), Kiefer (1993), Kiefer (1996) and Lombardo and Mazzitelli (1996). Recent work on such topics includes Banks et al. (2002), Banks (2008), Giraud and Serreau (2010), Koksma et al. (2010) and Mavromatos and Sarkar (2007).

39 exhibits one sort of property or another) will be important later on—and only then as derivative from the prior question regarding the furniture of the world as laid out by our most fundamental theory to date. The interdisciplinary nature of this project is such that certain issues addressed herein will speak to the reader with more or less force, depending on the questions at stake in the reader’s domain of inquiry or interest. I do not, for example, expect that all philosophers of physics will consider the nontological thesis to be as novel as non-physicists or metaphysicians might; likewise, the physicist might be more interested in the interpretation-neutral philosophical treatment of decoherence and its models and less concerned with my arguments regarding quantitative compositional mereology. Thus while the present work is indeed a cohesive whole, the valence of interest will shift from section to section or chapter to chapter, depending on one’s own interests. No doubt there may arise particular diﬀerences of opinion regarding my treatment or framing of particular topics. I can only say that I have done my utmost to preserve the integrity of the work while at the same time shaping the philosophical narrative in a way that translates across disciplines to as high a degree as possible.

40 CHAPTER 2

DECOHERENCE FOR NONSPECIALISTS

2.1 Introduction

In order to understand the process of decoherence and the formalism representing it, one must ﬁrst understand quantum coherence and its physical consequences. In turn, in order to understand coherence one must be familiar with related concepts such as entanglement, quantum superpositions and interference phenomena. The following chapter is divided as follows: I begin with a discussion of these three central terms: entanglement, quantum superpositions and interference phenomena. Though all three of these are at the root of quantum theory, they are frequently misunderstood by philosophers and physicists alike owing to the unintuitive conceptual nature of certain physical phenomena. Often these terms and their respective roles within the larger theory are not carefully distinguished, and they have not been considered from a philosophically informed perspective for many years by those currently writing about them, given their fundamental nature—in other words, these terms and their associated phenomena are more often then not taken for granted by those whose results (be they philosophical or physical) nevertheless depend to large extent upon such concepts.

41 There is nothing to be lost and much to be gained, therefore, in carrying out what may be to many a rather straightforward discussion of these concepts and their roles in decoherence, but with extra philosophical care. Insuﬃcient grasp of these particular concepts has gotten many into deep water, jeopardizing subsequent conclusions based on misunderstandings. From this discussion of entanglement, superposition and interference, the reader will be in a better position to grasp coherence, and consequently decoherence. Once I have presented a second-pass discussion of the process of decoherence (including arguments regarding its universality and behavior in certain limiting conditions), I will introduce the basic mathematical formalism employed for describing the dynamics of this process, called the density matrix formalism. My point in this chapter is to begin to convey the scope and implications of decoherence and prepare the reader for the next chapter, in which the four canonical models of decoherence will be analyzed in detail.

2.2 Preliminaries: Understanding quantum coherence

Quantum mechanics strictly speaking is done not in ordinary space, but in Hilbert space—a nonphysical space of two or more dimensions in which quantum states are represented by vectors. The necessity of representing quantum states in a Hilbert space is due to the fact that the probability amplitudes associated with possible quantum states contain both real and imaginary terms and therefore cannot be fully represented in a way that maps cleanly onto the three-dimensional space of everyday experience.

42 It is a mathematical property of Hilbert spaces that not only the vectors representing possible quantum states of a particular physical system exist within that Hilbert space, but in addition any linear superposition of those possible states also represents a possible state. This consequence of the linearity of Hilbert space is referred to as the superposition principle, and it is a fundamental feature of quantum mechanics. In discussing the paradox of optical isomers in the previous chapter, it was mentioned that Zeh was initially brought to this puzzle by considering degrees of freedom in molecules and the fact that despite the formal possibility of many different arrangements of such degrees of freedom, only a limited subset is ever observed. The same is true for many systems besides chiral molecules, and this strange mismatch between the mathematical formalism of quantum theory and experiment is codified by superselection rules. Superse- lection rules are not axiomatic, but are an attempt to describe what states are allowed and disallowed among systems in accordance with widespread empirical confirmation, despite the mathematical possibilities. After Zeh had written his famous paper of 1970 introducing the idea of system-environment interaction, he coauthored a paper that discussed the explanatory role decoherence might have in understanding the nature of superselection rules (Kübler and Zeh, 1973).1 To understand the superposition principle in quantum mechanics, consider the famous double-slit experiment, where a single quantum (a photon or electron, say) is sent through a barrier with two slits (whose spatial separation is on the order of the de Broglie wavelength of the quantum) and then detected on a screen

1Also see the discussion of superselection rules in Giulini et al. (1995) and Janssen (2008, p. 51 ﬀ.).

43 behind the barrier. The superposition principle tells us that since it is possible that the particle’s trajectory can be a vector corresponding to passage through slit

one (represented in Dirac notation as |ψ1i) or the path through slit two (|ψ2i), then it is also possible that the particle’s state can be described by the vector 1 |Ψi = √ (|ψ1i + |ψ2i), describing a superposition of “trajectories” through slits 2 1 and 2. It is crucial to note, however, that if the quantum state is a superposition (as in the case of a double-slit experiment), no straightforward physical interpretation can be assigned to the state. In other words, the superposition does not correspond to a classical statistical distribution and does not represent a case where the particle did travel through a single slit and we simply do not know which. Neither does the superposition simply describe an ensemble of component states (in our example, an ensemble including the slit 1 trajectory plus the slit 2 trajectory), but instead it describes an entirely new state of aﬀairs. This is empirically evident in certain cases where superpositions arise dynamically: these superpositions are observationally diﬀerent from the apparently mathematically equivalent of the superposition written as an ensemble of its components plus appropriate probabilities.2 Joos (Giulini et al. 1996, p. 2) explains this by giving the example of the K meson. The K meson is a particle that, when it exists in a superposition

with its antiparticle, gives rise to an entirely new particle—either the Klong or

Kshort mesons. The quantum system somehow occupies the |ψ1i component and

the |ψ2i component simultaneously, and not in a way analogous to an ensemble of

2Zeh makes this point in his chapter on the basic concepts of decoherence (Giulini et al. 1996, p. 7).

44 the components (Gell-Mann and Pais 1955 and Lee and Yang 1956). Obviously, this is a distinctly nonclassical state of affairs. Yet, in the example of the double-slit experiment, we do not typically measure the quantum system in a superposition of positions, though this is the expected state given the system’s mathematical description. Instead, because the measurement apparatus is a screen, which is a macroscopic device located in a “classical” basis—that of position3—we are only able to measure the quantum system in a basis of position, not of superpositions of position. What results from this is the emergence of an interference pattern on the screen. If one thinks about the double-slit experiment in a world without superpositions, the only possible quantum states for the particle would be through either slit 1 or 2. The results of repeated measurements of quanta sent through the double-slit screen one at a time and impinging on the screen would then be the standard Gaussian distribution centered behind each slit, corresponding to a statistical ensemble of these non- superposition states. This physical result indicates that superpositions must exist as possible states for quanta. The ontological sense in which these states “exist” is an interesting, and open, question. What is certain is that when we observe interference effects arising from quanta sent through a double-slit apparatus one at a time, this counts as empirical confirmation of the existence of superpositions, at least in the sense that a world without superpositions would not give rise to such measurement results.

3That is, the possible quantum states composing the screen itself are approximately diagonal in the position basis, which means we can distinguish the spatial positions of the composite systems nearly perfectly. As we shall see, even this fact (the macroscopic measuring device’s being in an apparently classical basis) is due to decoherence.

45 In order to understand quantum coherence, one must first understand the significance of phase relations in the states of quantum systems. Whenever a possible description of a quantum system includes superpositions, the relationships between various components of the superposition are called phase relations. In analogy with classical wave mechanics, one might think of phase relations as the degree of overlap between the waves constituting a wave packet, the packet’s parameters corresponding to those of the superposition.4 That this overlapping of waves occurs already within a single quantum’s Hilbert space explains why we detect spatial interference patterns when particles are sent one at a time through a double-slit apparatus, when by all classical accounts a series of independent quantum ought to give rise to a random distribution on a detection screen, as mentioned above. If the phase relations are well localized (which simply means they are constant with respect to one another), the system will exhibit interference effects, with the strength of this physical phenomenon depending on the degree of localization (constancy) of the phase relations. If the quantum system interacts with other degrees of freedom (e.g., those of the environment), the effect is to delocalize or smear these phase relations, effectively destroying interference. A coherent quantum state is simply a superposition in which phase relations between the component states are localized. In fact, quantum superpositions

4Though, keep in mind, that this analogy with classical wave mechanics is just that—an analogy. In classical mechanics, a wave packet is formally (and interpretationally) identical to an ensemble of individual wave functions. But as just described with the K meson example, we know that in quantum mechanical wave packets, though mathematically the superposed state is still described as an ensemble of individual component states, the resultant state is a diﬀerent thing than an ensemble and constitutes a new state of aﬀairs (a new particle in the case of the K mesons).

46 are properly called “coherent superpositions”, and are only referred to as superpositions for brevity’s sake. Thus decoherence is just another term describing the smearing of phase relations—the loss of coherence among phase relations of a quantum superposition that results in the suppression of interference terms within the basis of measurement.5 The term “local” in this definition is meant to convey the fact that coherent superpositions of quantum states are the closest analogue in all of quantum physics to a classical state—a state in which all physical observables are simultaneously well defined. Another way to say this is that the state of any classical system can be represented as occupying a single point in that system’s phase space. Instead of occupying a single point in phase space, a coherent quantum state can be represented by a minimal probability distribution in position and momentum (a typical phase space). The representations of coherent states of quantum systems are sometimes referred to as “minimum- uncertainty wave packets,” as on the wave picture they describe a physical state of least spread (while still obeying Heisenberg uncertainty) in phase space or in the basis of measurement.6 Coherent states are as “classical” as quantum states get, so to speak, since quantum states can never be more sharply defined than up to the limit imposed by the uncertainty relations.7

5Below I will discuss in what new, more general sense one can interpret measurement in light of decoherence. 6For example, a maximally coherent state in the energy basis (the basis of measurement typically adopted for microscopic systems) will still be represented as a tight Gaussian curve in energy, with the limit here being the uncertainty relation standing between energy and time. 7It is well worth noting a frequent misunderstanding regarding the nature of the Heisenberg uncertainty relations: they are not due to epistemic or experimental limitations. The limited deﬁniteness attributable to one or the other of a pair of degrees of freedom in the uncertainty

47 Thus far I have described certain aspects of quantum states in terms of the wave picture—that is, by analogy to classical waves, though I have used vector notation when introducing mathematical terms. The vector notation is typically associated with the matrix formalism and the language of linear algebra more broadly speaking, and I introduce aspects of it now as it will be helpful when discussing the density matrix formalism later on (which is, unsurprisingly, in the language of matrices). However, the wave picture is but one way of several of describing quantum phenomena that is closely analogous to classical wave mechanics; this correspondence is in fact the basis on which Schrödingerfirst developed a wave mechanics for quanta. However, just those subtleties of interpretation discussed in the introduction come into play when one realizes that under any particular description (be it the wave picture or in the language of linear algebra) the analogy with classical mechanics must break down. We have already seen a departure from classical mechanics in describing the obvious consequence of the superposition principle, that states like superpositions of position are equally valid possible states alongside normal position states. Perhaps the most significant departure from analogy with classical wave mechanics arrives in the form of entanglement. Entanglement is a state whose components (the pre-entangled subsystem states) cannot be factorized. That is, whenever two system interact they become entangled, and thus the initial, pre- interaction systems can no longer be represented independently but instead form a new state that is a non-separable combination of the initial systems. For example, relations is a feature of the world.

48 though interactions can be explained perfectly locally under quantum theory, the position states produced by this interaction cannot be described locally, because they are necessarily entangled states. Entanglement is often confused (or at least not disambiguated) from the term “coupling” in the literature, and this confusion leads to some misconceptions about the scope and nature of entanglement as a purely quantum phenomenon, as opposed to the broader class of phenomena related to coupling. Although coupling is, like entanglement, a result of interaction (between two or more systems, or between different degrees of freedom within a single system in some cases), it is unlike entanglement in that coupling is not a necessary consequence of all interactions, whereas entanglement is. Furthermore, coupling usually refers to classical energy transfer (e.g., thermal dissipation), whereas entanglement is a distinctly quantum mechanical state of affairs that does not necessarily correspond to energy transfer, though these dynamical processes often appear side by side.8 This confusion between coupling of systems and environments in the purely classical sense and entanglement as an independent, purely quantum state of affairs is in part why it took physicist so long to recognize decoherence as a process beyond normal thermal dissipation of a quantum system in interaction with an environment. Note that the property of being entangled is a binary property from a dynamical standpoint but may be said to have “strength” in terms of measurement processes. Entanglement itself does not have degrees of strength in the way that

8The theoretical proof for decoherence as a physical process independent of dissipation (and subsequent experimental veriﬁcation) involves one of the four canonical models that will be treated in detail in the next chapter.

49 mechanical coupling does. For example, the length of time during which systems interact thermally or the difference between energy states of two systems initially are parameters affecting the strength of classical coupling. However, once two systems have interacted, they are thereafter entangled in the sense that the states of those systems cannot be factorized any longer. In the world of physics, this signified the loss of one’s ability to describe systems as individuals. Obviously it is meaningless to talk of degrees of factorability—the state either is or is not able to be factored. When one hears references to degree or strength of entanglement, this alludes to the relationship among the amplitude coefficients, which in turn characterize the degree of distinguishability of various states. This sense of entanglement refers not to factorability but instead to the nature of the composite system with respect to measurement, i.e., to what degree of certainty one can infer certain properties of part of the entangled system’s degrees of freedom by measuring some properties (or set of properties) of the remaining degrees of freedom. Now we are in a position to understand why quantum entanglement—this strange, practically irreversible inseparability introduced by the interaction between two or more systems—is at the heart of decoherence. The consequence of entanglement is not only our inability to describe the initial quantum system independently of the environmental particle with which it interacts, but in addition, the phase relations of the initial system have been spread out over the additional degrees of freedom available via the entanglement. The entanglement of the system to the environment alters the Hilbert space in such a way that the

50 dynamics of the new entangled system incorporate the many degrees of freedom of the environment and allow them to influence the overall dynamics. The coherence of the original system is disturbed by the environment and can no longer be considered well localized in the basis of the interaction. This is yet another important difference between thermal dissipation and decoherence: the former process affects, albeit much more slowly, the combined state of the system-environment pair, whereas the latter process—while a result of the entanglement between the system-environment in all bases—only affects the phase relations within a particular basis (i.e., the basis of interaction between system and environment). It is this delocalization of phase relations that is called decoherence, as mentioned above. Let us put more meat on these skinny bones and equip ourselves with the mathematical and formal tools necessary to achieve a deeper understanding of this physical process. I begin by arguing for the ubiquity of decoherence by examining its scope and various limiting cases.

2.3 The ubiquity of decoherence

As should be obvious from the above discussion, the process of decoherence is deeply related to entanglement: not only is decoherence possible because of entanglement, but the process of decoherence itself causes further entanglement. The initial entanglement between the system and its environment results in the alter- ation of degrees of freedom available to the quantum system, which in turn results in the delocalization of phase relations, or decoherence of the quantum system.

51 As a result of this decoherence, the new entangled state describing the quantum system and the environment itself becomes entangled to a new environment, and so on—in a cascade of decohered systems begetting further entanglement. Certain assumptions must be made in order for models of decoherence to be developed given this information, starting with the ubiquitousness of uncontrolled entanglement (an assumption that has been touched upon already). In addition, one must assume the universal validity of Schr¨odingerevolution (to establish that this widespread and uncontrolled entanglement results in decoherence) as well as initially uncorrelated subsystems (which is related to the old assumption of isolated or closed systems). Each of these assumptions will be addressed in turn.

2.3.1 Assumptions

As quantum theory is the most successful physical theory to date, it is hardly a stretch to assume the validity of its axioms. However, not all physicists have been happy to assume the universal validity of Schrödingerevolution, specifically those who are adherents of wave function collapse interpretations of quantum mechanics.9 Since interpretations of quantum mechanics that change the unitary nature of the Schrödingerequation must be as empirically powerful as those that assume entirely unitary evolution and ample experimental evidence at various energy scales has yet to directly disprove unitary Schrödingerevolution, one can conclude that universal Schrödingerevolution is a fairly innocuous assumption.

9The exemplar of which is the Ghirardi, Rimini and Weber theory from their paper, Ghirardi et al. (1986). These authors add a non-unitary term to the Schr¨odingerequation in order to “collapse” the wave function into a single solution upon measurement.

52 It is not entirely truthful to say that everywhere and always unitary evolution of the Schrödingerequation is the case, and the reason for this is due to entanglement. In this way, the assumptions of widespread entanglement and of Schrödingerevolution are deeply related. Unitary evolution is destroyed locally when the system of interest interacts and subsequently becomes entangled to another system (say, the environment). At the scale of the newly created entangled system, unitary evolution is preserved; it is only when one is interested in tracking the dynamical evolution of the initial system independently that the mathematical description becomes non-unitary and (in most cases) frighteningly complicated. However, there do exist mathematical methods for approximately following the evolution of one of the entangled pair; this is the method of reduced density matrices, which will be treated in section 2.4 below. This method allows studies of decoherence to bypass the complicated state of affairs continually created by entanglement and to describe the measurement statistics of a system independently of its environment. In Zeh’s 1970 paper, he argues for the pervasiveness of entanglement by noting that the total wave function for a pair of systems will only in rare cases be found with both systems in definite states. He writes, “Any sufficiently effective interaction will induce correlations”; since macroscopic systems can effectively interact even at astronomical distances, “the only ‘closed system’ is the universe as a whole” (ibid., p. 73). Thus if one assumes the unitary evolution of the Schrödinger equation to describe sufficiently any microdynamics, one must accept the consequences of that universality: the ubiquity of entanglement, leading to the ubiquity

53 of decoherence. By way of warning, some care must be taken regarding this latter statement of Zeh’s about the universe as a whole being the only truly closed system. While this follows from uncontrolled entanglement (which follows from the dynamics of quantum interactions), claims about holism based on universal entanglement do not necessarily follow. While some may argue for holism in the sense that “everything is entangled with everything else,” all that is needed in order to establish the ubiquity of decoherence (and all that Zeh likely intended with his comment) is for each system to be interacting or have interacted with some other system’s degrees of freedom, and thereby exist in an entangled state. This is doubtless the case, for even a particle in the far reaches of the universe has interacted or is interacting with vacuum fluctuations in space or the cosmic microwave background radiation and therefore cannot be said to be a truly closed system. One might worry that if all systems are entangled with some other system, then the assumption necessary for studies of decoherence of an initially uncorrelated system and environment is untenable. Anglin et al. (1997) have demonstrated that the assumption of initially uncorrelated subsystems in models of decoherence does not affect the expected outcome of further entanglement generated by the process of decoherence.10 Though this paper is theoretical and employs a “toy” model, as we shall see in chapter 3, similar models of decoherence have nevertheless enjoyed strong empirical confirmation when applied to surprisingly disparate cases. In further support of this crucial ability to assume that a given experimentally-

10Thanks to Max Schlosshauer for drawing my attention to this paper.

54 defined system and its environment are previously noninteracting despite ubiquitous entanglement, one need only consider some fascinating recent biological research investigating the effects of decoherence on molecules and proteins. The theoretical groundwork for this research was laid in Briegel and Popescu (2009), and what is interesting for present purposes (though more about this work will be said in 3.6) is that the authors have predicted that many biological systems of interest—which are, by virtue of being biological, surrounded by very hot, noisy environments—can nevertheless be relaxed into energy ground states—a process that effectively “resets” the system-environment joint state to its pre-interaction, unentangled states. This allows one to assume initially uncorrelated systems, and thus allows for the application of decoherence and other models for the ensuing dynamical evolution. There are further underlying assumptions involved regarding the axioms of quantum theory, and although (as stated above) this theory’s empirical success confirms the validity of the axioms for most, there is one axiom—the Born rule— that is typically confused with various interpretations of quantum mechanics, and for that reason is often considered possibly contentious. This can be seen in one of the earlier papers by Joos and Zeh when they state that the use of a local density matrix “...presupposes the probabilistic interpretation leading to the collapse of the state vector at some stage of a measurement” (Joos and Zeh 1985, p. 224). The “probabilistic interpretation,” as I understand the authors to mean it, refers to the Born rule’s assignment of certain probabilities to the quantum system’s potential eigenstates. However, this is no more an “interpretation” than any

55 other axiomatic component of quantum theory. In fact, Zeh himself refers to this assumption as an axiom of measurement (Zeh 1970). Aside from this misleading reference to the Born rule in the above quote, Joos and Zeh claim that this leads to the collapse of the state vector upon measurement. The “collapse” of the wave function belongs to a particular interpretation (or class of interpretations) of quantum mechanics and is unnecessary for the formalism presented here. Again I stress: the Born rule is just an axiom and as such does not require metaphysical supplementation (nor may such a supplement exist). However, since many continue to misunderstand the implications (or lack thereof) of the Born rule, let us pause to consider it in more detail. As mentioned, the Born rule, sometimes referred to as the eigenvalue-eigenvector link, assigns a probability to each possible value obtainable through the measurement of a quantum system in a particular basis. That is all. The question of how Born’s rule works is a diﬀerent sort of question that may or may not have an answer that is entirely physical (i.e., non-metaphysical).11 To be painfully explicit, I stress that the Born rule itself does not entail speciﬁc metaphysical assumptions like the collapse of the wave function (as is claimed, for example, in Joos and Zeh 1985 and Janssen 2008). It is just a rule that assigns probabilities to outcomes, and does so successfully. Carrying the Born rule along with other assumptions of quantum theory leaves the discussion entirely interpretation independent, just as

11There have been some attempts to derive the Born rule from more basic principles in order to find an answer to the how question: cf. Rae (2009) for an overview of attempts, as well as Saunders (2004) and Squires (1990). At the end of the day, however, the dubious nature of many of these investigations leads one to believe that the Born rule is what it is, and what it is is an axiom of an extraordinarily well-confirmed theory. Attempts to derive the Born rule strike many as superfluous or misguided.

56 the empirical success of the theory is interpretation independent. In the following subsections, I will argue in greater detail for the ubiquity of decoherence by investigating various limiting cases and by considering the construction of Hamiltonians describing system-environment interactions, speciﬁcally the nature of the dynamics entailed by the total Hamiltonian.

2.3.2 Robustness, interaction Hamiltonians and the pointer basis

Recall our discussion of the problem of the preferred basis in the introductory chapter. There I argued (though below the support for this argument will be given in greater detail) that decoherence provided an answer to this aspect of the measurement problem by revealing why nature appeared to prefer certain bases to others in a majority of cases. In other words, it was claimed that the preferred basis of measurement (the so-called pointer basis) emerges dynamically when the system to be measured undergoes decoherence as a result of environmental entanglement. The pointer basis is determined by the construction of the Hamiltonian of the total system.12 One of the important achievements of Zeh was to recognize the implications of including the environment’s self-Hamiltonian as well as the interaction Hamiltonian in the construction of the total Hamiltonian and to note

12Hamiltonians are equations whose solutions are possible energies the system might take. A system’s Hamiltonian is considered its most fundamental mathematical description (alongside, perhaps, a system’s Lagrangian) for many reasons, one of which is this: once we have deﬁned a system’s Hamiltonian (in most cases the actual Hamiltonian is unknown, so physicists give effective Hamiltonians), its equations of motion or description under time evolution are obtainable by straightforward mathematical operations.

57 that the dynamics of the interaction were such that further entanglement would result. These adjustments to the basic equations (e.g., the Hamiltonian and the master equation that will incorporate it) describing the dynamical behavior of the quantum system in an environment give rise to increasing stability or robustness of the quantum system within a particular basis, i.e., the pointer basis. By looking at the construction of the total Hamiltonian, we can begin to see how, depending on which part of the Hamiltonian dominates the interaction, the preference for diﬀerent pointer bases emerge. The total Hamiltonian for a system-environment interaction is made up of the ˆ following components: HCS represents the self-Hamiltonian of the central system (the quantum system to be measured; “CS” will henceforth designate the central ˆ ˆ system), HEnv represents the self-Hamiltonian of the environment, and Hint is the interaction Hamiltonian. The total Hamiltonian is just

ˆ ˆ ˆ ˆ Htot = HCS + HEnv + Hint . (2.1)

In this construction, the self-Hamiltonians of both the CS and the environment characterize the intrinsic dynamics of their respective systems. In many real-world cases the total Hamiltonian can be simpliﬁed if the intrinsic dynamics of either the CS or the environment or both are negligible compared to the interaction Hamil- tonian. We will consider each of the three scenarios in which various components of the Hamiltonian dominate the total Hamiltonian: (i) the case in which the intrinsic dynamics of both the CS and environment are negligible with respect to the interaction dynamics (in which case the total Hamiltonian is approximately

58 equal to the interaction component); (ii) the case in which the environment is “cold”—with negligible intrinsic dynamics and having little or no perturbative effect on the CS (the total Hamiltonian can be considered approximately equivalent to the Hamiltonian of the CS); and (iii) the case in which none of the components signiﬁcantly dominates the total Hamiltonian. Before proceeding, however, a brief caveat regarding the mathematical foundations of quantum mechanics is necessary—in particular, the nature of commu- tations relations. Quantum mechanics is, in large part, written in the language of linear algebra precisely because this mathematical apparatus is constructed using entities called commutators that, as Born, Jordan and Heisenberg realized in 1925 (cf. Born et al. 1926), are the perfect tools for describing the peculiar, nonclassical ways in which variables in the physical world behave at the microlevel. Such physical variables (or “observables”—so-called because they correspond to the sorts of variables we are acquainted with from classical mechanics, such as position, momentum, energy and the like) are represented as operators in the algebra and thereby obey rules about which other variables they can be in communication (interact) with. For instance, if two variables are able to interact with one another, they are represented by commuting operators in the joint Hilbert space. Variables that do not see or interact with one another are represented by non-commuting operators in the algebra. To put this link between the linear algebraic representation and the physical world in a somewhat cute way, “commutation means communication.”13

13This all follows from a theorem in linear algebra stating that if two observables (say, the position coordinate of the system and the position coordinate of the environment) commute,

59 Let us now turn to the three interesting simplifying cases for the total Hamil- tonian of a composite CS-environment system and see in more detail how the preference for certain bases (the pointer bases) corresponds to the dynamics described in each particular case.

(i) Negligible intrinsic dynamics ˆ ˆ This is the limit in which Htot ≈ Hint. In the case of the interaction Hamil- tonian’s dominance, superpositions in the CS’s position basis are rapidly decohered owing to dependence of the interaction Hamiltonian on the position coordinates of both the system and the environment. The structure of the interaction Hamiltonian is such that dynamical evolution picks out a set of mutually commuting observables between the environment and the quantum system, and the system stabilizes in the basis described by these observables.

In the case of naturally occurring macroscopic systems, the interaction Hamiltonian typically has a strong position dependence owing to the 1/r2 force laws involved, and thus the set of mutually commuting observables is position (cf. Zurek 1981, 1982). Thus the quantum system remains most stable in this basis under interaction with an environment. This explains why position is indeed the basis in which macroscopic objects are typically then there exists a unique basis that is formed from eigenvectors common to both observables in the joint Hilbert space. For more details, any introductory text to quantum mechanics should have a section on commutation relations.

60 observed.14

(ii) Dominant intrinsic dynamics ˆ ˆ In this limit, Htot can be approximated by HCS. When the self-Hamiltonian of the CS dominates, the eigenstates of the self-Hamiltonian form the pointer

14The worry that this explanation for the preference of the position basis falls short of being a satisfactory explanation has been expressed to me on several occasion, for slightly different reasons. The general idea of one critique voiced by Don Howard focuses on the “quantum” verses “classical” description of the variables included in the Hamiltonian: since the r in the 1/r2 laws out of which the interaction Hamiltonian has been constructed is itself more correctly described in terms of quantum theory, an explanation in terms of this “classical” observable is unsatisfactory. My response to this critique is to say that the demand for an explanation whose explananda rely entirely on independent descriptions or definitions is too strong a demand in the case of physics: at some point we must bite the bullet and provide a definition in terms of some variable. Even were the explanation for the position basis’s prevalence to be given in terms of a quantum mechanical variable, the result at the level of observation would still be describable in terms of the usual classical observable. We would just know that a further quantum mechanical description of that same variable was available. Another criticism is a complaint about circularity, and it might go as follows: in order to explain the prevalence of the position basis for macrosystems, one goes to the dynamics of the Hamiltonian. But the Hamiltonian itself has the dynamics it does only because we constructed it that way—i.e., we constructed the interaction Hamiltonian in terms of the position variable; should it be a surprise that stability in terms of this variable is what we get back? While I agree that there is circularity here, it is not vicious. One might respond that this criticism does not properly take into account the fact that the interaction Hamiltonian, while it is indeed constructed by hand, is nevertheless perfectly valid from an empirical standpoint. Though this may strike some as bootstrapping, it is not for that reason unusual as far as explanations of physical phenomena go. Neither is the interaction Hamiltonian in the case of negligible intrinsic dynamics constructed for the express purpose of singling out the position basis as the best candidate for stability; we construct a Hamiltonian guided by experiment, and it is a consequence of this construction (and a consequence supported by empirical confirmation) that in this energy regime and with these overall dynamics for the system and environment, position falls out as the most stable basis under evolution. Furthermore, the structure of particular Hamiltonians now used to model system-environment interactions in various situations was decided long before the dynamics of entanglement and decoherence were well understood. Thus it might strike one as circular to use decoherence to explain what we already knew to be the case (e.g., the preference for the position basis with respect to macroscopic systems), though this impression in and of itself does not demonstrate the failure of decoherence to truly explain einselection. If one is still unconvinced regarding the dynamics of einselection, hopefully chapter 4’s discussion of particular models and the rates of decoherence with respect to certain degrees of freedom in a system over others will prove more persuasive.

61 basis.15 As the eigenstates of the self-Hamiltonian in quantum states are energies, it is in the energy basis that a system will become most localized upon interaction with a (relatively dynamically boring but non-negligible) environment.16

Indeed, for the same explanation as given in the ﬁrst simplifying case, one can explain the empirical fact that the energy basis is preferred for cases in which the intrinsic dynamics of the quantum system involve a much greater energy scale than the environment by pointing to the Hamiltonian and noting that a system so described will, under evolution, become stable in the energy basis.

This particular limiting case legitimizes certain simplifying assumptions used for deriving the master equation for certain situations in which the environment is comparatively slow. Obviously, environments whose energies are suﬃciently restricted relative to a given quantum system will rarely occur in nature but instead will be more frequently the situation in the laboratory.

(iii) The intermediate regime When the composite Hamiltonian is not signiﬁcantly dominated by the in-

15This case was ﬁrst studied in Paz and Zurek (1999). 16As Schlosshauer stresses (Schlosshauer 2007, p. 81), one cannot say that the environment is negligible in this case, as it is via interaction with the environment that the CS decoheres into a basis of energy eigenstates.

62 trinsic dynamics of either system or the interaction component, the evolution of the system-environment ensemble as a whole is subject to aspects of both limiting cases presented above, resulting in a situation analogous to quantum Brownian motion. Quantum Brownian motion involves considering a quantum system with continuous degrees of freedom immersed in an environment whose dynamics can be mapped onto a set of harmonic oscillators. More will be said about this important model in discussing the derivation of the Born-Markov master equation in section 2.4.3.1 and in the next chapter.

What occurs in this compromised situation is that the interaction Hamilto- nian describes the environmental monitoring or tracking17 of both position and momentum states of the CS. Thus the combined inﬂuence of the intrinsic dynamics and the interaction Hamiltonian lead to a pointer basis that is well localized in both position and momentum. Recall that this is the situation described by a minimum uncertainty wave packet in phase space and oﬀers the closest analog to classical mechanics. In cases where the pointer state is well localized in phase space (as opposed to either position or momentum), the recovery of quasi-Newtonian trajectories is possible. This is abundantly clear under the Wigner representation of a particular decoherence model (the oscillator-oscillator model) and will be treated in 3.4.2 below.

17These terms refer to information exchange between the environment and the system: the environment is said to “monitor” or “track,” or sometimes even “remember,” certain states of the quantum system, to a degree dependent on the environment’s own dynamics.

63 2.3.3 Minimal environments: Scattering-induced decoherence

Continuing our exploration of the scope of decoherence, let us next consider the case of decoherence induced by extremely minimal environments—i.e., environments comprised of particles scattering off of the system to be measured (this is as opposed to environments modeled as baths, where many environmental modes are interacting over time with the CS). Though a single scattering event is sufficient to create an entangled state, the number of scattering particles necessary to induce decoherence depends on the wavelengths of both the CS and the particle scattering off of it; the wavelength limits will be described in detail below. What will come to light as a consequence of investigating scattering-induced decoherence is not only the extreme ease with which decoherence begins but also the extreme rapidity with which it is carried out. This will support the ubiquity- of-decoherence arguments given above, for even in imagining a case of extreme isolation—say, a quantum floating freely in the nether regions of the universe— decoherence can be induced by interactions as minimal as vacuum fluctuations.18 Thus uncontrolled entanglement and decoherence demonstrate their ubiquity in natural situations. Even experimental results from laboratories with complicated engineering in order to isolate a system cannot escape decoherence forever. All these points shall be touched upon below.

18Indeed, some work has already begun on the question of decoherence induced by gravity or changes in the spacetime manifold; this is discussed brieﬂy in chapter 4.

64 2.3.3.1 Takes little prodding

In studying the case of a macroscopic object and the dynamical interaction between its position coordinate and that of a scattering particle, Joos (in his contribution to Giulini et al. 1996) demonstrates that many scattering processes result in the exponential damping of spatial coherence and that the strength of this damping follows a localization rate

Nv Λ = (k2σ ) , (2.2) V eff where the Nv/V factor is flux, k is the wave number of the incident particles, and σeff is the effective cross section for Rayleigh scattering events. Joos then provides some calculated values for Λ (ibid., p. 8). I replicate these data in Table 2.1; it gives one a general idea of just how effective scattering-induced decoherence is in the position basis for relatively large systems (in particular, it considers the localization rate of three molecules of different diameter when interacting with several sorts of environments, from cosmic microwave background radiation to the approximately isolated laboratory vacuum). Note, however, that these are theoretical estimates and are based on certain assumptions (e.g., no recoil of the CS and certain mass ratios between CS and scattering particles) that may alter the exact rates. Nevertheless, the extreme rapidity shown in these rates makes the point well enough.

That decoherence can be induced in such minimal environments as these re- conﬁrms the claim that decoherence is not a consequence of normal perturbation

65 TABLE 2.1

Λ (in cm−2s−1)

Central System: 10−3cm 10−5cm 10−6cm

dust particle dust particle large molecule

Cosmic background radiation 106 10−6 10−12

300 K photons 1019 1012 106

Sunlight (on earth) 1021 1017 1013

Air molecules 1036 1032 1030

Laboratory vacuum (103 particles/cm3) 1023 1019 1017

or classical disturbances; the scattering model is built on an assumption of no recoil on the part of the CS, which indicates that entanglement is the cause of the phase delocalization in the position basis. These calculations, though only theoretical, have proven close to the mark in certain experiments carried out since the original calculations were made by Joos and Zeh in their 1985 paper. Of particular note is a team of researchers at the University of Vienna (and part of Austria’s Institute for Quantum Optics and Quantum Information, or IQOQI) who have successfully produced interference eﬀects (that is, sustained coherence) for very large molecules. In Hackerm¨uller et al. (2003), the team describes experiments done on the large fullerene molecule

C70 and its interactions with various gases at diﬀerent pressures.

66 In the concluding section of their 2003, Hackermüller and coauthors make predictions based on their results as to the amount of pressure necessary for observing quantum effects in even larger objects. They state that for objects with the mass of, say, a polystyrene molecule (approximately 105 amu), one would need to slow the particle to a speed of only 10 m/s. The authors note that such velocities are impossible outside the laboratory and “rather demanding” in it (Hackermüller et al. 2003, p. 7). The difficulty with which decoherence is suppressed and coherence sustained in situations involving single scattering events is extremely telling.

2.3.3.2 Takes little time

As already discussed, early calculations and experiments confirming those calculations demonstrated the extreme time scale difference between dissipation—a thermal, non-quantum exchange—and decoherence. The most well-known publications making this point about the independence of these two dynamical processes are Zurek (1991), Anglin et al. (1997) and Leggett et al. (1987), the latter of which models the CS as a spin-1/2 particle. In Zurek’s paper, he demonstrates the widely disparate time scales on which these dynamics occur by stating the following: even were the relaxation (dissipation) rate approximately the age of the universe (a value he quotes to be approximately 1017 seconds), quantum coherence would be effectively destroyed for macrosystems in approximately 10−23 seconds (Zurek 1991, p. 14). Smaller quantum systems will have smaller rates of decoherence, but theoreticians understand that dissipation and decoherence rates are such that the two

67 processes are undoubtedly independent. Furthermore, one of the canonical models for decoherence—the spin-Boson model—is a model in which the CS is dissipation free and yet exhibits delocalization of phase relations in environmental interaction. As Zurek and others have calculated, the rate of decoherence (that is, the rate at which spatial coherence over the distance ∆x is damped) is proportional to Λ(∆x)2. One should not be surprised to see that the rate of decoherence depends on the localization rate (2.2) and even more strongly depends on the spatial separation between wave packets representing superpositions in the system’s Hilbert space. The larger the spatial separation, the more unstable the system and the faster the decoherence rate. This is confirmed by the comparative instability between macroscopic objects and quantum-sized objects when immersed in an environment. It was mentioned earlier that the effect of decoherence—suppressing the interference terms (delocalizing phase relations) in a particular basis—has attached to it a particular recurrence time. Experimentally this manifests itself as “periodic coherence revivals.” As explained by Kokorowski et al. (2001) following measurements of decoherence of atomic superpositions due to spontaneous photon scattering, periodic coherence revivals occur despite overall decay of coherence in proportion to distance. These coherence revivals occur when the number of scattering events is sufficiently low such that a single scattering event is sufficient for resolving the distance information about the CS (i.e., when the spatial difference between the superposition of the CS is on the order of half the photon’s wavelength). However, after a sufficient number of photons have become entangled to

68 the CS, such revivals of coherence cease (as the information being carried away by the photon environment is increasingly diﬃcult to retrieve). Recurrence times exist for all systems aﬀected by decoherence owing to the non-zero probability that at some future time the system will be found again in its initial state with initial values. The mathematical description of the process of decoherence is therefore constructed with periodic functions included to codify this possible revival of the system’s initial values. Of course, the exact parameters that allow one to calculate recurrence times for a given system depend on the strength of the system-environment interaction and on the nature of both. Schlosshauer (2007) shows that obtaining recurrence times small enough so as to be observable at all requires extraordinarily highly ordered initial states, among other unlikely values for initial parameters, such that interaction “in the wild,” so to speak, will inevitably entail recurrence times on the order of many lifetimes of the universe. He summarizes (ibid., p. 93):

Thus the loss of coherence from the system is typically irreversible for all practical purposes not only because of our practical inability to control and observe the environment, but also because the time scale for the recurrence of coherence is astronomically large in virtually all physically relevant situations.

2.3.3.3 Wavelength limits of decoherence

A useful illustration of the eﬀectiveness of environmental scattering for in- ducing decoherence is to consider the physical limits obtained in cases where the wavelength of the environmental particle is (i) small compared to the coherence length in the position basis of the CS and (ii) large by comparison. Instead of

69 talking about specific rates, we will think of these limiting cases of decoherence in terms of information to broaden understanding of the scope of decoherence. When speaking about information, the exact definition of the term is somewhat ambiguous, depending on the literature to which one refers. For our purposes in this section, it is only relevant that the reader consider the following: if the information stored in the environment about the CS (obtained via interaction and due to the nature of entanglement) is sufficient for telling a which-way or which- path story about the CS, then the mere fact that the sufficient information has been spread to the environment means that interference will be damped beyond the limits of observation. However, if upon interaction the environment does not obtain sufficient information to tell a complete which-way story, the CS will remain coherent enough to exhibit interference effects. Before moving to the two wavelength limiting cases (which correspond to cases of the environment obtaining sufficient or insufficient which-way information about the CS), one must be familiar with another parameter of the quantum system— its characteristic coherence length. Coherence length is just the distance in the system’s position space19 between the well-localized (hence coherent) wave packets representing substates of a spatial superposition state. We denote the distance between the wave packets ∆X = |X − X0|. Thus the (characteristic) coherence length is a parameter describing the maximum value of ∆X attainable without

19Although it is important to note that the term “distance” here is merely analogous. It is an exact expression in the wavelength limiting cases discussed below due to the fact that we are concerned with scattering-induced decoherence, which for reasons already given occurs in the position basis. However, one might equally well speak of coherence length (where length is also an analogy) being the “distance” between wave packets representing superpositions in any number of bases, , momentum, spin, superposition of position, etc.

70 effectively destroying interference upon interaction with the given environment. The two wavelength limiting cases for scattering-induced decoherence are then as follows. If the environmental particles interacting with the CS have wavelengths that are small with respect to ∆X, a single scattering event can completely resolve the coherence length. The complete resolution of the coherence length is tantamount to a situation in which the environment can obtain sufficient which-way information about the CS upon limited interaction with it, which in turn results in the delocalization of CS phase relations in that basis (and the destruction of interference phenomena). This is referred to as the small wavelength decoherence limit. The small wavelength limit imposes an upper bound on the rate of decoherence: complete resolution of the coherence length via a single interaction corresponds to maximal decoherence in the least amount of time. The short wavelength limit thus provides a ceiling by allowing a single particle’s interaction with the system to carry away full which-path information. Hence decoherence cannot be achieved more expeditiously, and we’ve defined the limit in this direction. Related to the small wavelength limit of the decoherence rate is a situation called decoherence saturation or over-resolution, in which the CS becomes entangled to an environment dense with particles whose wavelengths are small enough to resolve the CS’s coherence length. The second limiting case for scattering-induced decoherence is the large wavelength limit, wherein the wavelength of the environmental particle is larger than the separation ∆X. In this limit, decoherence will require stronger environmental

71 interaction, as each environmental particle can only gather partial information regarding the CS. In other words, this limit describes scattering interactions that do not resolve the coherence length due to their inability to provide complete which- way information. The large wavelength limit corresponds to a minimal rate of decoherence, as the time required for a sufficient number of interactions in order to fully resolve the CS increases as the difference between the environment’s wavelength and the system’s coherence length increases. Consequently, the greater the difference and the longer it takes the environment to obtain sufficient information, the longer it takes for the phase relations in the CS to be effectively destroyed: these two processes are one and the same. It is not difficult to understand why the large wavelength limit will occur rarely in natural situations. The reason for this is twofold: an increasing value for spatial separation in the CS corresponds to increasing instability, and on the other side of the limit inequality, increasing wavelengths become rarer and rarer in natural environments. Regarding the first, recall that as one assumes larger and larger ∆X one approaches the situation for macroscopic systems, who manifest this fact with correspondingly unstable superposition states and stable component states. These two factors work against one another to ensure the rarity of reaching the large wavelength limit ex laboritorium. Sufficient conceptual grasp of the process of decoherence and related issues (coherence, superpositions, phase relations and so forth) in addition to some rudimentary mathematical formalism (the construction of the total Hamiltonian, coherence lengths and so forth) are now in place, and we can describe in more detail

72 the particular formalism used most frequently to mathematically described the process of decoherence as a system evolves in time in interaction with its environment.

2.4 The density matrix formalism

In the above discussion of unitary Schrödingerevolution, I mentioned the fact that unitarity is only preserved at the level of the entangled system, meaning that in cases where there is entanglement, one cannot describe the constitutive subsystems individually, in terms of pure Schrödingerdynamics. Instead, the best one can do post-entanglement is obtain approximate statistical knowledge of one of the entangled pair (say, the CS) by calculating that system’s reduced density matrix. This method—more generally called the density matrix formalism—is the method for determining system-environment evolution I shall focus on, as much of the formative literature on decoherence, as well as the derivation of master equations for use in modeling, is done within this formalism. Furthermore, using a master equation to describe the evolution of a system in time requires the manipulation of a select few parameters, which in turn makes it easier to analyze just what assumptions are encoded in a given master equation associated with a given model. A different and powerful technique for studying decoherence that utilizes Feyn- man’s path-integral approach (most often referred to as the “restricted path integral,” or RPI, method) has become increasingly favored by experimentalists and

73 theoreticians of late.20 The issue with the RPI approach, however, is that while it is undeniably a powerful mathematical apparatus, one uses it at cost: the method of path-integrals relies heavily on certain calculation tricks that, though ingenious, invoke seriously disturbing metaphysical implications. For example, the RPI technique (and Feynmen’s path-integral approach generally) involves treating the time-evolution of a system as an integrand composed of all the possible trajectories (referred to as “channels” in decoherence research) the CS might undergo. Obviously, issues arise as to the exact meaning of the assumed trajectories in this method, as by studying decoherence we are necessarily dealing with inherently nonlocal, nonclassical, discontinuous entities. Thus one must be wary when considering certain philosophical conclusions made regarding decoherence as understood under the RPI method, and the metaphysical cost of this formalism over that of reduced density matrices were deemed too high given the purposes of the present work.21 Of course, similarly mistaken metaphysical claims can be made based on the density matrix formalism; as such, I shall do my best to ﬂag aspects of the formalism most susceptible to these misunderstandings in my description below. Because the current project demands full disclosure of assumptions, it is im-

20For an excellent introduction to the theory and experimental practice of the path-integral approach to decoherence, I refer the reader to Mensky (2000). In chapter 5 section 2, Mensky provides a nice comparison between the use of master equations, as we will employ in this work, and the path-integral approach. He argues that the two approaches are equivalent for nonselective measurements, which are precisely the sort of measurements going on in nature. 21For example, some supporters of particular interpretations of quantum mechanics have argued for their position over others from the path-integral approach, which is dubious. This occurs frequently in discussions of the consistent histories approach to decoherence.

74 portant to pause before engaging with the formalism and note that the use of density matrices to calculate master equations (and to understand the dynamics of decoherence) involves two assumptions. The first is that there exists a local interpretation for the density matrix. This assumption turns out to be rather harmless once one realizes that the theory of quantum is itself already a local theory, in the sense that the theory assigns unambiguous values to variables according to Schrödingerevolution. To posit a density matrix whose interpretation is local is not problematic, unless the density matrix describes an entangled state, in which case we know from the above discussion that a completely local description is not available. This problem is overcome by using the tool of the reduced density matrix; just how will be described presently. One might still worry about this assumption’s validity for times prior to the interaction between system and environment, as this entails the further assumption that the CS and environment under consideration are not entangled prior to their interaction with one another. This is worrisome when one realizes that of course the CS and environment so defined for the measurement at hand are nevertheless each immersed in their own environments and as such are already entangled. However, as mentioned above, careful study of this assumption and its implications has been carried out by Anglin, Paz, and Zurek (1997). The authors convincingly argue that the assumption of a previously noninteracting CS and environment is a viable one, especially in un-engineered or uncontrolled interactions, such as occur in nature. In particular, it is nearly always appropriate to

75 characterize the environment as occupying an approximately pure or uncorrelated state prior to interaction with the CS. This is due to the fact that uncontrolled environments and suﬃciently large controlled environments are by deﬁnition systems with an enormous number of degrees of freedom, all of which have been interacting among themselves for a comparably long time. Thus the environment is already itself decohered and can be said to be well localized. The second assumption the present formalism presupposes is the validity of the Born rule. This assumption has already been treated, and I have argued that it is misguided to understand the Born rule as a particular assumption when in fact its proper status is among the axioms of quantum theory.

2.4.1 Density matrices

Density matrices are a useful way to describe all the known properties of system ensembles. In situations of entanglement one cannot determine, due to the very nature of entanglement, the CS’s exact state. The density matrix of the entangled system can, however, provide complete statistical information about possible measurements performed on the composite system in some basis. Recall our discussion of operators above: operators are a way to pick out of a system’s Hilbert space the diﬀerent values obtainable for a given measurement and their associated likelihood (what is called the “expectation” value for a given measurement). Density matrices are just another mathematical representation of what is called the density operator. And, as stated, the density operator (what we will refer to as the density matrix) is a compact way of representing the complete

76 measurement statistics—including probabilities of obtaining certain measurement results—for a system characterized by either discrete or continuous degrees of freedom.22 In quantum mechanics, one sometimes speaks of “pure” versus “mixed” states, the former referring to a situation in which a system’s state is fully known and the latter referring to a case where the system’s state is not fully known, and thus can only be represented statistically. A pure state is mathematically represented by a single vector in a system’s Hilbert space (e.g., as |ψii) or by a superposition of such vectors (itself a vector). A mixed state refers to an ensemble of states. Both pure and mixed states are typically described using density matrices,23 though in the case of mixtures one must recognize that the nature of the mixed state is sometimes obscured when represented in density matrix form, as the same density matrix may be written any number of ways, depending on one’s choice of basis. For example, one might choose to represent the density matrix for a particular mixed state in that state’s eigenbasis so that the matrix representing it will be diagonal (i.e., will lack interference terms). Furthermore—and this point is crucial for my thesis—a mixed state characterized by a density matrix does not always represent a proper ensemble, which is to say an ensemble of a classical sort (where the system under consideration is known to occupy only one state among the ensemble of possible states, and we simply lack information regarding which).

22In section 2.5 below I will introduce the Wigner picture, or Wigner representation, as a slightly diﬀerent (and perhaps more insightful) way of mapping the evolution of systems undergoing decoherence. However, the Wigner representation is only applicable to systems with continuous degrees of freedom and is therefore somewhat more limited. 23See d’Espagnat (1976) p. 40 ﬀ. for a full proof.

77 Instead, when one is describing a system whose preparation is not fully con- trollable or known, one cannot say with any certainty that the density matrix for the state of the system represented is a proper classical ensemble, and one is merely ignorant of the single state occupied by the system. In fact, cases where the system under consideration is itself an entangled system—say, a central quantum system entangled with its environment—then one can assume the mixture is improper, as the system can no longer be said to occupy a single state of which we are ignorant and instead must occupy an entangled state. Again, though a given density matrix may be formally identical to that of a proper mixture, what we are dealing with in any case involving entanglement is an ontologically diﬀerent situation corresponding to what is called an improper ensemble.

If |Ψi represents a superposition of basis states |ψii, then the density matrix (and recall, it is nothing more than the matrix representation of the density operator, whose interpretation is well understood) gives the projection onto the superposition state with respect to the diagonal, or noninterference, terms:

X ∗ ρˆ = |ΨihΨ| = cicj |ψiihψj| , for i 6= j. (2.3) i,j The i 6= j terms—the off-diagonal, interference terms—embody the quantum coherence between the different components of |Ψi. Because these are the terms affected by the process of decoherence, they are precisely the ones we wish to track. Now consider the construction of the mixed state density matrix for a proper

78 ensemble:

X ρˆ = pi|ψiihψi| , (2.4) i

which is equivalent to (2.3) when i = j. Here, pi represents the sum of probabilities P under normalized conditions, which give the usual limits of pi ≥ 0 and i pi = 1. If we pause for a moment to consider the mixed state density matrix, we see that its construction as a sum over probabilities makes conceptual sense. In mixed states, we cannot calculate the expectation values straightforwardly because we lack knowledge of the exact state occupied by the CS. Thus we borrow from classical statistics the method of weighting values for each of the possible states, and this takes the form of a classical probability distribution.

The probabilities pi in (2.4) are thus epistemic probabilities, corresponding to the classical distribution and indicating no uncertainty with respect to ontology but only the limit of available knowledge of the system. The mixed state density

matrix represents a case in which the probabilities pi for all subsystems |ψii are zero, save one (whose probability is equal to unity). The switch from considering full Hamiltonians to speaking the language of density matrices (which is necessary in order to describe the evolution of the CS) might in a superﬁcial way be considered coarse-graining, as it is a move from fully known equations to statistical ensembles. Care must be taken when understanding this sort of coarse-graining, though, for it is something of a misnomer. It does not, as some have insisted, represent a loss of actual information to adopt the density matrix formalism. Indeed, the term “coarse-graining” does not carry ontological

79 weight in the way it seems to imply, for we still know as much about the entangled system as we could ever know statistically in addition to knowing that the density matrix represents an entangled system. As long as this latter fact is not forgotten, there is still much that can be said regarding the CS-environment interaction. Along these lines, not only must one be particularly careful regarding interpretations of ensembles (for not all ensembles represent identical epistemic positions with respect to the systems they describe), but one must also be wary of the fact that a mixed state density matrix and the density matrix representing a pure state superposition are similar in appearance, though representing distinct states of aﬀairs. The latter class of density matrices contains oﬀ-diagonal (interference) terms representing the phase relations among the component states constituting those superpositions. The superposition

X √ |Ψi = pi |ψii (2.5) i

24 is a case in which all subsystems |ψii are occupied. This means the probability distribution represented by pi in the above equation no longer corresponds to a classical statistical distribution, and we are no longer dealing with a mere epistemic limitation with respect to |Ψi.

24To be more ontologically precise at this juncture, I should instead say that the metaphysical situation described mathematically as such is not well understood and is in fact a great mys- terybut for the following point: we do know that this representation, however else it might be interpreted, does not correspond to any classical situation. The presence of interference terms excludes from the realm of possible ontologies those typically understood as classical—where a system really exists in just one well deﬁned state. We know this is not the case here. More will be said on this crucial metaphysical point as we progress through various chapters, for this thesis is, to some extent, just this point and the philosophical implications it entails.

80 Because we are here treating a superposition of states, we know there exists some degree of coherence between various components of the superposition, and therefore we expect the appearance of interference (or oﬀ-diagonal) terms in this system’s density matrix. Indeed, this is the case (where the indices i and j correspond to particular rows and columns in a matrix):

X √ X X √ ρˆ = pipj |ψiihψj| = pi|ψiihψi| + pipj |ψiihψj| . (2.6) i,j i i6=j Again, a pause to reﬂect on the meaning of the mixed state density matrix as opposed to the pure state density matrix for superpositions will be illuminating. We have already discussed the reason for the appearance of interference terms in the latter. For the mixed state, we now understand why there are no oﬀ-diagonal terms: there are no superpositions represented in this case. There is a single occupied state and therefore no interference terms to be described by the density matrix. In many real cases, the physicist calculates the density matrix for a CS without exact knowledge of how the system was prepared or what state it occupied prior to interaction. In such cases we cannot make an inference about the nature of the system based on the structure of the density matrix alone, as the appearance of a matrix corresponds to more than one possible physical situation. As a consequence, if we are given only a mixed state density matrix describing a system, we cannot be certain that the density matrix probabilities represent a perfectly classical statistical distribution, for we do not know if the initial system occupied a pure state or not. Schlosshauer explains,

81 Any nonpure density matrix [representing an improper or proper ensemble] can be written in many diﬀerent ways, which shows that any partition into a particular ensemble of quantum states is arbitrary. In other words, the mixed-state density matrix alone does not suﬃce to uniquely reconstruct a classical probability distribution of pure states. (Schlosshauer 2007, p. 42)

Before describing the aspect of the formalism most useful for decoherence— that is, reduced density matrices—there is another important tool that must be introduced called the trace operation. The trace of a matrix is deﬁned to be the sum of the diagonal elements of that matrix; under the normalization condition, the trace of a matrix must be equal to one. The trace of a density matrix is equal to one, which is expected under the deﬁnition of a pure state density matrix given in (2.3). In this case, the diagonal elements are each assigned a probability pi summing to unity, hence our expectation that the trace of such a matrix will also sum to unity. In both pure and mixed state cases, the trace of the density operator on a given observable Aˆ represents the mean value of that observable. In other words, if one multiplies the density matrix for a system with the matrix representation of an observable operator, the trace of the resulting matrix gives the average value for that observable. Formally,

hAˆ i = Tr(ˆρAˆ) . (2.7)

From these considerations we see that a trace operation over a density matrix is tantamount to a trace over degrees of freedom of system represented by the density matrix, and by deﬁnition it represents an average of possible values for observables

82 corresponding to the system’s degrees of freedom. An additional feature of the trace operation is that it is independent of any particular basis. This makes sense when we realize that no matter which basis one is working in, the sum of the probabilities assigned to all possible states in that basis must equal unity. This is simply another expression of the probabilistic constraints in our formalism. Since the trace is a sum over all possible measurement results on the system in some basis, we expect that, independent of which basis, the trace will always equal 1. Let us now turn to a consideration of the reduced density matrix, which is of particular importance for decoherence situations.

2.4.2 Reduced density matrices

The reduced density matrix allows one to describe entangled systems in particular: everything that can be known about the composite system can be discovered from local measurements on a single system using the reduced density matrix method. It does so by considering a partial trace. In a composite system, the partial trace allows one to trace over a subspace of the joint Hilbert space. For example, if A and B are interacting systems, the joint Hilbert space is the tensor product of their respective Hilbert spaces: HA and HB form a new space,

Htot = HA ⊗ HB. The partial trace is then a trace taken over one of the two subspaces instead of over the total Hilbert space, which is often impossible. The reduced density matrix for the system to be measured—let us call this

83 system A—is then deﬁned as

ρˆA ≡ TrB ρˆ , (2.8)

where TrB is the partial trace, or the trace of the total density matrixρ ˆ taken over just the Hilbert space of the unobserved subsystem B. As is the case with traces over a total Hilbert space, tracing over the degrees of freedom of the unobserved part of the composite system gives the mean expectation value over degrees of freedom of just that portion of the density matrix describing the subsystem of interest. Because of this, ﬁnding the reduced density matrix of part of a composite system is sometimes referred to as “tracing out” the degrees of freedom of the other subsystem. The application of reduced density matrices to system-environment entanglement (that is, the sort of interaction that causes the decoherence of the system) should be obvious at this juncture—simply let A represent the CS and B the environment in equation (2.8) above. Schlosshauer once again provides a succinct explanation of the reduced density matrix as it applies to decoherence (ibid., p. 45):

By tracing over (all, or a fraction of) the degrees of freedom of the environment of the system-environment density matrix, we obtain a complete and exhaustive description of the measurement statistics for our system of interest in terms of the reduced density matrix of the system. All inﬂuences of the environment on local measurements performed on the system will automatically be encapsulated in this reduced density matrix. Since the system is entangled with its environment, no individual quantum state can be attributed to the system itself. Therefore, the reduced density matrix is all we have available

84 to describe the statistics of measurements on the system, and this reduced density matrix is necessarily nonpure due to the presence of system-environment entanglement.

There are two things to note in this excerpt. First, it is imperative to understand fully Schlosshauer’s point that the CS’s reduced density matrix—obtained from the density matrix describing system-environment entanglement—is necessarily nonpure. What follows is that any interpretation that understands this mathematical formalism to represent classical ensembles (sometimes referred to as understanding the formalism as “ignorance interpretable”) with probability distributions will provide an incorrect description of system-environment entanglement and so, importantly, mistake the nature of the physics. Misinterpreting the reduced density matrix as ignorance-interpretable is perhaps understandable for cases in which the matrix is diagonal in a particular basis, rendering it formally identical to the density matrix of a proper ensemble. While in the latter case one can (depending on one’s knowledge of the system’s preparation) interpret the diagonal nature of the density matrix as indicating the system’s definite occupation of a single state from among the ensemble of possible states, the same cannot be said in known cases of entanglement such as this. Here, we expect the existence of off-diagonal terms because the reduced density matrix was derived from an entangled system, and we must therefore conclude that the diagonality of the matrix is superficial and in part contingent upon the basis within which the measurement was made. The takeaway lesson is that while it is correct to state that the reduced density matrix encodes all that we can know about a system-environment interaction, it

85 is not the full picture. We also know that the reduced density matrix describing one subregion (the CS region) of the composite Hilbert space of the entangled system is necessarily an improper mixture. As Schlosshauer states, density matrices are only “calculational tools for computing the probability distribution of a set of possible outcomes of measurements, but they do not specify the state of the system” (Schlosshauer 2007, p. 48; emphasis original). Let us pause to take stock of terrain thus far traversed. We have investigated the construction of total Hamiltonians for various system-environment interactions and considered various simplifying cases for Hamiltonians depending on the dynamics of components of a given total Hamiltonian. We have also become acquainted with the density matrix formalism as a means of concisely describing all available statistical information for quantum systems, and we have seen how reduced density matrices lend themselves beautifully to situations of system- environment entanglement, where we wish to obtain information about certain degrees of freedom but not others. We have accomplished much. But what we have yet to do is begin to understand the dynamics of decoherence, and this is the subject of the next section. Before moving on, however, we note that this last accomplishment—making use of trace operators and reduced density matrices—has introduced another instance of coarse-graining in the formalism of decoherence. I mention two points regarding the language of coarse-graining brieﬂy, as it is another source of some ambiguity in the literature. The coarse-graining introduced by choosing the density matrix formalism is (i) not true coarse-graining (as already discussed), in the

86 sense that it still represents all available knowledge of the entangled system, and (ii) is not necessary coarse-graining, as one might choose an alternate formalism for describe decoherence, in which no such coarse-graining occurs. Such a method is the RPI approach; but for reasons already given, the density matrix formalism (despite this “coarse-graining”) is comparatively metaphysically unencumbered. This is my primary reason for choosing it as a means to understanding decoherence and how it is modeled.

2.4.3 Master equations

The interpretation and derivation of master equations given below follows closely the excellent chapter 4 of Schlosshauer (2007). In general, master equations describe the time evolution of probable states of a system and are used extensively in both classical and quantum mechanical calculations. In the case of decoherence, we are interested in calculating the master equation of reduced density matrices describing the CS of our larger entangled system. There are two reasons motivating the use of master equations in decoherence:

1. Master equations allow us to calculate just those things we wish to calculate without the burden of excess baggage. Speciﬁcally, in cases of decoherence we are interested in calculating not the global system-environment state (nor is it possible in a majority of cases to do so) or the total dynamics but rather the dynamics of the CS alone. The structure of the master equation is such that, if derived from a reduced density matrix of the CS, it will yield all (and only) the desired information.

87 2. It is impossible in most real-world cases to fully determine the time evolution of an entangled density matrix. However, by the use of certain approximative methods, we can nevertheless derive a master equation that captures the evolution of such reduced density matrices, and this approach (approximative methods for deriving the master equation) is oftentimes the best that can be taken in a world of quasi-solvable situations.

The most generic form of the master equation is derived from the reduced density matrix of the CS and takes the following form:

ˆ ˆ † ρˆCS(t) = TrEnv ρˆ(t) ≡ TrEnv {U(t)ˆρ(0)U (t)} . (2.9)

The first portion of the above equation is simply a time-dependent version of the reduced density matrix according to the definition given in (2.8). On the far right-hand side, the operator Uˆ(t) is the time-evolution operator for the composite system. However, one can see that this calculation requires the determination of the total density matrix; this is nearly impossible to calculate in most system- environment models. But we know from the hints in point 2 above that there exist approximative methods available that give calculable master equations. Specifically, we will employ approximations that allow us to write our master equations in first-order differential equation form, as the usual master equations are generally written. For a more complete derivation of the master equation in first-order differential- equation form, I refer the reader to Duplantier et al. (2007), Giulini et al. (1996), Janssen (2008), Zeh (2002) and Zeh (2007). Our purposes require only conceptual

88 familiarity with the master equation in differential-equation form (so that when we adjust the master equation in particular ways for particular models in the next chapter, there will be sufficient foundational understanding of the formalism) and the tracking of various assumptions and approximations along the way. The general master equation for system-environment coupling in first-order differential-equation form has two components. The first component—the commutator of the Hamiltonian of the CS with the CS reduced density matrix— accounts for unitary evolution of the CS, while the second component describes decoherence. The equation is as follows:

d ρˆ (t) = −i [Hˆ0 , ρˆ (t)] + Dˆ[ρ ˆ (t)] . (2.10) dt CS CS CS CS

The prime in the Hˆ0 term is meant to remind us that this Hamiltonian of the CS is not the same as the free Hamiltonian for the system due to environmental perturbation. Schlosshauer (2007, p. 155) notes that the last term—the decoherence term—might include dissipative effects. We shall discuss this more below and in the following chapter. The decoherence (and possibly also dissipation) term is nonunitary, whereas the commutator term is entirely unitary by construction. Of course, our attention in what follows will be focused on unpacking the de- ˆ coherence term, D[ρ ˆCS (t)]. As we shall see, how this term is unpacked depends explicitly upon a few important parameters; in fact, Schlosshauer devises a mantra well worth remembering to explain the generality of the master equation in comparison to the various parameters (coefficients) it is assigned, depending on the situation. He titles a section as follows: “Master equations are general; coefficients

89 and spectral densities are specific” (ibid., p. 240). The most general of these assumptions are encapsulated in the particular version of the master equation called the Born-Markov equation, which will occupy us in the following section. However, before progressing to more detailed versions of the master equation, the mantra I’ve quoted from Schlosshauer mentions a final bit of formalism that will be crucial to our understanding of decoherence, namely, spectral densities. Spectral densities are a more general concept than their appearance in decoherence formalisms would indicate. An observable’s spectrum is composed of the set of possible eigenvalues for that observable. In decoherence, we are concerned not with the spectrum of possible eigenvalues of an environmental observable, but rather with their distribution in the Hilbert space—hence, the spectral density of an environment presents us with crucial data for calculating CS-Env interaction, though not necessarily of an observable’s eigenvalues. Spectral densities describe important properties of the environment—usually energy properties—and their construction depends on the choice of environmental modeling. For example, in the very common case in which the environment is modeled as a collection of noninteracting harmonic oscillators, the spectral density depends on frequency and can be defined as

X c2 J(ω) ≡ i δ(ω − ω ) . (2.11) 2m ω i i i i

The ωi’s represent individual oscillator frequencies, with their respect masses labeled mi. As we shall see in the next chapter, modeling an environment in

90 this way is completely general, for as the coupling between the environment and a quantum system is taken to the weak-coupling limit, all environments can be approximated as uncoupled harmonic oscillators. Thus, the spectral density along with various coeﬃcients (some of which will in turn depend on the spectral density and therefore the physical properties of the environment) are what specialize an otherwise general master equation.

2.4.3.1 Born-Markov master equation

The Born-Markov master equation is the most important of master equations for the wide generality and simplicity it achieves with reasonable assumptions. However, the Born-Markov equations do not apply in certain temperature regimes, in which case the dynamics are said to be non-Markovian. More will be said about non-Markovian dynamics in section 2.4.3.2. As regards the Born-Markov equation itself, the only term that diﬀers from (2.10) above is, as expected, the decoherence term. The unpacking of this term under the Born and Markov approximations requires a signiﬁcant amount of tech- nicalese, which I shall forego in favor of a more conceptual description of the dynamics capable of being explained by the present form of the master equation. This type of master equation is so named for the pair of assumptions upon which it rests: the Born approximation and the Markov approximation. The Born approximation allows us to consider the reduced density matrix of the environment as approximately constant for all time by assuming that (i) the system- environment coupling is relatively weak and (ii) the environment is large enough

91 to wash out minor internal fluctuations within the time period crucial for decoherence processes. Requiring the environment to be sufficiently large so as to wash out minor fluctuations is equivalent to assuming minimal back-action of the system on the environment. This assumption allows us to consider the environment as remaining in its initial state despite interaction with the CS. Treating the environment as approximately constant, taking its initial values, further allows us to remove time-dependent environmental states out of the total master equation, rendering it solvable (which would not be the case, save for rare simple cases, with a normal master equation, depending (as it should) on a fully known, time-dependent density matrix of the composite system). The Markov approximation might be called the “forgetful environment” assumption: we assume that changes in the internal dynamics (specifically, self- correlations) of the environment induced by the coupling interaction with the CS are negligible on the time scale of interest for the CS. In other words, the Markov approximation lets the decay rate of environmental self-correlations be approximated as much faster than the rate at which significant variation occurs within the CS, rendering these self-correlations trivial with respect to the CS’s dynamics (which we are most interested in, after all). The more obvious motivation for adopting the Markov approximation is as follows: we want to derive a master equation that is local in time and, as a consequence, takes the form of an easily solvable ordinary differential equation. To make a master equation time-local is to isolate and separate out any dependencies

92 the equation might have on times prior to the time of interest (in our case, this is the time of interaction). One’s ability to render the master equation independent

of prior times depends not only on the Markov approximation (i.e., τcorrelation

τCS) but also on a property of the trace operation: trace operations are invariant under cyclic permutations of the operators. Let us see how the approximation plus this property get us a master equation that is local in time (and therefore a nice, normal, solvable differential equation). The method for applying an effective Markovian approximation involves shrink- ing as much as possible the quantifiable expression for environmental self-correlation, thus making these correlations decay quickly with respect to the CS. This is done by assuming weak coupling (as we have for the Born approximation already) and a sufficiently hot environment. A hot, weakly-coupled environment behaves analo- gously to other stochastic processes classified as Markovian, in that the probability of a particular event (say, a measurement) is independent of all prior events. The ability to define our environment as behaving with approximately Markovian dynamics gives rise to the environment’s inability to remember; mathematically, this means the dynamics of the environment (specifically its self-correlations) arise but decay so quickly that the term in the master equation describing the environment’s intrinsic dynamics can be assumed to be independent of earlier times. This, then, results in a time-local master equation. In sum, our two approximations have allowed us to derive a master equation that depends on entirely known quantities, that is local in time (and subsequently in simple first-order differential form), and whose derivation was based on rea-

93 sonable assumptions. Cases where the Born and Markov assumptions will not be applicable include (i) slow, cold environments; (ii) strong system-environment coupling; (iii) when the environment is too small for effective washing out of internal fluctuations; and (iv) when environmental dynamics cannot be approximated as Markovian (the environment “remembers” its self-correlations). Note that situations (i) and (iii) are extremely uncommon in natural situations. Environments that are cold and/or slow enough to invalidate the Born and Markov assumptions are rare, and in the real world we are never dealing with small environments—our environments include, in truth, every degree of freedom in the universe save those of the CS. Thus, not being able to successfully apply the Born and Markov assumptions for reasons (i) and (iii) are concerns only for highly controlled laboratory scenarios. As for (ii), situations involving sufficiently strong coupling to render the Born- Markov master equation inapplicable are likewise rare in naturally occurring systems, as cases of weak coupling are the limit as environmental degrees of freedom approach infinity (this will be filled out in the next chapter). Non-Markovian environmental dynamics, though still comparatively rare in nature, nevertheless present more of a limitation than the rest; as such, we will pause to consider work with non-Markovian environments in more detail below. Before we do so, though, we need to acquaint ourselves with the most widely used class of Born-Markov equations: Lindblad master equations. Lindblad equations are simply Born-Markov master equations with an additional requirement: that the master equation ensure only positive values are

94 attributed to the CS reduced density matrix. This requirement is reasonable, given that in most cases we will want to associate the elements of the reduced density matrix with probabilities, and therefore we wish to avoid nonnegative values. In fact, the only reason that negative values are possible in the CS reduced density matrix in the Born-Markov master equation is due to the approximations employed to derive the equation. ˆ In this special class of master equations, the decoherence term D[ρ ˆCS (t)] from equation (2.10) can be given in Lie-bracket form as Dˆ[A,ˆ [A,ˆ ρˆ] ] (Mensky 2000, p. 118; my notation). The coeﬃcient Dˆ is a constant characterizing the strength of the system-environment interaction, the density matrixρ ˆ is the total density matrix, and the operator Aˆ corresponds to the coordinate of the CS being (continuously) measured by the environment in a given interaction. If the master equation for a CS is exactly solvable, the elements of the reduced density matrix are trivially nonnegative. Thus requiring that an equation take Lindblad form is only necessary because of the assumptions made under the Born-Markov derivation of the master equation. Because hardly any interesting system-environment couplings can be described by exactly soluble master equations, instances of Lindblad equations frequent the experimental literature.

2.4.3.2 Non-Markovian dynamics

In cases of strong system-environment entanglement or with environments that remember (i.e., retain information regarding self-interactions that complicate the dynamics of the overall system), the assumptions necessary for employing Marko-

95 vian dynamics are violated. A well-known example of non-Markovian dynamics is the superconducting qubit, a fundamental part of the burgeoning work done in quantum computing. It is necessary in situations such as these to model one’s environment in a less simplified manner, but as the next chapter will demonstrate, such difficulties have been successfully overcome. The limit of Markovian dynamics is reached when all coefficients of the master equation are time-independent. This is not a frequent state of affairs outside of the lab. Yet even in the regime of non-Markovian dynamics, one can still employ d a general master equation ρˆ (t) = Kˆ (t)ˆρ (t), where Kˆ is a time-dependent dt CS CS super-operator (an operator on an operator). Though this equation is unlike the Born-Markov master equation precisely because of the time-dependence of this operator, Kˆ is not fully dependent on time, in the sense that it is independent of times prior to the interaction.

2.5 An alternate perspective: The Wigner representation

In situations where there is continuous measurement, or monitoring, of the CS by the environment, one can apply the Wigner equations to model the dynamics of the system-environment instead of applying the reduced density matrix formalism described above. One reason why the Wigner formalism is attractive is owing to its visual similarity to comfortable classical probability distributions. However, the Wigner function of a system’s density matrix can take negative values (which for obvious reasons cannot be the case with a classical probability distribution), and this complicates interpreting the picture as representative of probabilities in

96 the usual sense. However, integrating the Wigner function over a single variable instead of over an entire phase space (as in the form of the equation below) will yield an entirely positive probability distribution for the other, conjugate variable. The equation that generates the picture—which is a three-dimensional phase space mapping of the system (the axes being position, momentum and probability amplitude) is

Z ∞ 1 ıpy y W (x, p) ≡ dy e ρˆCS(x ± ) , (2.12) 2π −∞ 2 where the variables x and p may represent any pair of conjugate variables (though in analogy to classical mechanics, the Wigner mapping is usually down in a traditional phase space with position and momentum), andρ ˆ is the now-familiar reduced density matrix of the CS. Perhaps a more obvious reason for using the Wigner representation is due to its ability to provide a clear visualization of position-momentum correlations in the CS’s phase space. In such a (classical) phase space mapping, the Wigner function displays the component states (think of the states “dead” or “alive” in the case of Schr¨odinger’scat) as two minimum-uncertainty Gaussian peaks that oscillate (incorporating the equation’s negative values), and whose separation in the position-momentum plane represents the coherence length of the CS in either dimension. Between the two states exist smaller Gaussians representing the superpositions of the component states. What is particularly neat about the Wigner representation is that when it is set to interact with an environment, the evolution of the system makes the process of decoherence particularly vivid: the interfer-

97 ence terms—the smaller peaks between the two minimum-uncertainty peaks—are extremely rapidly suppressed in amplitude until they cannot be resolved at the scale of the two component-state peaks. The two component state peaks, while exhibiting some movement in the momentum-position plane, remain largely coherent. This, then, is a visual representation of the effects of decoherence; it is not difficult to see why, watching animations of the Wigner representation, one is struck by the apparent utter destruction of the off-diagonal peaks. It is quite a powerful demonstration of the effectiveness of decoherence even in cases of weak CS-environment interaction. The most useful application of the Wigner representation with respect to decoherence applies to one of the four canonical models taken up in the next chapter, in which both the environment and the CS have continuous degrees of freedom and are considered dynamically equivalent to harmonic oscillators. This model— called the oscillator-oscillator or quantum Brownian motion model—describes an exceptionally large array of dynamical situations, and the success of this model (and of the others) will be important for substantiating any future metaphysical claims made on the basis of decoherence.

98 CHAPTER 3

MODELING DECOHERENCE

3.1 The general practice of modeling

Before getting into a description of the four models, it will be worthwhile to pause momentarily and consider what roles models play in physics. Understanding the general nature of models in scientiﬁc research is supposedly straightforward: in order to speak meaningfully about observations, measurements and the like, physicists must impose divisions on the phenomena to organize them in understandable subunits (i.e., in terms of systems, subsystems, environments and the like). Physicist N.P. Landsman paraphrases this necessary restriction of the quantum physicist in particular when he states,

The essence of a “measurement”, “fact” or “event” in quantum mechanics lies in the non-observation, or irrelevance, of a certain part of the system in question. ... A world without parts declared or forced to be irrelevant is a world without facts. (Landsman 1995, pp. 45–46)

The experimentalist and the theorist alike must actively choose which physical features will be relevant and which irrelevant, and that this choice was made, and how, must not be forgotten. Usually these choices are encoded in the parameters describing particular models. Thus the empirical success of a particular model is

99 evaluated in part based not only on its applicability in the domains within which it was constructed, but furthermore in the model’s ability to extend beyond its initial parameters and describe increasingly disparate phenomena. The issue of models in the philosophy of physics is and has been an active and broad arena of investigation and discussion; I will look briefly at a particular account of models—that of Nancy Cartwright’s—which has garnered much interest in the field and has the merit of being a metaphysical account of models and not merely an epistemic or realist/antirealist treatment. I find her account of models on the whole to be a cautionary tale about what types of metaphysical theses can or cannot be inferred from the success of physical models, and I think decoherence in particular has something interesting to say as to why Cartwright believes models are tricky sources of metaphysical support. In discussing Cartwright’s account, I hope to make a few things clear that relate directly to the present chapter and the subsequent chapters, whose philosophical strength rests on the decoherence models to large degree. I am, first, to give the readers a sense of the received view of the role models play and how philosophers of physics have generally understood the relationship between models and metaphysical views; second, to describe one of the most forceable critiques of this view to date, and in doing so, third, demonstrate that the models of decoherence, if taken as an example of models and metaphysics and their relationship, strongly support the spirit of Cartwright’s view while at the same time disagreeing heartily with the letter of it.

100 3.1.1 Cartwright on models

Entire papers and whole chapters of books have been written either by Cartwright herself or those responding to her on the particular issue of models. The main thrust of her view, however, can be stated rather briefly: Cartwright holds that the world is dappled, in the sense that various sciences and their respective laws do not form a unified whole subject to intertheoretic reduction but instead are “patchwork.”1 The theories themselves are severely limited in scope, owing to the necessity that the laws composing them contain ceteris paribus clauses. The effect of such clauses is to restrict validity of the laws such that they obtain only “when a nomological machine is at work” (Cartwright 1999, p. 25). Cartwright also con- nects this limitedness to models of theories more directly: in the introduction to Dappled World she writes (p. 12),

... [M]y investigations into how basic science works when it makes essential contributions to predicting and rebuilding the world suggest that even our best theories are severely limited in their scope: they apply only in situations that resemble their models, and in just the right way, where what constitutes a model is delineated by the theory itself.

Indeed, this understanding of models as only deﬁnable from within the framework of a given theory greatly restricts how successful such models can be; their accuracy, for example, cannot apply beyond the domain of the parent theory, by construction. Thus claims about a model’s success can hardly extend beyond the

1In particular, see Cartwright (1983, 1994a,b, 1999) as well as Hartmann et al. (2008) for various critiques of Cartwright and her replies to them.

101 walls of the laboratory.2 Thus far, these statements of Cartwright’s seem to sit perfectly well alongside the lessons of decoherence. That it took so long after the advent of quantum mechanics for physicists and philosophers to consider decoherence a distinct physical process worthy of investigation is testimony to how quickly we forget the idealizations that go into the creation, articulation, modeling and testing of laws. It is precisely the realization that the Schrödingerequation is only truly valid for closed systems—and furthermore, that owing to quantum interactions being wildly different in nature than dissipative interactions, no system (save the universe as a whole, possibly) is truly closed—that is responsible for the current interest in decoherence. This is fine for Cartwright; interestingly enough, she turns to the Schrödingerequation to demonstrate her point about the idealizations inherent in even our most successful theories. That idealizations inevitably play a part in our sciences isn’t news to anyone; what is controversial is the extreme restrictions Cartwright believes this limitation implies. Specifically, she wants to argue against fundamentalism, where fundamentalism is defined to be the unwarranted privileging of concrete facts (“legitimately regimented into theoretical schemes”) about behavior in “highly structured, manufactured environments” (Cartwright 1999, p. 24) almost to the exclusion of concrete facts obtained in other ways. Fundamentalists are those

2In her response to Hoefer (Hoefer 2008) Cartwright does clarify this notion of a model’s applicability being dependent on whether it is understood internally or externally with respect to the laboratory, saying that of course certain models remain valid when applied beyond the literal walls of a laboratory; nevertheless, she maintains that it is not far beyond the literal walls that the parameters needed to deﬁne a given model will no longer strictly apply, and one ﬁnds oneself still a long way from being able to base strong metaphysical claims on the model.

102 who think facts derived from the lab belong to a uniﬁed scheme and are exemplars of how nature really works (ibid., pp. 24–25). It is this type of attitude that Cartwright opposes when positing the “patchwork” nature of laws, thereby also resisting various types of theory reduction that have long been in vogue. Hence a curiosity: Cartwright’s method of arguing for her patchwork view of laws is certainly sympathetic to the things I’ve been emphasizing in my thesis so far. And yet, the conclusion I draw from the models of decoherence is exactly that which Cartwright meant to disable—fundamentalism, particularly modes of quantum fundamentalism. Why do decoherence models seem to escape Cartwright’s conclusions?

3.1.2 Responding to Cartwright

The general lesson one might draw from Cartwright’s dappled-world thesis is that when one takes into account the deeply idealized nature of our laws, the necessary restrictions of model applicability outside the theory for which they were developed, and the psychological pressure to transfer undue warrant to facts obtained from within certain theories, then one realizes there is no one scheme that correctly describes physical phenomena—not even quantum theory—and our metaphysics should reﬂect this fact. The real world, in other words, does not conform to any of our physical laws for these reasons, and we should not expect it to. The fascinating thing about models of quantum decoherence in this light is that they were constructed as a result of physicists’ realizing that in the domain

103 of quantum mechanics more and different assumptions are at stake (owing to the oddness and pervasiveness of quantum entanglement as compared to other physical interactions); the consequence of these realizations was to develop models that explicitly took into account just those idealizations Cartwright claims restrict other models (e.g., that unitary evolution of the Schrödingerequation only approximately describes non-isolated systems). The result, as we shall see, is the overwhelming empirical confirmation of quantum mechanics. In other words: models of decoherence just are models of the world resulting from dropping idealizations, and the well-confirmed picture these models give us back is that of a thoroughly quantum mechanical world. In decoherence models, we see the con- vergence of the uncontrollable, everyday world with precisely those tools—models of quantum theory—that Cartwright has considered unable to give the everyday world to us. One need not go to decoherence models to take aim at some of Cartwright’s statements concerning the role of models, however. In a response to Cartwright on behalf of fundamentalism, Carl Hoefer writes the following:

Cartwright sets the core of the [fundamentalism/anti-fundamentalism] dispute out very clearly: What may we induce, from the success of our physical theories...? Her answer seems to boil down to this: You can induce that the theories truly describe those systems that have been shown to ﬁt the core interpretive models of the theories, and nothing more. Notice how dangerously close her answer is to the following: We have reason to think that the laws of a physical theory hold only in those cases where we can show that they hold. But this is not so much a principled restriction on induction, as a ﬂat unwillingness to induce anything at all! Much depends, obviously, on how reasonable and

104 principled the dividing line Cartwright oﬀers really is. A fundamentalist thinks that the range of (approximate) truth of the Schr¨odinger equation goes quite a bit further than the list of cases where it can be explicitly demonstrated and that this is a reasonable inductive conclusion to draw from the successes of QM. (Hartmann et al. 2008, p. 316)

In this response we get a sense of the received view—perhaps one might even call it the “strong view” of models, which might be characterized by a further comment of Hoefer’s (ibid., pp. 317–318):

Where we are clever enough to be able to test this theory [QM] and this equation [Schrödinger’s],they seem to be correct. But—aside from this question of what we are clever enough to be able to model and treat with a theory—there seems to be no very relevant difference between matter inside the labs and matter outside the labs. A hydrogen atom in a spectrometer is, plausibly, much the same as a hydrogen atom floating in your living room. The simplest hypothesis would seem to be that if there are mathematical laws governing these things in one setting, then the same laws govern them everywhere. (Emphasis original)

Hoefer seems to suggest in this response to Cartwright that the power of models, despite the genesis, development and testing of said models within a controlled laboratory setting, can nevertheless plausibly be extended to matters outside the lab. This stronger view of models does appear to be supported by the work of decoherence (pace Cartwright); this point is one the reader should bear in mind as the models are discussed in more detail. To summarize, one of the many points Cartwright wishes to make, especially regarding the role of laws and models in informing our metaphysics, is that our models and our laws are undeniably successful but (i) only within the severely

105 limited domain that helped produce them and (ii) only under the auspices of the theory for which they were constructed. Both of these qualifications importantly undermine the model-metaphysics link by casting serious doubt on the strength of inference that can be drawn from the empirical successes of such models to their ability to describe the behavior of systems “in the wild.” However, I have argued that the success of decoherence models need not (and cannot) be subject to Cartwright’s qualifications by the nature of decoherence itself: discovering and understanding decoherence as an essential part of the overall dynamical process was a consequence of taking quantum mechanics out of the laboratory in the first place—or perhaps more accurately, of realizing that via ubiquitous quantum interaction the world brings itself into the laboratory and, like a petulant child, refuses to be ignored. In this way, my taking the models of decoherence and their empirical success (under reasonable, minor assumptions—considered so not least owing to the sheer breadth of systems for which the models seem to work rather well) as crucially supporting certain metaphysical conclusions remains not only impervious to Cartwright’s scheme but perhaps even strengthened by it—for it is by studying just how and why our idealizations in quantum theory break down that the universality of the theory is further demonstrated.

3.2 An overview of the four canonical models of decoherence

A large majority of real-world interactions can be modeled surprisingly well using only four models that combine diﬀerent properties of both the system and

106 the environment. An overwhelming percentage of physically interesting systems and environments ﬁt into just two categories: those systems (or groups of systems, i.e., environments) with discrete phase spaces (whose dynamics can be mapped onto that of a two-state system) and those with continuous phase spaces (or with dynamics analogous to an harmonic oscillator, i.e., with a continuous phase space). The four canonical models for decoherence are constructed by all permutations of these two types of systems: two models have two-state (or approximately two- state) central systems and two have continuous, harmonic oscillator-like central systems, while the corresponding environments are modeled by a collection of spin-1/2 particles or as a collection of harmonic oscillators. Thus the four models are (listed by CS type-environment type) the oscillator-oscillator model, the spin- oscillator model (also called the spin-boson model), the spin-spin model and the oscillator-spin model. With these combinations we can describe, with surprising accuracy, the process of decoherence as it occurs in most conditions of interest. I will discuss each of these four models in turn in this chapter, grouped by environment type. I present them in this manner because, as we shall see, oscillator environments ﬁt the widest range of naturally occurring phenomena. This in turn is due to the fact that, in the limit of weak CS-environment interaction, all systems can be represented as some central system weakly, linearly coupled to a set of harmonic oscillators. Additionally, the other type of environment—that of discrete coordinates—turns out to be appropriate only for modeling decoherence when the environment is at low enough temperatures that one would not expect to encounter such an environment in nature.

107 Since the project of this dissertation is to investigate the process of decoherence especially with respect to ontology, though we will still be interested in learning all that we can about this process in any temperature regime, we will nevertheless rely more heavily on those models representing natural states of affairs as opposed to highly engineered laboratory setups. Thus the focus of this chapter is on oscillator environments, their characteristics and assumptions, and oscillator environments entangled with both a spin-1/2 CS and a harmonic oscillator CS. After presenting the general characteristics of the environment and then the specifics of both the spin-osc and osc-osc models, I will discuss theoretical and experimental predictions and confirmations of these models. The second major part of the chapter will then focus on spin-environments generally and explore in some detail (though not as much as with the oscillator environments) the specifics of both spin-environment models along with experiments done testing their validity under various parameters. Before moving to the main discussion, I have provided below a brief overview of the major characteristics of certain CS models and environment types so that one may see at a glance the major regimes of application, basic construction and so forth. Each item will be explained in more detail in the body of the chapter.3

3Note that sometimes a particular environment will be referred to as a “bath,” which indicates that the environment (whether it be bosonic ﬁeld modes or a collection of two-state systems) is in thermal equilibrium. As we shall see, much of the theoretical work done on decoherence involves bath environments. This is a rather innocuous assumption, however, when one considers that any naturally occurring environment has typically been interacting thermally with its own environment, leaving any relatively local environment in approximate thermal equilibrium.

108 Central system models

• For a CS with continuous coordinates (usually position and momentum coordinates)

– Dynamics mappable onto those of a harmonic oscillator in a potential well

– Strength of CS self-Hamiltonian is approximately equal to interaction Hamiltonian, leading us to expect decoherence in the CS’s phase space4

– Most widely applied decoherence model of all—the quantum Brown- ian motion model—considers the CS to be analogous to a quantum Brownian particle in a harmonic potential

• For a CS with discrete (or approximately discrete) coordinates (i.e., a two- state system)

– Dynamics mappable onto those of a spin-1/2 particle, where two energy minima in a potential well (a double-well potential) are separated by a barrier that, depending on the parameters, may require introduction of a non-negligible tunneling term in the CS self-Hamiltonian (to codify the possibility of the CS quantum tunneling from one minimum to the other)

– Intrinsic dynamics in these models are more complicated owing to possibility of quantum entanglement within the system itself (e.g., the

4Recall the discussion on simplifying Hamiltonians from 2.3.2.

109 translation coordinates may become entangled with other degrees of freedom, like spin)

– Extremely useful model for quantum computing, due to ease of mapping binary language (a unit of which is called a “qubit”) onto a CS that eﬀectively has just two states (corresponding to the CS’s occupying one or the other of the energy minima)

– This model is used whenever the CS is a fermion (e.g., an electron or proton), as fermions are two-state systems with respect to spin (“spin up” or “spin down”)

Modeling oscillator environments:

• A collection of harmonic oscillators corresponds to an approximate continuum of delocalized (i.e., with overlapping wave functions) bosonic ﬁeld modes5

• Gaseous environments are often best suited to this model; the dynamics of gases behave like bosonic ﬁelds

• Coherence and energy of a CS in such an environment will leak practically irreversibly, resulting in (respectively) decoherence and thermal dissipation

5Bosonic fields have dynamics that are easily mappable onto a potentially infinite set of independent quantum harmonic oscillators. In particular, bosons are symmetrical, and more than one boson can occupy the same energy state (which is a key feature if one wishes to describe up to infinitely many such particles all occupying the same approximate energy state, as is necessary in cases of thermal equilibrium

110 of the CS and causing the wave function of each environmental ﬁeld mode to become increasingly spatially widespread (delocalized)

• An extraordinarily powerful way to characterize environments due to its being the weak-coupling limit. That is, in this limit, all CS-environment interactions can be modeled successfully as a CS linearly coupled to a harmonic oscillator environment

• With this type of environmental model, we assume that the coupling strength of each individual mode scales with the number of oscillators in the envi- √ ronment as 1/ N. This scaling assumption is necessary to ensure a well deﬁned thermodynamical limit as N approaches inﬁnity

• Usually it is assumed that the inﬂuence of any one environmental oscillator upon the CS is negligible. In other words, the intrinsic dynamics of the environment (encoded in the self-Hamiltonian of the environment) are negligible compared to the interaction Hamiltonian or the CS’s intrinsic dynamics. This assumption of asymmetric inﬂuencing will be crucial for modeling real-world scenarios

Modeling spin-1/2 environments:

• This model is extremely useful in highly engineered experimental situations, as it best models phenomena interacting at low temperatures (that is, the really low temperatures necessary for superconducting and quantum computing). However, this model of the environment (because it functions well

111 only within the low-temperature regime) does not have wide applicability in nature, which is, on the whole, “hot and bothered”

• Why is this model good for low temperatures? In this regime, the total Hamiltonian is best described as interacting most strongly with well localized environmental modes (as opposed to the delocalized modes modeled by the harmonic oscillator/bosonic field environment). In other words, the wave functions associated with very cold environmental modes are confined to small spatial regions and are describable in a finite-dimensional Hilbert space with a finite energy cutoff. These modes, being highly localized, are therefore best characterized as an environment with discrete energy states

• The interaction between CS and such highly localized cold environmental modes is typically weak compared to the intrinsic dynamics of the CS. This corresponds to the case described in 2.3.2 in which the total Hamiltonian can be considered approximately equivalent to the CS self-Hamiltonian. We therefore expect the preferred basis in such environments to be energy eigenstates of the CS, whether that corresponds to the continuous range of energies associated with a harmonic oscillator or the discrete states available for a CS that is mappable onto spin-1/2 particle dynamics

• In the weak-coupling limit of CS-environment, the dynamics of a spin-1/2 environment reduce to those of a bosonic ﬁeld (oscillator environment)

• There is debate as to whether spin environments should exhibit strong scaling dependence, as is assumed in the oscillator-environment case to maintain

112 reasonable thermal limits. In some instances, a spin-coupling scaling on the order of 1/N is applied to spin baths

The discussion of each type of model is meant to limit technicalities and carefully explore philosophical assumptions, limitations and scopes of the models as well as the treatment of a given model in extant literature. As such, I will not provide lengthy presentations of the master equations for each model in what follows. Instead, I will begin with a brief review of general master equations based on work already done in chapter 2 but will add considerably more detail regarding the meaning of various coefficients and two other crucial parameters of the models—the noise kernel and dissipation kernel. Once the generalities are on the table, I will proceed to the presentation of the models and briefly introduce (along with model-specific Hamiltonians) the special master equations and associated parameters for each model. The purpose in describing carefully these aspects of each model again comes from the overall aim of the thesis: to understand the metaphysical ramifications of decoherence. As discussed in the introductory section, the extent to which metaphysical claims can be substantiated by the physics is crucially dependent on the match between theory and experiment—a relationship in which models serve as arbiters.

3.3 Master equations: From general to speciﬁc

Of course there is no one master equation that is appropriate for all four canonical models (and perhaps many more that aren’t appropriate for a single

113 model!); however, as explained in the previous chapter a lot of good work can be extracted in particular from the Born-Markov general master equation. Thus I begin with a treatment of it and from there we shall become more model-specific. As in the prior chapter, I shall focus not on the mechanics as much as on the assumptions and dependencies of the equations crucial for building models.6 Recall Schlosshauer’s mantra that master equations are general, spectral densities and coefficients (of the master equation) are specific. In the Born-Markov master equation, both the decohering and dissipative forces of the environment are completely described by taking an integral over various kernels. In turn, these kernels are described by the spectral density of a particular environment (recall from chapter 2 that the spectral density is the equation that describes the relevant characteristics of an environment). Thus we expect different spectral density equations for oscillator and spin environments; as a consequence, we expect that the kernels for oscillator and spin environments will be different, reflecting the difference in the construction of the spectral density. Let us first take a step back: in deriving master equations, one of the key elements in need of specification is the environmental self-correlation function, which characterizes the relationship of various dynamics within the environment to other processes, specifically noise and dissipation at time τ after interaction with the CS. Thus, environmental self-correlation describes the intrinsic dynamics of

6For full derivations of various master equations, see Schlosshauer (2007, chapters 2, 4 and 5) or Breuer and Petruccione (2002, chapter 3).

114 the environment. The environmental self-correlation function is typically deﬁned

C(τ) ≡ ν(τ) − iη(τ) , (3.1) where ν(τ) is the noise kernel of the environment, deﬁned as

Z ∞ ω ν(τ) ≡ dω J(ω) coth cos(ωτ) , (3.2) 0 2kbT and the η(τ) represents the dissipation kernel of the environment, deﬁned as

Z ∞ η(τ) ≡ dω J(ω) sin(ωτ) . (3.3) 0

There are a few things to notice about both of these environmental kernels. The ﬁrst is that both are integrals over all possible environment spectral densities—these functions describe the lay of the land, so to speak, with respect to either noise or dissipation. Second, both kernels depend of time τ. Time τ = 0 corresponds to the initiation of CS-environment interaction, and we are interested in the behavior of the environment for this moment as well as all times after interaction: τ ≥ 0. The third item of note is that both kernels contain periodic functions. The fact that periodicity is built into the dynamics of the environment at the very start accounts for the existence of coherence recurrence times, as discussed in the previous chapter (section 2.3.3.2). Every kernel involved in the master equations for our four canonical models of decoherence contains at least one periodic function, because we know that recurrence times (called Poincar´e recurrences) are physical possibilities for the dynamics of any system; if left to

115 evolve for a suﬃcient amount of time, all systems will (it is postulated) return to their initial state. Though these recurrence times may be extraordinarily large— in most decoherence cases, many orders of magnitude greater than the lifetime of the universe—they are nevertheless important features of the equations. A fourth thing to notice is that the noise kernel has temperature dependence

(as we see by the 2kbT denominator in the hyperbolic cotangent in equation (3.2)), whereas the dissipation kernel does not. Noise is a characterization of the thermal interactions in the environment, and thus we expect that our equation for the intrinsic dynamics of the environment, C(τ), describe what is happening thermally among environmental modes for all times after the interaction. The ﬁnal thing to note is that both the noise and the dissipation kernels depend on the spectral density of the environment (which in turn depends on what is termed the “normal frequency,” about which more will be said shortly). That both kernels depend on spectral density makes sense, since the spectral density is constructed to describe the dynamics of the environment in as much as we can know them, and this construction ought to include known classical eﬀect like those due to noise and dissipation. In turn, the dependency of the spectral density on a particular frequency ω is bounded by the weak-coupling limit: as coupling strength goes to zero, all environments interacting with a central system will assume dynamics similar to that of a set of harmonic oscillators.7 To construct the most general environment

7That all environments interacting with a central system can be treated like a set of harmonic oscillators in the weak-coupling limit was ﬁrst realized by Feynman and Vernon (1963). For a concise discussion of their argument, I refer the reader to Stamp (2006).

116 spectral density, ω takes the value of a normal, or unperturbed, simple harmonic oscillator: hence “normal frequency.” Recall from the previous chapter that the spectral density for an environment of independent harmonic oscillators is written

2 X ck J(ω) ≡ δ(ω − ωk) , (3.4) 2mk ωk k with k = 1, 2, 3..., indexing over individual environmental oscillators. In the most simpliﬁed situation we assume that the individual mass mk, frequency ωk and

2 damping constant ck diﬀer from each other only negligibly and so drop the index.

The delta function δ(ω − ωk) takes a value of 1 for any nth oscillator whose frequency ωn is equal to the normal frequency ω, and a value approaching 0 for oscillators whose frequencies diﬀer from the normal mode increasingly. In other words, the delta function assigns weights to various environmental modes assuring that as the spectral density approaches a smooth limit (and the sum becomes an integral) the value does not exceed one. When constructing models for particular environments, one considers three principle regimes of spectral densities, each assigned diﬀerent frequency values (indicated by the variable ω): the regimes are called ohmic, sub-ohmic and supra- ohmic. Ohmic spectral densities are the default regime, so considered due to the observational fact that oscillator environments exhibit nicely behaved linear dependence on constituent frequencies in this regime. The models described below for oscillator environments typically employ an ohmic spectral density, as the models tend to break down or become quite complicated in the sub-ohmic and supra-ohmic regimes. These latter two regimes correspond to nonlinear depen-

117 dence of the spectral density on frequency. This complication “trickles up” to the master equations describing such environments, and the models are less easily solved. More will be said about this parameter and the limitations on models it presents in the next section. As one increases the strength of the CS-environment coupling in one’s model, the spectral density can more simply be described (in the limit of well-localized, spin-1/2-mappable environmental modes with discrete degrees of freedom) in terms not of frequency but of energy. Indeed, in most spin-1/2 environment models the environment is considered to be in thermal equilibrium (a spin bath) whose spectral density can be deﬁned in terms of environmental self-energies ∆k (k = 1, 2, 3..., indexing over modes, or individual oscillators) thus:

X 2 J(∆) ≡ gk δ(∆ − ∆k) . (3.5) k Notice the formal similarity between the above equation and the original spectral density described in (3.4) with respect to the delta functions, and both having coupling constants. When this spectral density (or indeed any spectral density whose parameters have been altered to reﬂect the details of a given model’s environment) is used in place of (3.4), diﬀerent noise and dissipation kernels will result. This will in turn alter the construction of the master equation, the most general of which—the Born-Markov equation—can be expressed in terms of the

118 two kernels as follows:

d ρˆ (t) = − i[Hˆ , ρˆ (t)]− dt CS CS CS Z ∞ n ˆ ˆ ˆ ˆ o dτ ν(τ)[X, [X(−τ), ρˆCS(t)]] − iη(τ)[X, {X(−τ), ρˆCS(t)}] . 0 (3.6)

Here we use shorthand notation for commutators and anti-commutators: [A,ˆ Bˆ] ≡ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ AB − BA and {A, B} ≡ AB + BA, respectively.ρ ˆCS(t) is the reduced density matrix of the central system familiar to us from the previous chapter. Finally, Xˆ represents the position operator of the CS, which appears as both time dependent and time independent in the above equation. Now that we understand more clearly the effect of altering the spectral density according to a specific model, what of the coefficients mentioned in Schlosshauer’s mantra? There are four coefficients that help to characterize the master equation with different central systems: the frequency shift coefficient Ω,˜ the momentum damping γ, the normal diffusion term D and the anomalous diffusion term f. The first two of these coefficients depend explicitly on the noise kernel, and the latter two depend on the dissipation kernel. In this way, all four of the coefficients used for tailoring master equations for a given model are also dependent on the definition of the model’s environmental spectral density via the noise and dissipation kernels. All that remains to specify in our general master equation is the Hamiltonian. As I describe the various types of models below, I will begin by constructing

119 the total Hamiltonian and introducing certain simpliﬁcations (in accordance with those discussed in 2.3.2). Though I will not then give the complete master equation for all four canonical models, I will do so for the most important model for our purposes: the quantum Brownian motion model, which is the oscillator-oscillator model. I will give the full master equation, including a description of the form of the four coeﬃcients in this model, so that the idea is clear in most general cases of environments how the formalism plays a role in the constructing and testing of models.

3.4 Oscillator environments

As described above, the basic features of an oscillator environment are as follows. We assume that the environment is made up of a possibly infinite set of harmonic oscillators. As is often the case in spin-osc models, the environment is referred to as a bosonic field with various modes. This is due to the fact that bosons have symmetric wave functions, and as such multiple bosons can occupy the same energy state. This is an important feature of an oscillator environment as we wish to be able to model all the oscillators (or modes) in a similar state prior to measurement. One might alternatively think of the oscillator environment as a collection of bosons sitting in a harmonic potential well (that is, a potential well describing the possible dynamics of a harmonic oscillator). These ways of understanding the environment give rise to equivalent dynamical situations. Bosonic fields are mappable onto systems with extended or delocalized modes, as is the case for oscillator environments. Models with oscillator environments

120 will be applicable to extended mode environments like collections of photons, phonons and conduction electrons (Hines and Stamp 2008, p. 543). That this environment is spatially extended, or delocalized in the position basis, is not due to interaction among modes; the individual modes do not interact with one another in this model and only weakly interact with the CS. As discussed in chapter 2, this will allow us to assume when constructing the total Hamiltonian of a given oscillator-environment model, that regardless of the choice of CS model, the self- Hamiltonian of the environment will be negligible by comparison. The assumption of asymmetric influencing of the CS on the environment will become important in certain temperature and frequency regimes, as we shall see below. When the environmental modes occupy normal frequencies and the temperature is warm but not hot (that is, within the range of naturally occurring temperatures), then one usually assumes a coupling dependence that scales as 1/N, with N being the number of environmental oscillators. This limit is imposed for thermodynamic reasons and to guarantee that the oscillator environmental dynamics remain within the weak-coupling limit. Note that the oscillator environment functions as the weak coupling limit by decree; that is, we set the parameters of the environment in this case so that it matches the thermodynamic limit as the number of oscillators in the environment approaches infinity—remember that infinitely many environmental modes corresponds to a truly closed system. Despite the weakness of the typical CS-Env interaction on oscillator-environment models, the interaction with the CS is sufficient to introduce delocalization of environmental modes, which is why this sort of environment is sometimes considered

121 to be an approximate continuum, in that the environmental dynamics are best modeled as continuous (over all environmental modes) and not as discrete. In other words, the wave functions corresponding to each mode of the environment are increasingly spatially widespread. Since extended ﬁeld modes in oscillator environments imply the dynamical independence of individual modes from one another, we write the corresponding self-Hamiltonian as a sum of (possibly inﬁnitely many) noninteracting harmonic oscillators:

X 1 1 Hˆ = p2 + mω2q2 . (3.7) Env 2m k 2 k k Here, the particular oscillators interacting with the CS are indexed by k = 1, 2, 3..., and are assumed to occupy ground state prior to measurement (this corresponds to a relatively thermally “calm” environment). The indices have been dropped on the mass terms m and frequency ω because we assume for simplicity that all the environmental oscillators have negligibly different masses and frequencies. The variables p and q denote canonical coordinates (e.g., momentum and position) of the kth oscillator. Bosonic field models are appropriate especially for gaseous environments, radiation fields and other environments with temperatures that are warm and that are best described with continuous degrees of freedom. This makes oscillator environments the single most widely applied environment model for decoherence investigations and the model most interesting for those interested in the nature of systems “in the wild,” which will typically interact primarily with radiation

122 ﬁelds or gases. We now discuss two diﬀerent ways of modeling central systems immersed in environments such as these.

3.4.1 Osc-osc models: Quantum Brownian motion

The quantum Brownian motion model just is the most general oscillator- oscillator model, applied to situations analogous to classical Brownian motion: we model the quantum CS as a Brownian particle whose degrees of freedom are continuous and that is immersed in a warm environment composed of non(self- )interacting harmonic oscillators that weakly interact with the CS via random scattering events.8 To say that the CS is a Brownian system is to indicate that it can be approximately described dynamical as a spatial superposition of two coherent, maximally localized position states. The self-Hamiltonian of the CS can be written in the usual manner of single harmonic oscillators as

1 1 Hˆ = Pˆ2 + MΩ2Xˆ 2 , (3.8) CS 2M 2

where M is oscillator mass, Ω the frequency and variables Pˆ and Xˆ designating two canonical coordinate operators. The interaction Hamiltonian for this model

8Caldeira and Leggett were among the ﬁrst to develop a decoherence model for a quantum Brownian particle in a ﬁeld of weakly damped harmonic oscillators in Caldeira and Leggett (1983, 1985). In these papers, they use a particular method to construct their model developed by Feynman and Vernon already in the early 1960s (Feynman and Vernon, 1963).

123 then takes the initial form

2 X (~r − ~ck) Hˆ = γ q exp − , (3.9) int k l2 k where γ is the coupling constant for the interaction, qk is some canonical coordinate of the kth oscillator interacting with the CS, ~r represents the coordinate of the CS being monitored by the kth oscillator, ~ck is the spatial location of the kth (interacting) oscillator, and l is the range of the CS-Env interaction. Two notes on this interaction Hamiltonian are in order. First, the canonical coordinate qk of the kth interacting oscillator need not be position; one might, for instance, choose to measure in terms of an electromagnetic variable. Second, the exponential factor

9 in this equation represents a driving force fk(~r)(t) of the kth oscillator on the CS. One can see by examining the variables upon which (3.9) is dependent and by recalling the discussion of simplifying approximations for Hamiltonians in chapter 2 that when both the CS and the environment are described as harmonic oscillators in a potential well, the dynamics of the interaction Hamiltonian (for times after its initially unentangled state) are such that the set of mutually commuting operators are those of position. The general idea as explained in the previous chapter can now be made more speciﬁc: although the canonical coordinate ~r of the CS might represent (say) an electromagnetic variable instead of position, the CS will nevertheless be most easily measured—most robust—with respect to its position states because that is the variable whose operator commutes with (i.e., that “sees,” and thus responds to) individual environmental modes, who are mod-

9 ˆ P As such, equation (3.9) can be expressed in more compact form as Hint = k γ qk exp [fk(t)].

124 eled in terms of their position coordinates, ~c. Thus, in the oscillator-oscillator or quantum Brownian motion model, the environment is said to continuously monitor the position coordinate of the CS; when we take the partial trace over position coordinates of just the environment (considered a subsystem of the total entangled CS-Env system), we obtain complete statistical information in the position basis regarding measurements preformed on the CS. From the total Hamiltonian and using the equations of motion we derive the master equation for the reduced density matrix in the osc-osc model as follows:10

d ˆ 1 ˜ 2 ˆ 2 ˆ ˆ ρˆCS(t) = − i[HCS + MΩ X , ρˆCS(t)] − iγ[X, {P, ρˆCS(t)}] dt 2 (3.10) ˆ ˆ ˆ ˆ − D[X, [X, ρˆCS(t)] ] − f[X, [P, ρˆCS(t)] ] .

When this equation is broken down we see that it isn’t as complicated as it first appears. There are four separate terms, each attached to one of four coefficients— Ω˜ 2, γ, D and f—whose conceptual meaning can be stated straightforwardly (see below). All four coefficients in this master equation are integrals over time τ, from zero to infinity, describing all times after and including interaction between the CS and environment. The coefficients and the spectral densities are where the specifications are plugged in to the general master equation and how the four models are differentiated. Thus if one can understand the basic construction and assumptions involved in the most basic case of the osc-osc master equation (3.10),

10For a more detailed derivation, see Schlosshauer (2007, chapter 5) and Breuer and Petruc- cione (2002, chapter 3).

125 one has understood a majority of the conceptual aspects of all four canonical models. As we shall see in later sections, this master equation only varies slightly in form depending on the model. In the oscillator-oscillator model currently under consideration, the four coefficients from the master equation will take the following forms—all of which include periodic functions, in accordance with our expectation that Poincarérecurrence constrains evolution. The Ω˜ 2 coefficient is defined to be the frequency shift experienced by the CS reduced density operator (described by the familiar reduced density matrix) as it evolves, and it is dependent on the dissipation kernel η(τ) (and thereby dependent upon spectral density but independent of temperature). The γ in the above equation quantifies energy dissipation over time experience by the CS by characterizing momentum damping. Unsurprisingly, this dissipation- related term is also dependent upon the dissipation kernel (and, by extension, the spectral density, but not temperature). The D coefficient represents diffusion in momentum space and is dependent on the noise kernel, ν(τ). It is often referred to as the normal-diffusion coefficient in literature on quantum Brownian motion. The coefficient f, representing anomalous diffusion of the density matrix, is also dependent upon the noise kernel but is usually considered negligible compared to normal diffusion processes. The two diffusion coefficients, since they are described in terms of the noise kernel, are thereby (i) dependent on spectral density and (ii) dependent on the environment’s temperature. This makes sense if one thinks about it: diffusion is a physical process characterized by the thermal state of the environment, whereas momentum

126 damping (i.e., friction) and shifts in the frequencies of the oscillators will not be influenced by thermal activity of the environment. Together these four coefficients are what transform the general master equation for evolution of a CS’s reduced density matrix into a specific master equation described by one of the canonical models and in accordance with stipulated parameters. Since the master equation (3.10) is general, let us dissect it a bit more and investigate the four commutator terms before describing the oscillator-oscillator model specifically. 1 The commutator −i[Hˆ + MΩ˜ 2Xˆ 2, ρˆ (t)] describes the time-reversible uni- CS 2 CS tary dynamics of our central system qua harmonic oscillator, where the oscilla- p tor has a frequency of Ω2 + Ω˜ 2, and thus Ωthe˜ frequency shift coefficient— represents the natural frequency of the CS with a Lamb shift contribution. The Lamb shift is a rather complicated phenomenon; it is sufficient for our purposes to characterize it as the contribution to the dynamics made by vacuum fluctuations in the radiation field.11 ˆ ˆ The second commutator term, −iγ[X, {P, ρˆCS(t)} ], describes the momentum damping of the CS as its reduced density matrix evolves. Recall that the γ coefficient quantifies the amount of dissipation (as a function of time τ). This term as a whole describes damping of momentum as an effect of interaction with the environment and thus depends on spectral density (which, recall, characterizes the environment into which the momentum of the CS is dispersed). Momentum

11For more detail on the role of the Lamb shift in deriving master equations, see Breuer and Petruccione (2002, chapter 3); also see their chapter 12 for the (more signiﬁcant) role of the Lamb shift in open quantum electro-dynamical calculations.

127 damping, recall, is independent of the temperature of the environment. This will be important to keep in mind as we analyze the appropriateness of the oscillator- oscillator model in various situations. ˆ ˆ −D[X, [X, ρˆCS(t)] ] is the normal diﬀusion term, and it is here presented in a condensed form called Lindblad double commutator form. The term as a whole is crucial for understanding the process of decoherence, as it models the environmental monitoring of the CS’s position coordinate (with position operator Xˆ) over time. The spelling out of the double commutator form helps us see this more

0 2 0 straightforwardly: the term can be expanded as D(X − X ) ρˆCS(X,X , t). The first factor, D(X − X0)2, is just the rate of decoherence (recall that the coefficient D represents diffusion in phase space, and that the distance between positions X and X0 represents the coherent separation of the superposition state in position.12 The decoherence rate is multiplied by a density matrix dependent on both spatial

0 0 positions X and X as well as on time; this factor—ˆρCS(X,X , t)—describes the spatial localization that occurs in the model due to environmental scattering. The term as a whole (in Lindblad double commutator form) makes conceptual sense as the portion of the master equation describing the environment’s monitoring of the CS: it contains all the relevant information about the rate of decoherence due to environmental monitoring and scattering and the resulting localization of the CS in the position basis.13 ˆ ˆ The remaining double commutator term, −f[X, [P, ρˆCS(t)] ], accounts for anoma-

12Recall the description of coherence length in 2.3.3.3. 13Again, see 2.3 for the discussion of environmentally selected bases through decoherence.

128 lous diffusion (which hopefully is unsurprising news, given that the coefficient f is called the anomalous diffusion coefficient). This term can usually be dropped from the master equation, as it is often negligible compared to the normal diffusion term. Now that the preliminary, general description of the master equation has been explained, we can begin our discussion of the osc-osc model to which it belongs. One of the reasons for the importance of the quantum Brownian motion model is that, if the quantum coupling is linear (as is typically the case14), the master equation is exactly solvable and has the added feature of being local in time.15 Finding a model that is both exactly solvable and whose dynamics are governed by a time-local master equation is rare; hence the usefulness of the quantum Brownian motion model. One of the major assumptions necessary for the quantum Brownian motion model is that the CS experience negligible recoil from environmental scattering events. This is only a reasonable assumption when the relative masses of the CS and the environmental particles form a large ratio. This requirement is not always realistic, and thus some caution is necessary when referring to this particular model to make generalizations about the behavior of CS-environment decoherence

14Although Anglin, Paz, and Zurek (1997) have argued that the eﬀects on the CS of an environment whose intrinsic dynamics are nonlinear can still be approximately accommodated in this model. 15An equation that is local in time can be considered a snapshot of the system’s evolution at a particular time; it is as opposed to a global equation whose solutions depend on integrating over the history of the system’s evolution. The beneﬁt of a time-local equation, therefore, is that one need not know the entire evolutionary history of the system in order to proceed with one’s calculations.

129 more generally. For example, there are cases in which the force exerted on the CS by the environment during the scattering interaction will result in recoil of the CS signiﬁcant enough to suppress the eﬀects of decoherence. Anglin et al. (1997, p. 3) describe this phenomenon roughly as “the adiabatic [non-thermal] dragging of the high frequency bath [environment] degrees of freedom.” Such a case corresponds to that of supra-ohmic spectral density described above. What this means is that high energy modes of the environment that would ordinarily remain impervious to the dynamics of the CS instead become (non-thermally) coupled to it, and thus these higher-order environmental terms must be accounted for when constructing the interaction Hamiltonian. Obviously this greatly complicates things—in fact, to such an extent that supra-ohmic environments require their own model. The supra-ohmic environmental model is needed instead of the normal quantum Brownian motion model (with normal, ohmic frequencies) when the environment’s spectral density is high at large values of ω, resulting in nontrivial higher order terms in the description of the environmental coupling to the CS. Likewise, sub-ohmic environments require special models that account for some CS recoil from scattering events with the environment. In terms of naturally occurring environments, however, supra-ohmic as well as sub-ohmic environments are extremely rare, as spectral densities in these regimes correspond to environments of harmonic oscillators whose frequencies are well within the ultraviolet range (in the supra-ohmic case; extremely hot environments) or in the deep infrared (in sub-ohmic cases; extremely cold environments). This explains the applicability of the quantum Brownian motion model to environments with warm temperatures

130 (i.e., not cold enough so as to render inaccessible many of the environmental energy modes and call for sub-ohmic spectral densities, but not so hot as to require supra-ohmic densities). Thus the quantum Brownian motion model, widely employed due to its comparative formal accessibility and exact solvability, by that same turn is dependent upon somewhat strict assumptions, one of which is no recoil for the CS. Neverthe- less, the model has been considered successful in a number of ways. For example, a study of harmonic oscillator environments with quantum Brownian particle central systems in near-classical situations carried out by chemists at the University of Texas (Bittner and Rossky 1995) concluded that even when the model yields relatively long quantum decoherence time scales (corresponding to cases of low temperature and also weak coupling environments), the effects of decoherence must be taken into account “in order to make realistic predictions of condensed phase phenomena” (ibid., p. 8142). Myatt et al. (2000) contains the results of three experiments on trapped ions in harmonic oscillator reservoirs. In particular, the authors simulated the decoherence of a charged atom (ion) when entangled to a hot reservoir (modeled as a collection of harmonic oscillators) by applying “noisy” voltages to ion-trap elec- trodes in two experiments, while in a third they simulated a zero-temperature reservoir using laser cooling to damp the ion’s motion. They conclude (ibid., p. 273): “The cases considered demonstrate a quadratic dependence of the rate of decoherence on the size of the superpositions, demonstrating the difficulty in generating truly macroscopic superpositions, such as that of ‘Schrödinger’scat.’”

131 Exactly as we expect, given the rates generated by the osc-osc model. Many other such examples of experimental success could be given, but let us move now to a particular characterization of the osc-osc model via Wigner functions.

3.4.2 The Wigner approach to osc-osc models

In the previous chapter I introduced the Wigner approach as an alternative method to the master equation derivation for models of decoherence. The Wigner approach, unsurprisingly, arrives at a slightly diﬀerent (but perhaps more intu- itive) picture of the time evolution of a quantum system in an environment, and so we continue our investigation of the quantum Brownian motion model under the Wigner representation. Consider the CS to be representable as a pair of Gaussian peaks separated in phase space (where the two horizontal axes are position and momentum, with the vertical axis representing phase or probability amplitude of the wave functions) with smaller peaks between them, representing phase relations in superpositions of the two major peaks. Just such a representation is applicable to cases involving quantum Brownian central systems, or systems experimentally referred to as Schr¨odingercat states.16 If this representation of the phase space of our CS is then evolved forward in time while allowing it to interact with a weakly coupled

16In creating Schrödingercat states, the two major peaks correspond to two different coherent states that are approximately orthogonal (or opposite in phase) with respect to one another. The middle, smaller peaks then represent phase relations between all linear superpositions of these two maximally coherent states. For a relatively gentle introduction to experimental work on Schrödingercat states, I refer the reader to Raimond and Haroche (2005).

132 harmonic oscillator environment, the effect is that the smaller peaks representing phase relations (or interference terms between the two maximally coherent states) are rapidly damped, while the component state Gaussians remain more or less intact (there is some small rotational motion, but it is trivial compared to the damping of the interference terms). What we see in this three-dimensional map is what appears to be two peaks that are completely isolated from one another in phase space, though we know the interference terms still exist (though unobservable at any reasonable scale of representation, say the size of a computer screen or published page). But of course, we expected this damping of the phase relations via interaction with the environment. What else can this method demonstrate? Mapping the CS in its phase space using the Wigner equation (hence “the Wigner representation”) presents the data in such a way as to make certain comparative analysis quite clear. For instance, when modeling quantum Brownian, it is obvious that the CS becomes delocalized more quickly in the position basis than in the momentum basis. Decoherence of the interference terms still happens in both position and momentum at a rate much greater than the rate of normal, thermal relaxation of the CS due to its presence in some environment, as we expect. But being able to notice and then quantify the different time scale of delocalization in the position versus the momentum axes is new. However, we realize that we can explain this feature of the model based on our understanding of the Hamiltonian’s construction and therefore of the strength with which the environment monitors and subsequently couples to different degrees of

133 freedom. We already know that in the model of quantum Brownian motion, the mutually commuting degree of freedom between system and environment is that of position; hence, we expect the interaction to most deeply aﬀect the CS with respect to its position. Again, the interaction Hamiltonian of the osc-osc model is dependent upon position, resulting in direct environmental monitoring of the CS in the position basis. This in turn leads to rapidly spatially decohered states of the CS. The momentum, on the other hand, is not a component in the interaction Hamiltonian, which explains the slower rate of decoherence in this basis. The environment no longer couples directly to the CS in a way that causes rapid decoherence or in a way that allows the environment to monitor the CS. Instead, decoherence in momentum superpositions occurs indirectly, via the environment’s interaction in position. One of the crucial outcomes of the Wigner picture of quantum Brownian motion is that fully decohered central systems, or the “environment superselected pointer states” resulting from the CS’s interaction with the environment are mapped as minimum uncertainty Gaussian wave packets—that is, as well localized in both momentum and position as is physically possible (given the constraint of the uncertainty relations). These maximally coherent, approximately deﬁnite states are considered quantum analogues to points in a classical phase space, whence quasi-Newtonian trajectories can be recovered.17 However, one must never forget that this is merely an analogy, an idealization that treats entire Gaussian

17What it means to recover quasi-classical or quasi-Newtonian trajectories from wave functions in phase space simply means that the center of mass of the Gaussian can be said to follow an approximately classical trajectory in certain limits. This limit, associated with Ehrenfest’s theorem, will be treated in more depth in the following chapter.

134 distributions as single spacetime points. The CS remains a system whose position and momentum cannot be described deﬁnitely—and moreover, whose interference terms are still existent though damped. There are no such things as points, full stop. At the quantum level, the best approximations to classical particles are still Gaussian packets with imperfect localization and with phase relations (albeit delocalized ones) indicating that, ontologically, what cannot be said is that these two peaks are in any way representative of classical entities. The Wigner representation makes such lessons abundantly and visually clear.18

3.4.3 Assumptions in the osc-osc model

Recall that, so far, we have focused on the quantum Brownian motion model with ohmic environments, which correspond to relatively weak coupling/damping; we have also focus on moderate temperature regimes (corresponding to typical room temperatures). What happens with the model when we increase interaction strength? In other words, what happens when the harmonic oscillator environment is not weakly damped (as in the weakly interacting case) but is instead modeled with increasingly strong damping eﬀects on the CS? In addition, how well does this model apply in the temperature limits? In Anglin et al. (1997), Anglin, Paz and Zurek argue that the quantum Brow- nian motion model (or more generally, a continuous CS linearly coupled to a ﬁeld of bosonic modes) is too toyish, and as such many of the phenomenological gen-

18To see animations of decoherence in the Wigner representation, I refer the reader to the following: http://www.youtube.com/watch?v=Kb_6AwpPusA and http://www.youtube.com/ watch?v=fBHsQIsUY_s&NR=1.

135 eralizations drawn from theoretical work on the model should be considered with skepticism. Their perception of the model as too toyish is based on the construction of the model using various assumptions, such as no recoil of the CS and linear CS-Env coupling. Both of these assumptions are appropriate for normal temperature regimes but break down in low-temperature (on the order of micro- to milli-degrees K or less) environments and extremely high-temperature ones (significantly above room temperature, which is approximately 300 K). It is widely agreed that the quantum Brownian motion model is not easily applicable, or ill suited, for such temperature regimes. However, we note that if our concern is to understand the nature of decoherence in naturally occurring systems, the sorts of temperatures one need consider fall almost entirely within the regime of temperatures for which the osc-osc model is successful. More specifically, low-temperature environments such as those modeled well by spin baths (to be described presently) are, as mentioned, approximately zero K or slightly above. Zero degrees Kelvin corresponds to -459.67 degrees Fahrenheit. No such temperature is ever measured on earth. Things are a bit trickier in the supra-ohmic regime, as has been briefly discussed already. Anglin et al. do agree that the osc-osc model adapts well to high temperatures, and it is only at extremely high temperatures that the decoherence rate becomes significantly different. They conclude generally that the system becomes saturated by decoherence in such a temperature regime and that the rate of saturation increases the greater the initial distance between the component states. They write simply, “decoherence saturates at large distances” (ibid., p. 4049), and

136 as a result the rate of decoherence plateaus instead of increasing linearly. What this means in physical terms is that as one moves from the microscopic regime (in which the distance in phase space between component states of the model CS is correspondingly small) to macroscopic regimes (with great distances in phase space between component states of the CS, corresponding to “classical” situations), the rate of decoherence does not continue to rise but levels out. Again, at what level the decoherence rate saturates depends on the particular system and environment under consideration, but the main point of the authors is that a consistently growing rate derived from microscopic models cannot be assumed for all types of models, including more realistic ones. This in turn aﬀects whether one can use decoherence to neatly explain the transition from quantum to classical scales; Anglin and coauthors argue that the model dependence of the decoherence rate renders this task of explaining the quantum-to-classical transition much less straightforward than usually believed. Perhaps the lesson to take away from the critiques presented by Anglin et al. is not their worry about the toyishness of the quantum Brownian motion model (which has seen considerable empirical veriﬁcation in the years since this paper was published), but rather their conviction that certain general phenomenological lessons drawn from these highly idealized models have nevertheless become standard lore in the decoherence literature (ibid., pp. 4041–4042). However, the content of the so-called “phenomenological lore” all pertains to the rate of decoherence. The authors describe these unwarranted generalizations as follows:

From the bulk of previous theoretical studies of decoherence, one might

137 be tempted to deduce three significant principles concerning the rate of decoherence: one can define a simple decoherence time scale which is valid at least for linear systems at high temperature; the rate of decoherence of classically impossible “Schrödinger’sCat” states is always set by the fastest time scales present; and the rate of decoherence increases with the square of the distance between the two branches of such Cat states. (Ibid., p. 4041)

The authors go on to concede that these claims have indeed been shown correct in the early experimental work of Brune et al. (1996), but they further point out that this does not guarantee the truth of such claims for all future experimental work (ibid., pp. 4041–4042). While their point is well taken—and indeed, their primary aim is in keeping with the point of this chapter to warn readers that attention to the speciﬁcs of each model is crucial for understanding the subtle phenomenology of decoherence—the bits of generalization they are worried about in particular have to do with speciﬁc decoherence rate calculations. Three points can be made with respect to this worry. First, even though the numbers for decoherence rates in early theoretical work were somewhat question- able, it was still understood by all parties involved that the rate—whatever the actual number be—is still many orders of magnitude greater than the applicable rates of thermal dissipation or system relaxation, demonstrating conclusively that (i) decoherence is independent of any classical process, and is uniquely quantum mechanical, and (ii) decoherence occurs too rapidly to be observed in a prodigious number of cases, and in practically all cases of uncontrolled entanglement, as in nature. Second, there is a great deal of experimental work that has been carried out since Anglin et al. published their cautionary paper in 1997 that has helped

138 to refine calculations of decoherence rates and confirm its dependence on model construction, while at the same time empirically verifying the general lessons of importance for present purposes, to wit, my points (i) and (ii) above.19 Third, to the best of my knowledge no similar worries regarding ill-founded generalizations generated by the theoretical community have been expressed since the mid-1990s. All of these points confirm the dependency of the available understanding of decoherence on the given model, while at the same time generally affirming the strength of the very same models, as evidenced by the watershed of experimental work on decoherence even within the last decade.

3.4.4 Spin-osc models

The present model is usually referred to as the spin-boson model, because the environment is assumed to take certain values for its frequency and temperature parameters such that its dynamics are analogous to delocalized modes of a bosonic ﬁeld. In particular, the behavior of a gas environment is best modeled as a bosonic ﬁeld in this manner (in that the independent molecules of the gas behave like bosons in a harmonic-oscillator potential), thus making this model (appropriate for a fermion like an electron in a gas) widely applicable. Leggett et al.’s 1987

19Several research groups (in particular, groups working on developing Schrödingercat states using large fullerene molecules associated with the Institute of Quantum Optics and Quantum Information, or IQOQI, and groups in Paris who work with quantum cavities) have confirmed the osc-osc model as widely applicable and accurate. For some results out of the IQOQI collaborations, see Arndt et al. (1999), Arndt et al. (2002), Brezger et al. (2002), Hackermüller et al. (2003), Hackermülleret al. (2004) and Hornberger et al. (2003). For results from the Paris collaborations, see Bertet et al. (2002), Brune et al. (1992), Brune et al. (1996), Davidovich et al. (1996), Raimond et al. (2001) and Raimond and Haroche (2005). Also see Bernu et al. (2008) and Monroe et al. (1996).

139 paper is the seminal work investigating spin-boson models theoretically; much of what follows is based upon their work.20 Spin-boson models involve a CS whose dynamics can be mapped onto those of a spin-1/2 particle, or more generally, those of any two-state system with a potential containing two minima (called a double-well potential). In other words, we use a spin model for the CS when its behavior can be described as a particle sitting in a double-well potential, where one well (or potential energy minimum) is separated from the other by a barrier of sufficient height such that the system is effectively “either-or”—the energy it would take the particle to tunnel from one well into the other renders this possible state so miniscule (owing to the comparative height of the barrier between the energy wells) that the system’s dynamics are effectively discrete and can really only be considered in either one energy state or the other.21 One extremely useful feature of the spin-boson model is that it is exactly solvable. However, this solvability comes at quite a cost in terms of complexity. Another beneficial feature of the spin-boson model is that in its simplified form (which amounts to neglecting the tunneling term when constructing the CS self-Hamiltonian) this model gives decoherence processes in the absence (or near absence) of dissipation. This is important for establishing the independence of these physically disparate processes. The major paper motivating and constructing a Hamiltonian for the spin-boson model and theoretically demonstrating the

20For more details on this model, I refer the reader to a series of papers by Caldeira, Leggett and Chakravarty: Caldeira and Leggett (1983, 1985), Chakravarty (1982) and Chakravarty and Leggett (1984). 21A truly discrete two-state system would be one in which the barrier was inﬁnitely high.

140 independence of decoherence from dissipation is the aforementioned Leggett et al. (1987) paper. The self-Hamiltonian for the environment of the spin-boson model is just that of the ﬁeld of harmonic oscillators, given by (3.7). For the CS, one can choose to ignore the tunneling term as mentioned; I have nevertheless given below the CS self-Hamiltonian with the tunneling term for completeness. Adding in the tunneling term, while it does not alter the construction of the interaction Hamil- tonian, does generate intrinsic dynamics within the CS (i.e., the entanglement of magnetic degrees of freedom with continuous degree of freedom like position, momentum, etc.), an important consideration that will change the various parameters in our master equation. The intrinsic dynamics of a spin-1/2 central system with nontrivial tunneling possibility is represented as follows (following Leggett et al. 1987):

1 1 Hˆ = − ∆ σˆ + σˆ . (3.11) CS 2 ~ 0 x 2 z

Both of theσ ˆi matrices represent Pauli spin matrices, each indexed over all three spatial dimensions. We choose to represent the system in a basis that corresponds neatly to our double-well modeling of the CS. In other words, we choose a basis in which the possible eigenvalues ofσ ˆz will be |0i and |1i, values that correspond to the system occupying either the left well or the right one. The term is called the bias or the detuning parameter, and it describes the diﬀerence in energy between the ground states of the two potential wells. In the limiting case of well symmetry (that is, when the ground state energies of both wells are equal and the bias

141 approaches zero), the tunneling term dominates the intrinsic dynamics, and the 1 resultant dynamics of the CS are simpliﬁed by dropping the σˆ term. 2 z The initially uncorrelated interaction Hamiltonian for the spin-boson model (irrespective of whether we’ve included a tunneling term in the self-Hamiltonian of the CS) is constructed as

ˆ X Hint =σ ˆz ⊗ ckqˆk . (3.12) k

Here the Pauli matrixσ ˆz deﬁning our discrete spin basis for the CS is linearly

coupled to some coordinateq ˆk of the kth interacting environmental harmonic oscillator. When we analyze the total Hamiltonian for the spin-osc model, we expect there will be no energy exchange—that is, we expect no thermal dissipation to result from system-environment interaction. We expect this because the environment’s self-Hamiltonian does not commute with the self-Hamiltonian of the CS or the interaction Hamiltonian as the latter two contain spin vectors while the environment does not (remember our cute saying from chapter 2, “no commuting, no communicating”). This is extremely important, as it indicates that any eﬀect the CS has on the environment upon interaction cannot be due to thermal exchange—i.e., dissipation—and that, instead, the dynamics described by the total Hamiltonian are due solely to the process of decoherence. The Born-Markov master equation for the spin-boson model is derived in similar fashion to that of the quantum Brownian motion. As such, it depends on many of the same parameters already discussed, including the noise and dissipation kernels and the spectral density of the environment. With respect to the spectral

142 density, many of the same dependencies on the frequency regime best characterizing the environment (ohmic, sub-ohmic or supra-ohmic) must be carefully considered. In particular, Leggett and coauthors stress that in order to construct a relatively general total Hamiltonian for this model, the domain of validity must be restricted to within light or moderate damping environments. Heavy damping alters the situation (cf. their appendix A). More concretely, all three of the following parameters—~∆ (the matrix tunneling element), kBT (associated thermodynamical constant) and (the detuning parameter)—must have values very small compared to the height of the barrier in the double-well potential. This makes sense when one wants to safely approximate the dynamics of the CS as a particle with truly discrete degrees of freedom, when most of the systems modeled this way are in fact systems with continuous degrees of freedom that just happen to have potentials with two minima and therefore can map easily onto the dynamics of a truly discrete, two-state system. In other words: the approximation of discreteness breaks down when the energy required for transitioning between potential energy minima is realizable. However, these parameter requirements are met in many cases. As Leggett et al. state in their 1987 paper, the condition of having the above three parameters take values much less than that of the barrier between the two potential minima is “fulﬁlled for the vast majority of cases of interest in physics” (ibid., p. 8). Cases wherein this three-pronged condition is not met include certain systems employed in chemistry (for example, see Leggett and Garg 1985); the question of what other

143 model might best fit such cases is as yet an open question. A few further assumptions specific to the spin-boson model include generaliz- ability to Hilbert spaces of higher dimension: in mapping our CS onto a double- well potential we necessarily restrict ourselves to a two-dimensional Hilbert space (corresponding to the two main vector states available to the CS). This particular question of whether or not the model can successfully be applied to Hilbert spaces of dimension three or greater is answered optimistically by Leggett and coauthors, who argue that their results will still obtain. A brief catalogue of some of the experimental work done using spin-osc models includes investigation of macroscopic quantum coherence, as well as the exploding new field of quantum computing. The successes of this model of decoherence can be found in Hines and Stamp (2008), Landau (1996), Raimond and Haroche (2005) and Richter (2000).

3.5 Spin-1/2 environments

Spin-1/2 environments—or more generally, environments whose degrees of freedom take discrete (or approximately discrete) values—are extremely useful for highly engineered experimental situations as they best model phenomena interacting at extremely low temperatures (that is, anywhere from a few milli-K above zero to the sometimes sub-zero K temperatures used for superconducting and quantum computing). This model of the environment, because it functions well within the low-temperature regime, does not have wide applicability in nature; therefore, my discussion of spin-environment models will be relatively brief

144 compared to that of oscillator-environments.22 These environments are typically referred to as spin baths, though the former description is more general. Spin baths, recall, are environments at thermal equilibrium. Since such models are applicable in extremely low temperature regimes, it is not unreasonable to assume that the very cold and approximately discrete, highly localized environmental modes will, once reaching thermal equilibrium, remain there or thereabouts. Very cold environments entail limited thermal interaction. One might begin to understand why low-temperature environments are best mapped onto discrete, well-localized, approximately (spin) bath dynamics. For one, it is in the low-temperature regime that a central system interacts most strongly with well-localized environmental modes (as opposed to the delocalized, approximate-continuum generated by a harmonic oscillator environment). In other words, the wave functions associated with with very cold environmental modes are confined to small spatial regions and are describable in a finite- dimensional Hilbert space with a finite energy cutoff. These modes, being highly localized, are therefore best characterized as an environment with discrete energy states. There is debate as to whether spin environments should exhibit strong scaling dependence (as the number of environmental spins approaches infinity), as is assumed in the oscillator-environment case to maintain reasonable thermal limits.

22Remember that absolute zero, or zero Kelvin, is equivalent to -459.67 degrees Fahrenheit. To give the reader an idea of just how cold this is, the lowest temperature ever measured on earth (at, unsurprisingly, an Antarctic scientiﬁc outpost) was 184 K, or -128.6 degrees Fahrenheit (Vos 2011).

145 In some instances, a spin-coupling scaling on the order of 1/N is nevertheless assumed for spin bath modeling. The theoretical framework for both canonical spin-environment models is derived mainly from results of boson field environments, which is natural considering the fact that such environments are the weak-coupling limit—thus we want our spin environments to converge onto oscillator environments in this limit. In addition, the mathematics for oscillator environments is easier and, due to the wide applicability of such models, also much better understood. The details of how one maps a spin environment onto a harmonic oscillator environment is less important for our purposes here than to understand the assumptions it takes to allow such mapping. We already know in large part that the mappability of spin environments onto a bosonic field is due to the analogous construction of the spectral densities for both environments. Schlosshauer (2007), among others, has shown that when one considers the high energy limit for a spin environment, the assumptions that allowed us to describe the environmental modes as discrete and well localized are no longer applicable; though the limit is not a smooth one, we are not being unreasonable in assuming that, as the field modes become increasingly delocalized, the spectral density will approximate that of a bosonic field in ground states. Taking the energy or temperature limit to the other extreme gives us the spectral density described above in equation (3.5), which in turn characterizes the noise and dissipation kernels and therefore master equations applicable to spin environments. In fact, the master equations for both the spin-spin model and

146 the osc-spin model described below just are the Born-Markov master equation for quantum Brownian motion, but with the new spectral densities (and therefore kernels and coeﬃcients) and, of course, with the appropriate Hamiltonians, which will be described shortly. The intrinsic dynamics of a spin-1/2 environment (assuming thermal equilibrium, i.e., a spin bath) are described by the following self-Hamiltonian:

X 1 Hˆ = ∆ σˆ(i) , (3.13) Env 2 i x i where ∆i represents the tunneling matrix element, similar to the case of the spin- 1/2 central system for the spin-osc model. One can also see by comparison to the self-Hamiltonian of the spin-1/2 CS that for a spin bath we have assumed the detuning parameter is set to zero; this is part of the assumption that the environment is in a thermal equilibrium, and the ground states of each eigenstate of an environmental particle are thus symmetric. Castagnino et al. (2010) have argued that spin bath models do not support the assumption of widespread entanglement. However, their argument is a case of generalizing based on a specific model: while the considerations of these authors may be true in highly idealized laboratory settings involving extremely complicated engineering (such as we know to be the case for application of spin baths, owing to their validity only in extreme low temperatures), this focus severely limits to specific models the sorts of generalizations the authors wish to make about all decoherence processes regardless of system-environment details. Now let us consider specific spin environment models, starting with the most

147 common of the two: decoherence in spin-spin situations.

3.5.1 Spin-spin models

Upon reﬂection, it is understandable why a spin-spin model should be a highly useful one, as it will be the model most appropriate for capturing the dynamics of physical systems in which the environment is low in temperature and the CS is a fermion. This model describes well instances of cold (laboratory engineered) environments with strong CS-Env coupling. This model, like that of quantum Brownian motion, can be solved exactly, given the adoption of reasonable values for the relevant parameters (cf. Cucchietti et al. (2005) and Dobrovitski et al. (2003)). As in the spin-boson model, the self-Hamiltonian of the CS is described by (3.11), and the interaction Hamiltonian initially takes the uncorrelated form

ˆ X (i) Hint =σ ˆz ⊗ giσˆz . (3.14) i

The constant gi represents coupling strength between the CS and the ith environmental spin. The entanglement that will arise from this interaction takes place between the general coordinate of the CS (in this case, the spin-z component of the CS whose eigenvalues are either “spin up” or “spin down”) and each individual spin of the environment.

When ∆0 is very small compared to the energy distribution in the environment, the CS has weak intrinsic dynamics. This is the precise physical situation found in cases of superconductivity, where we have a qubit coupled strongly to a very cold

148 environment. As mentioned in the introductory section, though this particular low-temperature spin-1/2 model is extraordinarily useful for laboratory situations, these sorts of system-environment interactions are nonexistent in nature. This will be interesting to keep in mind with respect to our ontological question. The experimental literature for spin-spin models is vast, especially given the recent experimental focus on quantum information via quantum computation and superconducting—both of these experimental regimes make use of spin bath models. I will give but a few examples of current research being done. Already in 1981, Voss and Webb studied macroscopic quantum tunneling in a Josephson junction— an apparatus central to SQUIDS (superconducting quantum interference devices) and superconduction more generally (see Voss and Webb 1981). More recently, Yu et al. (2002) have revisited earlier investigations of Josephson junctions in diﬀerent frequency regimes. Among their results was a new lower-limit for decoherence times in the macroscopic superposition states obtained with Josephson junctions. What this indicates is that by using the spin-spin decoherence model to engineer the initial quantum state in the Josephson junction more cleverly, these researchers were able to maintain a macroscopic superposition state (i.e., they were able to maintain coherence among phase relations in the superposition) for longer time periods, resulting in a longer time scale for coherence. This has significant implications regarding the possibilities of constructing quantum computers. Yu and coauthors were furthermore able to experimentally conﬁrm decoherence rates in this frequency regime expected from their earlier research (cf. Han et al. 2001).

149 SQUIDS, superconducting and the modeling of Josephson junctions in particular are large areas of interest in recent experimental physics, and in all these situations spin-spin (and sometimes spin-Boson) decoherence models are key to interpreting system dynamics and creating better apparatuses.23

3.5.2 Osc-spin models

Harmonic oscillator central systems entangled with spin baths are the rarest of the four canonical models. They are applicable in new areas of research like the construction of quantum-electromechanical systems (referred to as QEMS) and in constructing very small ion traps, on the order of micrometers (Schlosshauer 2007, pp. 282–288). The total Hamiltonian for such a model is constructed (as well the reader can guess) by using the self-Hamiltonian of the harmonic oscillator in (3.8) and the self- Hamiltonian of a spin bath given by (3.13). The interaction Hamiltonian must of course take a form that will, after interaction, reﬂect the fact that entanglement in this case occurs most eﬀectively between the position coordinate of the CS oscillator and each environmental spin, as follows:

ˆ ˆ X (i) Hint = X ⊗ giσˆz . (3.15) i A complete derivation and analysis of the Born-Markov master equation for osc-spin models has been carried out in Schlosshauer et al. 2008. These authors

23To get a more general idea regarding the role of this model in testing decoherence, I refer the reader to several excellent collections on decoherence: Blanchard et al. (2000), Duplantier et al. (2007), Giulini et al. (1996) and Namiki et al. (1997).

150 state their motivation as stemming from recent interest in QEMS and micrometer ion traps, and find that the osc-spin model exhibits quite different temperature dependencies compared to the quantum Brownian motion (or other osc-osc) models. Specifically, the authors found that the disparity in temperature dependencies between osc-osc models and the osc-spin model with which they are concerned can be explained by the extreme rapidity with which the environment (of many two-state systems) saturates due to decoherence (ibid., p. 022111-7). Recent experimental work based in part on the findings of Schlosshauer et al. (2008) include Fa-Qiang et al. (2009), Jacobs (2009), Venkatesan et al. (2010), Wei-Ci et al. (2010) and Wesenberg et al. (2011). A good introductory paper on QEMS that involves discussion of decoherence as a physical process tested in various temperature regimes, see Blencowe (2004). The number of experimental works published since Blencowe and even in years since Schlosshauer et al. published their 2008 paper on osc-spin models has experienced prodigious growth. The nature of these experiments is along the lines of what has been seen thus far: using the general master equation for a particular model, various researchers set parameters in accordance with their experimental setup and find results in good agreement with predictions. But what of decoherence models being experimentally tested for nonmechanical systems? In other words, do the four canonical models (and variations thereon) remain plausible, useful, fecund and so forth when applied to organic systems?

151 3.6 Decoherence models in biological & chemical systems

In 2.3 I mentioned in passing recent theoretical work in biology done by Briegel and his team at Innsbruck. More speciﬁcally, Briegel and his team are interested in studying the migratory sensitivities of birds, and how decoherence might explain the particular physical orientation (in states that should be, but are apparently not, entangled) of those molecules in the animals’ brains that corresponds to polarization and therefore direction with respect to the earth’s magnetic orientation.24 In this research (whose theoretical foundations were laid in Briegel and Popescu 2009), the investigators apply speciﬁc toy models describing entanglement, decoherence and re-entanglement in biological systems like proteins. Briegel and Popescu concede that their models are so toyish as to be nonexistent in the real world; nevertheless, they are sure that the models demonstrate quite forcibly that the basic dynamical idea of entanglement, decoherence and then re-entanglement as the model predicts in open, driven systems (which describes nearly all biological systems at some scale) is real. They write,

[A]lthough our speciﬁc toy models may well have very little direct relevance, we are conﬁdent that the processes we described (entanglement pumping, resetting, etc.) are to be found in a way or another; the same applies to the idea of molecular cooling. Ultimately, the power of biological evolution coupled with the fact that biological organisms are open, driven systems, may open the door for many unexpected quantum phenomena. (Ibid., p. 18)

Work of both the theoretical and experimental sort has blossomed on the basis of these works in just a few years, and it would be impossible to describe the

24Also see Cai et al. (2010), Gauger et al. (2011) and Hartmann et al. (2006).

152 work here. The point to be taken away is that the inclusion of decoherence in constructing increasingly refined models of biological systems has generated entirely new paths for investigation in recent years, and much of the early work that started this trend is based upon adoption of the canonical decoherence models, with parameters tweaked to describe the biological system of interest.25 More recently, models incorporating decoherence for studying the structure and dynamics of DNA have been carried out and have been experimentally confirmed. Based on the theoretical groundwork in Senthilkumar et al. (2005) and Zilly et al. (2009), Zilly, Ujsághy, and Wolf (2010) conclude good agreement with such models. They report that though the models incorporating decoherence were developed with ohmic spectral densities and linearity of the CS, the authors successfully adapted the model to experimental values for conductance in a DNA double helix. In particular, Zilly et al. devise a model in which decoherence occurs in randomly selected subregions of the molecular sample instead of continuously throughout. Nevertheless, the authors assume that entanglement between a DNA base pair to its environment (including the remainder of the DNA helix) is sufficient to introduce decoherence to such a degree that, as they write, “coherence is completely destroyed on these bases pairs” (ibid., p. 3). This assumption allowed the authors to assign a local energy distribution function to describe the base pair as opposed to a nonlocal one (because decoherence has effectively destroyed the nonlocal dynamics). As a consequence of these

25The reader might ﬁnd of interest in particular the following recent biological papers (both theoretical and experimental): Asadian et al. (2010), Cheng and Fleming (2009), Collini et al. (2010), Engel et al. (2007), van Grondelle and Novoderezhkin (2006), Hwang and Rossky (2004), Jang et al. (2007), Lee et al. (2007), Mohseni et al. (2008) and Wilde et al. (2010).

153 decoherence-based assumptions, they could calculate conduction between different segments of the DNA strand. Their experimental results were in good agreement with this model, thereby lending considerable weight to such assumptions. Thus the flexibility of decoherence models manifests itself in important molecular investigations, as well as affirming the crucial adoption of models sensitive to decoherence in understanding and explaining dynamics at various scales. Another arena of expansive growth regarding decoherence research and inclusion into modeling is again in molecular chemistry, this time with respect to condensed phases and non-adiabatic systems (in other words, time irreversible dynamics). One of the earliest groups using decoherence for understanding and modeling systems such as these is research from the Rossky Group at the Uni- versity of Texas, Austin. A few examples of the works generated by this group are Bittner and Rossky (1995, 1997), Del Buono et al. (1994), Motakabbir et al. (1992), Prezhdo and Rossky (1998), Rossky (1998) and Schwartz et al. (1996), to name but a few.

3.7 Conclusion

Many claims are made in the literature of both physics and philosophy that consider the decoherence models too general and therefore discount their plasticity for successfully describing incredibly disparate situations. Our investigations in this chapter have shown such generalizations to be unsubstantiated.26

26Undue warnings about the toyishness of decoherence models can be found in ?, Lombardi et al. (ming) and Stamp (2006).

154 At the same time, comparably numerous claims are being generated from physics and philosophy sources that base their arguments on highly engineered situations, and from decoherence as studied in these particular cases make general conclusions about decoherence in diverse, less-controlled settings. This, too, is unwarranted given the theoretical and experimental work to date: what seems to really be the case is that exploring model sensitivity by exploring limiting cases of various parameters (e.g., extreme temperature regimes and coupling strength) indicates that we have a much better idea of decoherence and its effects than such general claims originating in studies of highly specific, highly controlled models of decoherence present; as such, these claims ought not be considered a source of significant worry. If the applicability of various decoherence models to systems in the wild is still subject to some doubt, I refer the reader to the preceding section describing fascinating current research wherein investigators are applying decoherence models to biological systems. This speaks to the power of decoherence models in a way that, in my opinion, trumps the strength of the models as tested solely on physical systems. For one, it is much more difficult to refute success of models as applied to chemical or biological systems, which (at least when compared to typical physics models) can in no way be considered toy, for they are the very definition of systems in the wild. Decoherence research in such arenas is only just beginning and promises to yield an abundance of fruit. The confidence with which scientists from institutes and universities the world over are taking up models of decoherence to aid in their various queries is yet another merit of the models: even

155 as we know the phenomena of decoherence to be far from perfectly understood and likewise much progress can be made with respect to these original models, they are already exhibiting such a high degree of fecundity in ﬁelds outside of physics that passing the models oﬀ too quickly as too idealized to teach us about the world is a mistake. The next chapter will begin to unravel certain philosophical claims being made, using as central the lessons we have gained so far from our study of current theoretical and experimental work on decoherence. In particular, chapter 4 argues for the nontological thesis—that there is no metaphysically robust sense in which “classicality” refers.

156 CHAPTER 4

WHENCE “CLASSICALITY”?

It is now increasingly being realized that the conventional treatments of the classical limit are flawed for a simple reason: they do not represent any realistic situation. The assumption of a closed macroscopic system (and thereby the applicability of the Schrödingerequation) is by no means justified in the situations which we find in our present universe. Objects we usually call “macroscopic” are interacting with their natural environment in such a strong manner that they cannot even approximately be considered as isolated, even under extreme conditions. This observation opens up a new approach to the understanding of classical properties within the framework of quantum theory.

Joos, Introduction (p. 2) to Decoherence and the Appearance of a Classical World in Quantum Theory

4.1 Introduction to Part II of this thesis

The purpose of the second part of this thesis is to explore the philosophical consequences of decoherence processes, speciﬁcally regarding the question of interlevel relations. Recall the nontological thesis introduced in chapter 1, stated as “there exists no classical world.” Now that the appropriate knowledge of the

157 physical processes is in place, we can begin to examine the nontological thesis and evaluate whether or not one can define “classicality” in a way that cuts metaphysical ice. This is the goal of the present chapter. Chapter 5 then examines a more general version of the nontological thesis which says that, ontologically, there exist no levels tout court. If there are no levels yet quantum theory remains universally valid, one must conclude that there is only a quantum world. But the nontological thesis does not merely reduce to an endorsement of quantum fundamentalism, though that is indeed what follows from the claim that there are no levels tout court and that physical phenomena can be described in quantum- theoretical terms—including the explanatory strength decoherence brings to the table. More succinctly, quantum fundamentalism (understood as a metaphysical position and not as methodological, intertheoretic or otherwise) is necessary but not sufficient for a metaphysics constrained by the nontological thesis. Beyond quantum fundamentalism (which is by no means a globally accepted position and thus requires its own arguments), the nontological thesis entails the impossibility of “cutting nature at its joints,” to use a well-worn philosophical phrase. No matter how one might choose to parse the world’s configuration space into subsystems according to one’s ontological view, in cutting the world at all one has thereby introduced falsehoods into one’s ontology. This is due to the ubiquity of nonlocal quantum correlations: no region of space can be truly isolated from the remainder and labeled an entity in its own right. The lesson of decoherence is that there is no sense in tracing out the quantum systems that apparently (or even theoretically) compose everyday objects, because in a deep way the physical

158 information giving rise to the phenomenological experiences one might have in seeing, touching, moving said object does not exist within the perceived spatial- temporal domain of the object. Indeed, though we may never directly observe or be able to observe the nonlocal features of the quantum systems giving rise to the physical experience of the object, we know from decoherence that they cannot be absent. In chapter 5 I will discuss various attempts from the literature in philosophy of physics to “slice the world” into levels and then characterize interlevel relations in the familiar language of reduction, supervenience and emergence. We will see not only that the nontological thesis stands in its more general form as a view about the nonexistence of levels in ontology, but furthermore, that in light of this consequence terms like “reduction,” “supervenience” and “emergence” do not appropriately describe the way the world is. The ﬁnal chapter of metaphysical analyses is concerned with one particular consequence of the nontological thesis that has serious ramiﬁcations for contemporary debates in metaphysics and meta-metaphysics. Namely, while chapters 4 and 5 argue from decoherence against the existence of interlevel relations, chapter 6 is concerned with arguments from decoherence against the existence of the intralevel relations necessary for or implied by part-whole talk. In other words, chapter 6 argues that if there can be no parts and wholes in one direction, then there is no sense in speaking of parts and wholes in any direction, and this entails the nonexistence of mereological parts and wholes.

159 4.2 Deﬁning classicality

What does it mean to say something is classical? A ﬁrst-pass answer might be something like the following: what is classical is that which behaves classically; that which behaves classically is that which obeys classical mechanics; and that which obeys classical mechanics is classical. I am perfectly content to concede that there are regimes of phenomena appropriately treated according to classical methods or by some combination of classical and quantum calculational techniques. But our ability to handle objects a certain way does not translate into an argument regarding the metaphysical status of distinct “classical” or “quantum” worlds, entities, behaviors and so forth. Nevertheless, it has often been and continues to be claimed that the apparent, seemingly deep aptness of describing certain energy or size regimes in terms of classical mechanics or semiclassical mechanics has metaphysical implications. Convincing though our ability to apply certain classical techniques or methods—or even metaphors—with the degree of success with which we do—is not thereby a valid argument for the metaphysical primacy of said methods, techniques or metaphors. The success of decoherence models in describing and explaining the dynamics of uncontrolled system-environment interactions gives reason to deny claims whose metaphysical thrust (implicit or otherwise) runs contrary to the thesis of quantum fundamentalism The purpose of the chapter is to argue more explicitly for the nontological thesis: there is no classical world in virtue of there being no way in principle to divide a quantum world (or quantum behavior, entities, properties) from a classical world (or behavior, entities, properties). In other words, there can be no

160 metaphysically meaningful definition for “classicality.” The title of this chapter is a question posed to those whose metaphysical views involve particular definitions (explicit or otherwise) of classicality that contradict the nontological thesis directly. These are of the general sort “classicality is that which is not quantum mechanical.” To define classicality as such will conflict with the nontological thesis in different ways, depending on the ontic strength one attaches to the definition. For example, one might embrace whole-heartedly the definition, in which case it entails the existence of non-quantum entities. This is an outright denial of quantum fundamentalism, and since the nontological thesis embraces quantum fundamentalism, such definitions impact the nontological thesis as well. Obviously the strong definition of classicality as not-quantum mechanical (and the denial of quantum fundamentalism this claim entails) is not the option commonly chosen, though in section 4.3 I will briefly engage with Cartwright once again. Cartwright’s pluralism is about as close as one gets in the philosophy of physics literature to a strong denial of quantum fundamentalism with respect to ontology, i.e., leaving aside those who disfavor quantum fundamentalism as a methodology or explanatory method or the like. I reserve a fuller discussion of anti-quantum fundamentalists for chapter 6, as it is in metaphysical positions one most frequently encounters this view (or something near enough to be worrisome). Stances that are less strong ontologically are those wishing to maintain quantum fundamentalism while at the same time employing a definition of classicality as “not-quantum mechanical” to the extent that those holding such positions demand “classicality” have a deeper meaning than mere explanatory or pragmatic

161 fertility. Most of the present chapter is concerned with this reading of classicality. One can only maintain such a position, I argue, by claiming that in certain limits or given certain parameters, there can be delineation between quantum-ness and classicality—that there are places where the theories overlap, and therefore places where one might successfully cut nature at some joint. These attempts do not succeed, and I present arguments to this end in section 4.4 below. There I describe in some detail the positions of Wiebe, Ballentine and Emerson. They are quantum fundamentalists (as I read them) and yet they demand that “classicality” deliver something for their account that goes beyond a phenomenological isomorphism with quantum mechanics, and that validates a strong reading of the correspondence principle. I conclude that the committment of these three authors to the ensemble interpretation of quantum mechanics is what causes their downfall, in that it pushes them to associate similar metaphysical readings to empirically indistinguishable situations that, given the physics discussed in this thesis so far, we know are in fact not metaphysically similar situations. After discussing this particular failure at a “middle ground” definition of classicality, I will catalogue several popular attempts at defining classicality in ways that try to recover more from the definition than appearances. I show that these ways, without exception, fail. The lesson to be learned here is that one cannot expect definitions of classicality based on what it is not (i.e., quantum mechanical) to succeed beyond phenomenology. Any attempt, no matter how slight, to stray toward a stronger ontological reading of the concept gets one into trouble

162 on fairly short order. Section 4.5 considers the very weak end of the spectrum occupied by those who wish to understand the concept of classicality in an entirely non-metaphysical way. In this section, I engage with two recent proposals in the philosophy of physics literature that do not contradict quantum fundamentalism and involve sufficiently weak senses of classicality to escape the criticisms of previous sections. The two accounts I address are those of Batterman and Bokulich, as presented in their respective works (Batterman 2002 and Bokulich 2008b). I believe it will serve us to consider, in exploring these accounts, just how easily one can slip into stronger readings of classicality. Thus this section will serve as a caution for those who might wish to engage with accounts like Batterman’s or Bokulich’s yet avoid the suite of metaphysical problems that enter as soon as “classicality” is taken to be a term with ontic traction. I end this chapter by discussing briefly the epistemic circle mentioned in chapter 1: by application of classical concepts and laws of motion the groundwork for quantum mechanics could be laid; now via decoherence we see that it is in virtue of quantum mechanical behavior that the world could be explained as “classical” to begin with. I argue that this circularity is not vicious, and in doing so hope to banish forever from the literature certain misunderstandings about the definitions of classical operators, classical concepts, and the like.

163 4.3 Denials of quantum fundamentalism

Though certain interpretations of quantum mechanics entail positions that appear ill at ease with quantum fundamentalism,1 in keeping with my avoidance of the interpretation discussion I will focus on Cartwright. In truth, it is primarily within the ﬁeld of metaphysics that one encounters frequent denials of quantum fundamentalism, but since this issue will be treated thoroughly (albeit in a slightly diﬀerent guise) in chapter 6, what follows is a discussion that remains within recent philosophy of physics. Even here it will not be hard to demonstrate problems exist for the pluralist in light of decoherence, if that pluralism is understood metaphysically.

4.3.1 Cartwright once again

Though Cartwright’s general metaphysical thesis regarding the non-unitary, non-fundamental nature of scientific theories was treated to some extent in the previous chapter, it is worthwhile to invoke her name once again, this time focusing on her arguments specifically against quantum theory as the fundamental theory. I must begin by admitting that I am deeply mystified by Cartwright’s understanding of quantum mechanics, particularly as it relates to classical mechanics. Her argument against the universality of quantum theory go beyond her general arguments against the universality of any theory (perhaps because the perceived threat from that corner is strongest), and the discussion to this end takes place

1Don Howard reminds me that many proponents of Bohmian mechanics fall under this description.

164 primarily in the ninth chapter of Dappled World (Cartwright 1999). In this chapter, Cartwright tries to avoid the common reductionist view that classical states reduce to quantum states by insisting that both classical and quantum states can be understood to hold simultaneously, without contradiction. This notion coupled with her claim that there exists no general way of relating quantum properties to classical properties (and the damning observational fact that classical properties show up all over the place, in and out of the lab) leads her to conclude that quantum theory isn’t special. Like all the rest of science, it is severely limited in its domain of application. My ﬁrst comment is to note that Cartwright begins with the assumption that there exist distinct classical and quantum realms. This assumption can be seen lurking underneath statements like the following:

Though quantities represented in classical and quantum physics are mutually constraining... they are diﬀerent quantities exhibiting be- haviour that is formalised diﬀerently in the two theories. We should not expect to be able to represent classical quantities in the quantum formalism, nor the reverse. In particular we should not expect to be successful in the familiar attempt to represent macroscopic classical quantities by a set of commuting operators in quantum theory. (Cartwright 1999, p. 218)

Her statement that classical quantities should not be expected to be representable in the quantum formalism only makes sense if she already understands those quantities in some deep way as non-quantum mechanical. In this way she can be read as assuming the very thing she is demonstrating—the failure of quantum theory to be universal owing to the independent existence of classical quantities. The second comment I wish to make is with respect to her claim that both

165 classical state descriptions and quantum mechanical state descriptions can be true of a system at the same time. To try to understand what this might mean apart from the measurement problem and apart from the issue of interpretations of quantum mechanics is diﬃcult, as Cartwright frames her approach to the classical- quantum transition in just these ways, as per tradition. Even barring these issues, it remains a simple matter of fact that there exist systems—at very diverse energy, size and other scales—that exhibit undeniably quantum mechanical behavior, or are usually observed in a quantum mechanical state and not a “classical” one. The reverse cannot be said: as we shall see in the discussion of semiclassical mechanics below, there are no classical or semiclassical states, phenomena, systems, etc. that cannot also (usually more accurately) be described in quantum mechanical terms. How, then, can there be “no incompatibility” between the two types of states, as Cartwright insists? The worry that Cartwright tries to capitalize on in her arguments against the universality of quantum theory is the perceived lack of explanation for abundant classical phenomena. In a response to Cartwright’s metaphysical position, Michael Esfeld goes to some length describing how it is that none other than decoherence can ﬁll this perceived explanatory void regarding the appearance of the world as “classical.” Esfeld’s description of decoherence supports what has already been said frequently throughout this thesis when he writes

We know what the world is in itself: namely, an unimaginably complex network of quantum relations of entanglement. And it seems that we can know in-principle how this way the world is in itself is connected with the way in which the world appears to us, decoherence being the clue. (Hartmann et al. 2008, p. 331)

166 This is precisely what has been argued thus far—that one of the most significant effects of the ubiquity of quantum correlations and subsequent decoherence processes is to render empirically indistinguishable what physics says is the case (that physical systems have prodigiously damped yet still existent interference terms) and what appears to be the case (that the world has classical components). If decoherence answers the puzzle of the abundance of (apparently!) classical phenomena, then at least one significant motivation for adopting Cartwright’s account is stripped away. It must be noted that Cartwright’s position includes an importantly different understanding of metaphysics under which the world is only truly described in instances of model-system isomorphism. The models themselves provide the interpretive framework, but this framework is theory-specific only; outside of theory in which it was constructed, a model has no interpretation and no metaphysical relevance. It is in this way that Cartwright justifies a metaphysical picture entail- ing multiple, overlapping “ways the world can be”—the same bit of world might be described equally validly under different model-system isomorphisms, by different theories. This seems to be the method of argument as applied in the case of quantum states versus classical states. Thus one might read Cartwright’s account rather weakly, and take her metaphysics to verge on the sort of story wherein classicality is just a useful notion in the laboratory, or within certain size and energy regimes. There is obviously some important talking-past occurring in this case, and my reading Cartwright’s view as posing a threat to the nontological thesis is only true if one adopts a stronger

167 deﬁnition of what counts as metaphysics in the ﬁrst place. If one chooses the latter reading of Cartwright, one confronts the philosophical arguments coming from decoherence as stated by Esfeld and the work of this thesis so far.

4.4 Attempts at a middle ground

I characterized this somewhat amorphous middle ground position regarding classicality as consisting of attempts to draw more from deﬁnitions of this concept than the physics presented in this thesis allows. The arguments I give against these sorts of positions come in two parts. In the ﬁrst, I focus on an arena of particularly lively debate regarding the universality of quantum mechanics that is found in the literature on semiclassical methods and quantum chaos. The supposed challenge of classical chaos to quantum fundamentalism is motivated by Batterman as follows (Batterman 1993, p. 52):

Classical chaos is apparently a genuine and ubiquitous phenomenon. Given that quantum mechanics is supposed to be the more fundamental theory, it should be able to explain the observed classical behavior of dynamical systems, regardless of whether that behavior is chaotic or not. (Emphases original)2

2Batterman goes on to list another motivation for examining chaotic systems, namely his belief that since classical chaos supposedly undergirds assumptions in classical statistical mechanics, there is also “a need, just as in the classical case, to ground certain a priori probabilistic or statistical assumptions of quantum statistical mechanics” (ibid., p. 52). I, for one, do not share the conviction that quantum statistical mechanics requires any explanatory help. But even this aside, Batterman’s motivation is telling in that it assumes non-fundamentality of quantum mechanics by implying that further grounding is necessary for assumptions in quantum mechanics. Though Batterman is silent as to which aspects of the theory constitute its assumptions, the axioms of the theory have borne themselves out in nearly century of subsequent experimental veriﬁcation.

168 Thus we should expect to find either a quantum counterpart to classical systems so that the correspondence principle (under Batterman’s interpretation) and its limits are preserved, or we should expect to find underlying dynamics in quantum mechanics that describe the appearance of chaos in the semiclassical and classical regimes. The challenge is understood by Batterman and others to be unmet by quantum mechanics, which leads several authors to conclude that quantum mechanics is not fundamental, and that the chaos found in semiclassical systems can only be interpreted and understood in terms of a separate dynamics. Though most of these arguments are not meant to be metaphysical (for the interesting reason that most of those presenting such arguments are, in their purview, straightforward reductionists qua materialists), many of their views end up pointing in that direction. The metaphysical implications often sneak in under the ruse of arguing for explanatory strength, utility, and approximate truth. I will engage to some extent with this literature in order to demonstrate that such semiclassical behavior can indeed be explained at a still deeper level by quantum theory (with or without the added insight of decoherence, depending on the argument at issue). The second part of my engagement with middle-ground definitions of classicality comes in catalogue form: I list (and subsequently critique) many of the most popular attempts at defining classicality in a way that is more robust than mere explanation or pragmatism. This includes considering various limits like Ehrenfest’s theorem, which was mentioned briefly in a prior chapter.

169 4.4.1 Wiebe, Ballentine and Emerson

I begin by noting that many of the objections to claims about the necessity of decoherence for explaining the appearance of classicality often take decoherence to be an interpretation of quantum mechanics or a particular package of assumptions, whereas I understand decoherence to be merely the process of delocalizing phase relations in the quantum system upon interaction with its environment. Understanding decoherence thus minimally already makes moot many of the objections presented by those who disagree with me, as these authors still believe quantum mechanics plays a necessary (if not suﬃcient) role in the appearance of classical behavior. Thus, in as much as their accounts endorse quantum fundamentalism and as I understand decoherence as nothing over and above the axioms of quantum mechanics applied globally, our conclusions are not at odds with one another. However, since several of these authors operate under an understanding of decoherence that, unfortunately, propagates untruths regarding this process, it will be worthwhile to correct some of these ideas and in doing so set the record straight regarding conclusions made in the present work. I begin by discussing Wiebe and Ballentine (2005) and the response to this paper given in Schlosshauer (2008). After this, I turn to a discussion of Emerson’s PhD thesis (2001), which argues for a claim similar to that of Wiebe and Ballentine but from slightly diﬀerent grounds.

170 4.4.1.1 The Hyperion dispute

In 1984, Wisdom, Peale and Mignard published a paper titled “The chaotic rotation of Hyperion,” in which they evaluate the dynamics of one of Saturn’s moons and predicted that it would be found to be tumbling chaotically in its orbit owing to its odd shape (described on p. 23 of Bokulich 2008b as “approximately three times the size of the state of Massachusetts and... roughly the shape of a potato”) and coupled with the inhomogeneous gravitational force experienced by the satellite due to the presence of Saturn and other moons. The criterion used to decide whether or not a system’s dynamics can be classiﬁed chaotic is the existence in the system’s phase space of exponential rates of deviation of trajectories with similar initial conditions. The degree of these deviations is given by the system’s characteristic Lyapunov exponent. Wisdom et al. calculated the Lyapunov exponents for Hyperion and determined that they qualify as chaotic. They write:

As tidal dissipation drives Hyperion’s spin toward a nearly synchronous value, Hyperion necessarily enters the large chaotic zone. At this point Hyperion becomes attitude unstable and begins to tumble. ... It is expected that Hyperion will be found tumbling chaotically (Wisdom et al. 1984, p. 137).

In Zurek and Paz (1997), the authors investigate the dynamics of Hyperion from the perspective of decoherence (the paper is titled “Why We Don’t Need Quantum Planetary Dynamics”). They argue that, on generous assumptions for certain estimated parameters of the Hyperion system, the moon should nevertheless be in “a very nonclassical superposition, behaving in a ﬂagrantly quantum

171 manner” (ibid., pp. 370–371). They continue:

In particular, after a time [on the order of the correspondence breakdown time of non-chaotic motion, calculated in the paper] the phase angle characterizing the orientation of Hyperion should become coherently spread over macroscopically distinguishable orientations the wave function [of Hyperion’s center of mass] would be a coherent superposition over at least a radian. This is certainly not the case, Hyperion’s state and its evolution seem perfectly classical. Why? The answer... is provided by decoherence. (Ibid., p. 371)

In other words: if quantum mechanics provided the correct microphysical description of Hyperion, the moon should have been observed as occupying a superposition of position bases at some point. Since this has not been the case, one must either concede that quantum mechanics fails to provide the correct description or conclude that the full explanation of Hyperion’s behavior requires taking decoherence into account. Zurek and Paz (and again in Zurek 1998) already gave their answer to the dilemma, arguing that since Hyperion is entangled with its environment decoherence necessarily occurs, suppressing the coherent spreading of the system’s wave function in all but the stablest of bases and giving rise to a situation wherein Hy- perion’s motion in the preferred basis (the position basis—in which our measurements are indeed performed on Hyperion) is empirically indistinguishable from a counterfactual situation in which Hyperion’s dynamics are not inﬂuenced by decoherence. Wiebe and Ballentine, among others, wish to resist this conclusion by claiming that decoherence is not necessary in order to describe the divergence between classical and quantum mechanical predictions for Hyperion’s behavior (and, by

172 extension, of other chaotically behaving systems). In their 2005 paper, Wiebe and Ballentine are responding primarily to Zurek’s 1998 paper when they argue that the classical-quantum divide is better delineated by looking to the Liouville limit and not Ehrenfest, as Zurek had done. Ehrenfest’s theorem is concerned with the apparent classical motion of the centroid of a quantum system’s wave packet (to be discussed more thoroughly in the next subsection). Note the role here of appearance—already the discussion of the classical-quantum divide in Wiebe and Ballentine is couched in terms of epistemic limitations. Wiebe and Ballentine instead argue that the Liouville regime is the more appropriate limit in which to study the correspondence or the transition from classical to quantum (or vice versa). The Liouville regime is that within which quantum probability distributions for a particular measurement are approximately equivalent to the classical probability distributions satisfying the Liouville equations. Given that these authors choose the framework of probability distributions within which to evaluate classicality and quantum behavior, they are able to state that “decoherence converts a pure state into a mixed state, but it does not produce localization of the position probability density, and so it has no significant influence on the breakdown of Ehrenfest’s theorem” (Wiebe and Ballentine 2005, p. 2). They conclude that “the effect of decoherence is to destroy a fine-structure that would be unobservable anyway” (ibid., p. 12). And perhaps most crucially for my purposes, Wiebe and Ballentine end by saying that while decoherence may aid in reducing the apparent differences between the classical and quantum regimes, “it is not correct to assert that environmental decoherence is the root cause of the

173 appearance of the classical world” (ibid., p. 15). One must note that both sides in this debate—Zurek (1998) and Zurek and Paz (1997), as well as Wiebe and Ballentine (2005)—have assumed from the start that a distinction between classical and quantum behavior exists. This is due to these authors’ arguments existing in the plane of epistemology and not ontology. In other words, the debate over Hyperion as I’ve understood it is a debate concerning the conflict between the predictions for this moon’s chaotic behavior and its observed behavior. As such, the force of my nontological thesis is somewhat at cross-purposes with this debate. Nevertheless, the conclusions of these authors have implications extending beyond mere epistemology in as much as they take quantum mechanics and its predictions to have varying degrees of meaning, rang- ing from the epistemic to the ontic. This will become more clear by looking at Schlosshauer’s 2008 response to Wiebe and Ballentine, in which he makes this point and others to the effect that decoherence is indeed a necessary part of any discussion on the transition to classicality. Schlosshauer begins by reminding his readership that if it weren’t for decoherence one would always be able to choose to measure a chaotic system (or any system) in a basis that would reveal the system to be in a superposition (Schlosshauer 2008, p. 748). However, because we know there is decoherence in the world, the consequences are such that the possibility of measuring, say, Hyperion in a basis that would expose its persistent non classical behavior is effectively extinguished. Again: this follows trivially from the principles of quantum mechanics.

174 Zurek’s 1998 paper demonstrated that the effect of decoherence on Hyperion is to locally “degrade” the superposition the moon is really in into an apparent (improper—i.e., nonclassical) ensemble of minimum (or approximately minimum) uncertainty wave packets in the position basis. Herein lies the main problem for the Louiville regime approach used by Wiebe and Ballentine: the Louiville regime is that within which the quantum probability distribution for, say, a position measurement of Hyperion’s center of mass corresponds to the classical probability distribution for such a measurement. Ignoring entanglement for the time being, there already appears to be an underdetermination of ontic states of affairs by empirical states of affairs that arises when one takes the quantum system’s probability distribution to be the only relevant information, and one does not know the initial state of the system at issue. This is because at the level of statistics an improper ensemble is identical to a proper ensemble, and thus one could not say whether the ontic situation represented by the ensemble was a proper mixture (that is, the system really only occupies one of all the possible states in the probability distribution and we are merely ignorant of which) or an improper mixture (in which case the system is actually in a superposition of possible states described by the probability distribution). The lessons of decoherence understood as simply a quantum mechanical process emphasize that entanglement cannot be neglected, especially in a case of uncontrolled system-environment interaction like that of Hyperion and its sur- roundings. Thus, the problem of underdetermination at the level of the statistics dissolves, and we must conclude that the system is in a highly nonclassical state,

175 and therefore that the probability distribution for a position measurement of Hy- perion must indeed by an improper ensemble of possible states, and not a classical one. Herein lies the problem for Wiebe and Ballentine, whose choice of understanding the dynamics of various systems in terms of Louiville regimes and therefore to consider dynamics at the statistical level and with respect to a single observable, thereby operate under a false assumption that the probability distribution on the classical side, though apparently approximately equivalent to the probability distribution on the quantum mechanical side within the Louiville regime, nevertheless represents a vastly diﬀerent ontological situation. Thus it is simply incorrect to explain the relationship between quantum and classical behavior in terms of probability distributions, as the success of such an explanation demands that one assume a classical interpretation of the quantum probability distribution from the start. When one is careful to track the implications of one’s claims with respect to epistemology and appearances versus what we know to be the case at the ontic level (in spite of appearances), decoherence does address all that cries out for explanation. As Schlosshauer writes (Schlosshauer 2008, p. 798):

The superposition initially conﬁned to the satellite is rapidly dynamically delocalized into the composite satellite-environment system via environmental entanglement. This implies that there exists no local measurement that could be performed on the satellite that would in practice reveal the presence of the superposition.3

3Furthermore, Schlosshauer claims, the in-practice impossibility of measuring Hyperion in a basis that would yield a nonclassical measurement result is “rather insensitive” to the particular

176 As described in chapter 2, the reason that the position basis is the appropriate basis of measurement for a macroscopic system like Hyperion in because such a basis, under the dynamical inﬂuences of the environment, is prodigiously more stable than any other basis. In this way, the fact that position is a classical observable is a derivative fact—it is a result of the underlying dynamics that certain observables are “given to us” as more appropriate in various energy regimes and in certain environments than others, and hence named “classical.” This argument will be described more fully below. Even further assumptions can be discovered within the Wiebe and Ballentine discussion, and Schlosshauer continues by pointing out that in restricting their analysis to the level of statistical ensembles, the authors have tacitly endorsed a view—since shown inappropriate in light of decoherence—often called the ensemble interpretation of quantum mechanics.4 Schlosshauer states (ibid., p. 799) that such an interpretation entails problems beyond that of matching probability distributions. Namely, adopting this interpretation “implies that the entire formal body of quantum mechanics (for example, a probability amplitude) has no direct physical meaning” (ibid., p. 800). In other words, we know more about the dynamics than such an interpretation allows for; furthermore, it is precisely the information lost in an ensemble interpretation that supports the arguments of the pro-decoherence side of the Hyperion dispute. model of decoherence applied (ibid., p. 798). This is in full agreement with the conclusions reached in the previous chapter. 4Ballentine himself is in fact responsible for one of the major works on this interpretation: his 1970 paper titled “The Statistical Interpretation of Quantum Mechanics” (Ballentine 1970).

177 In summary, the attempt of Wiebe and Ballentine to negate the importance of decoherence in the Hyperion dispute fails based on a misunderstanding of decoherence as an interpretation instead of as merely a physical process deriving from the principles of quantum theory, in tandem with the adoption of a particular ensemble-interpretation of quantum mechanics that renders highly problematic their choice of the Louiville regime as the arena in which to settle the quantum to classical transition question. Schlosshauer summarizes nicely how, on these assumptions, Wiebe and Ballentine were able to conclude as they did:

Once the problem of the quantum-to-classical transition is reduced to the comparison of quantum and classical probability distributions for a single observable (which... is deliberately chosen such as to make sense also in the classical setting), environmental interactions and the resulting decoherence processes will play but a minor role when compared to a framework in which measurability and the existence of quasi-classical observables is to be explained from within quantum mechanics. (Ibid., p. 801)

4.4.1.2 Emerson’s thesis

Emerson’s PhD thesis of 2001—directed by Ballentine—includes a chapter written with Ballentine defending a point similar to the one made above by Wiebe and Ballentine—that classicality comes about without signiﬁcant contributions from decoherence. In the introductory chapter to the dissertation, Emerson writes the following:

The central focus of the work in this thesis is to demonstrate that, for the chaotic motion of some macroscopic systems, standard quantum mechanics may be unable to reproduce Newton’s laws of motion over experimentally relevant time scales. Moreover, contrary to the con-

178 clusion of a recent argument by Zurek [1998], I will provide evidence that this speciﬁc breakdown of quantum-classical correspondence is not circumvented by taking into account the decoherence eﬀects of a stochastic environment. (Emerson 2001, p. 3)

There are a few things to note already: first, when Emerson states that standard quantum mechanics ought to reproduce Newtonian mechanics, we want to make sure that this claim remains mechanics-centered. In other words, that the discussion with respect to the correspondence principle remains between the mechanics and not, say, behavior. While quantum mechanics was designed to correspond with classical mechanics in certain regimes (and thus that Newtonian laws of motion would emerge), the same cannot be said of classicality tout court, or even classical behavior. The reason is already given in part in the prior section with the little demonstration that mechanics aside (and perhaps even in some cases in the mechanics) classicality cannot so easily be defined. Thus it will be important to track Emerson’s understanding of the relationship between these two regimes, as it might very quickly become confused, rendering this assumption about what their relationship ought to be like in the first place. A second thing to note is along the same lines: from Emerson’s description of his own thesis, he takes the correspondence principle as something to be preserved at all costs. It is unclear why this ought to be so. For one thing, a majority of physicists view quantum mechanics as a more fundamental theory than classical mechanics (i.e., more fundamental in the sense that it explains a great variety of phenomena including those described by classical mechanics). Therefore, if we should at some point find a violation of the correspondence between the two the

179 response should not primarily be to save the correspondence principle at all costs, nor should it be to jettison or consider inadequate the explanations provided by quantum mechanics. If anything is jettisoned at all, it should certainly not be quantum mechanics. Note that I find it quite easy to explain the difficulty with the correspondence principle when extended beyond the heuristic regime in which it was most likely originally intended. The explanation is that it is doomed to fail, because it does just what decoherence has shown us cannot truly be done: it tries to delimit the classical and quantum regimes. If there are no regimes to begin with, we can let lie the correspondence principle and surrounding disputes as to its nature or application, resting happy in the fact that it serves as a heuristic device and nothing more than that. It was crucial for developing the deeper theory of quantum mechanics, but by no means are we—having since progressed beyond classical interpretations of the world—still beholden to the principle. Hopefully this argument will become clearer in the remainder of this chapter. Perhaps the most crucial note is that in order to make sense of the correspondence principle when discussing the dynamics of various systems one has to first assume that there exist two separate worlds or regimes that can stand in the correspondence relation with one another. I don’t deny that this relation might have meaning—perhaps even prove useful—in instances where its employment doesn’t extend into metaphysics. But trouble ensues when one attempts to glean more from the principle than heuristic insight. As mentioned in the introduction to this chapter, though authors discussing the quantum-classical relation typically

180 intend their arguments to remain within the arena of epistemology (e.g., a “for all practical purposes”-flavored discussion), I find that the conclusions or results of such arguments stray somewhat dangerously into ontological matters. Another caution with respect to Emerson’s overall project regards his choice of case studies as chaotic systems. Earlier in our discussion of Hyperion, chaos was defined to be exponential sensitivity to initial conditions. This is a general definition. Emerson’s more careful definition is precisely what gets him into the weeds. He states that extreme sensitivity will for his purposes be characterized in terms of Lyapunov exponents (a term mentioned above). However, Emerson defines Lyapunov exponents in the following problematic way: “This property [of extreme sensitivity] is normally identified with an exponential divergence, on average, in the separation between initially nearby classical trajectories” (ibid., p. 7). As we shall see below, apparently classical trajectories—even in the Ehrenfest limit—are nevertheless constrained to obey Heisenberg’s uncertainty relations, and can never represent continuous Newtonian paths traced by a true point. One should beware of conclusions regarding the nature of classicality or, more specifically, the inadequacy of quantum mechanics to describe a system whose definition is already dependent upon classical trajectories or laws of motion conforming to Newton’s laws. As mentioned, part of Emerson’s thesis is taken from a paper published with his advisor, Ballentine (Emerson and Ballentine 2001). In this section of his thesis, Emerson considers two interacting spins and takes the classical limit to be derivable as the quantum number approaches infinity. Already one might choose to

181 quarrel with this definition given certain problems with understanding this particular limit (more will be said about the particulars of this limit below). Though this limit is by itself considered insufficient by Emerson, he supplements this criterion with the Lyapunov exponent. This, too, may contribute to confusion regarding the assumed distinct quantum and classical behaviors of the model. Also of note is the claim at the start of the chapter (p. 37) that the results of studying the interacting-spin model shows the subsistence of the correspondence principle even for chaotic systems, and furthermore that “this demonstration of correspondence is obtained for a few degree-of-freedom quantum system of coupled spins that is described by a pure state and subject only to unitary evolution” (Emerson 2001, p. 37). One need hardly go farther that this sentence in order to show the true toyness of any model that considers a system to evolve unitarily, when decoherence has taught us that the system itself cannot be considered properly isolated and to evolve unitarily; it is only at the level of the system plus environment that unitary Schrödingerevolution remains valid. My final comment regarding this chapter is also independent of the model and instead has to do with reliance upon the Louiville regime as the appropriate regime within which to characterize the quantum-classical transition. Of course this definition is not new from Ballentine’s end, considering the similar tack he takes with respect to the Hyperion dispute. Unfortunately for Ballentine and for Emerson, the same critiques remain, namely that a comparison of mere probability distributions makes rather large assumptions about the interpretation of the body of quantum mechanics, in particular by limiting the discussion necessarily

182 to the level of epistemology or, if you prefer, phenomenology. In other words, the Louiville limit will indeed save the phenomenon, if the phenomenon to be saved is the correspondence principle and if the correspondence principle is understood in a manner similar to that of Emerson. However, we know better (and decoherence has helped us in this) than to attribute ontic meaning to the appearance of statistical ensembles. Though one might argue that since Emerson has stipulated that we begin with the system in a pure state in this model, one might thereby be legitimated in expressing the resultant mixed state as equivalent to a classical statistical distribution. But once again this argument commits the very mistake decoherence aims to correct for—namely, the assumption that conclusions obtained from truly toy models neglecting the still, small point about ubiquitous entanglement can nevertheless be applied to real-world models. This is false. Though the approach adopted by Emerson in his thesis is in many ways similar to the same critiques of decoherence and quantum fundamentality that were expressed in the Hyperion debate, the former is illuminating in that it commits many of the errors predominate in dissenting literature regarding quantum mechanics, namely, that certain assumptions about the metaphysical relationship between classical and quantum mechanics are often assumed ab initio.

4.4.2 Other attempts at a middle ground that won’t do

Below I catalogue the most popular attempts at deﬁning classicality, and demonstrate that in no case is a robust delineation made between classicality and nonclassicality. Some of the deﬁnitions overlap in terms of the microphysi-

183 cal dynamics (i.e., saying that classicality is lack of interference phenomena is in some ways the same as deﬁning classicality in terms of diagonalization of system matrices); I nevertheless treat the arguments separately for the sake of clarity.

• Classicality as defined by Ehrenfest’s theorem Ehrenfest’s theorem states that quantum mechanics and classical mechanics intersect when the wave packet representing the quantum system in phase space is sufficiently localized and its center (the average position and momentum) follows a Newtonian trajectory. Ballentine, Yang, and Zibin (1994) argue that one’s ability to apply Ehrenfest’s theorem to a system is neither a necessary nor sufficient condition for defining the system as classical. They demonstrate the failure of Ehrenfest’s theorem as a sufficient condition for classicality by discussing the example of a harmonic oscillator. Though the equations of motion for a quantum harmonic oscillator can be approximated as those of a classical harmonic oscillator in certain conditions, certain other features of the quantum and classical harmonic oscillators remain distinct, in particular the specific heat of these oscillators (a function of thermal equilibrium energy) (ibid., p. 2854). Thus, Ehrenfest’s theorem cannot get one entirely over to the “classical regime.”

To demonstrate the failure of Ehrenfest’s theorem as a necessary condition for a quantum system’s being considered classical, the authors give several examples of systems that can be characterized as classical independently of the equations of motion. One such example is looking to probability

184 distributions in phase spaces of classical and quantum mechanical systems: the authors state that the centroid no longer need follow a classical trajectory as in the Ehrenfest limit since “it is ultimately the non-commutativity of operators ... that is responsible for the diﬀerent evolutions of the classical and quantal probability distributions” (ibid., p. 2855). They conclude as follows (ibid., p. 2858; emphasis original):

We have shown that Ehrenfest’s theorem is neither necessary nor sufficient to characterize the classical regime in quantum theory. Ehrenfest’s theorem asserts that, for a sufficiently narrow probability distribution, the mean position in the quantum state will follow a classical trajectory. However, generally speaking, the classical limit of a quantum state is not a single classical trajectory, but an ensemble of trajectories. The averages and higher moments of the quantum and classical probability distributions often agree in situations where Ehrenfest’s theorem is not applicable. There- fore Ehrenfest’s theorem is not applicable. Therefore Ehrenfest’s theorem does not define the conditions for classical behavior.

They extend their results to various attempts to deﬁne classicality in the arena of quantum chaos. First, they argue that Goggin et al. (1990) fail to appropriately deﬁne classicality in expressing quantum commutators in terms of classical Poisson brackets in tandem with higher-order corrections to Ehrenfest’s theorem. The claim is that classicality is obtained if and only if such higher-order corrections vanish; Ballentine et al., however, show that not all of these higher-order corrections do vanish in cases that are typically considered classical.

185 Another variation on Ehrenfest’s theorem as a deﬁnition of the classical regime that Ballentine et al. say fails is that of Lan and Fox (1991). Lan and Fox apply Ehrenfest’s theorems to Hamilton’s equations, but Ballen- tine et al. again state that “although Ehrenfest’s theorem does indeed break down much sooner for chaotic motions than for regular motions, the degree of agreement between the classical ensemble and the quantum state is comparable for both chaotic and regular domains.” Q.E.D.

• Classicality as the limit ~ → 0 As many have already demonstrated,5 simply letting ~, the quantum of action, approach zero is insuﬃcient for a deﬁnition of classicality due to the fact that in many cases this limit is a singularity. Why a singularity is an issue for interpretations of limits is poignantly described by Berry (and given as quoted in Bokulich 2008b, p. 15) as analogous to the following:

Biting into an apple and fining a maggot is unpleasant enough, but finding half a maggot is worse. Discovering one-third of a maggot would be more distressing still: The less you find, the more you might have eaten. Extrapolating to the limit, an encounter with no maggot at all should be the ultimate bad-apple experience.

In other words, there is something intrinsically diﬀerent about this limit that cannot be evaluated from a purely numerical standpoint, and this fact is encoded in the necessity of using entirely diﬀerent algebras when performing calculations that are classical versus those that are quantum mechanical.

5E.g., see Batterman (1995), Batterman (2002), Berry (1994), Berry (2001) and Bokulich (2008b) for fuller discussions of the failure of this deﬁnition of classicality.

186 The fact that quantum mechanics relies importantly on commutation relations and the uncertainty relations and not so classical mechanics is a fact often forgotten when one focuses on the quantitative aspects of the limit

~ → 0. Nevertheless, the importance of these diﬀerences between quantum and classical mechanics remains, no matter the size of Planck’s constant.

• Classicality as the limit n → ∞ As the variable n, called the principle quantum number, becomes increasingly large, it represents a physical situation in which the discreteness of a system’s possible energy states can be increasingly approximated as continuous due to decreasing gaps between the energy states. However, since as the quantum number for a particular system approaches inﬁnity it remains

constrained by the uncertainty relation ∆E∆t ≥ ~/2, it follows that this limit (i) is only approximate and as such (ii) cannot provide a metaphysically deep division between the classical and the quantum regimes.

Further discussion of the inadequacy of the n → ∞ limit can be found already in Messiah (1965), in which he argues that, while classical behavior will emerge as the quantum number increases, this is not necessarily so. Liboff (1984) has also argued that this limit is insufficient to recover classical behavior for much the same reason discussed in the previous paragraph. Liboff shows using two case studies of physical systems that the discreteness characterized by the uncertainty principle cannot be overcome, and a purely classical continuum of energy (or any other quantal property of a system) is

187 impossible.6

• Classicality as ~ → 0 and n → ∞ Combinations of various parameters in the limit will not provide one with an adequate description of classicality either as (given the descriptions of the limits individually treated above) the problems of the individual limits are not solved by conjoining them. For example, a deﬁnition of classicality

as the joint limit ~ → 0 with n → ∞ lacks the metaphysical robustness sought due to the ever-present, never violable uncertainty relations, coupled with the strange and often singular behavior of Planck’s constant at the zero limit. Thus one must not seek refuge here either.

• Classicality as definiteness Recall our earlier discussion of an aspect of the measurement problem we called the problem of general outcomes. There I stated that decoherence demonstrated why this question was ill posed: namely, that through decoherence we understand there are no definite outcomes in any uncontrolled situations, and rarely in controlled ones. The observation of measurement outcomes as apparently definite is due to the increasing impossibility of measuring the system in any unstable basis, as the environment einselects a basis as robust via interaction with (and decoherence of) the system of interest. The fact that the dial on the machine appears localized is itself a conse-

6Both Messiah’s and Liboﬀ’s discussion of the n → ∞ limit are summarized nicely in Bokulich 2008b, pp. 17–18.

188 quence of quantum decoherence, as this device interacts with its environment uncontrollably and therefore decoheres into its own stable basis—typically position (for reasons described in previous chapters). As such, classicality certainly cannot be defined as definiteness, for we know that definiteness is superficial (due to quantum effects) and not a metaphysical truth.

• Classicality as involving significant magnitudes or mass A first attempt at defining classicality is often made in terms of magnitude— that is, defining as classical only systems involving a large number of molecules or weighing a certain amount. Granted, this very simple (albeit naive) definition does seem to classify objects correctly with respect to a successful application of the mechanics. However, it isn’t hard to find counterexamples systems that cannot be treated classically yet which are relatively large or involve a large number of degrees of freedom. For instance, imagine a system of large potential number of degrees of freedom where many of those degrees are “frozen out” so as to be inaccessible in a certain energy or temperature regime.

Then there are examples of very large objects that must be treated using quantum mechanics. One such example (given by Joos in Giulini et al. 1996, p. 135) is the Weber bar, which is a tool for measuring gravitational waves. The bar itself must be quite massive (on the order of tons) for the detection of the waves, and yet the sensitivity of the device is such that it must detect displacements on the order of 10−21 meters, or approximately 50

189 billionths the radius of a ground-state Hydrogen atom. As such, the weighty Weber bar must be treated as a quantum oscillator in order to appropriately characterize its behavior, despite its size.

We should also remember the variety of observable macroscopic quantum effects; a partial list includes circulation quantization in superfluid helium, the Josephson effect in superconducting (as mentioned in the previous chapter) and, of course, a very trivial sort of answer: lasers. Even the commonplace red, blue or green laser-pointer employed mundanely during business presentations are examples of quantum effects observed at a comparably large scale—with our own eyes, in this case.

An even more convincing argument against this sort of deﬁnition can be given by simply noting, as above, that any physicist will agree that even very large systems might be described by quantum mechanics, only it is impractical to do so on a purely calculational basis. That is, treating large systems (large in terms of degrees of freedom or mass, etc.) as classical by using classical mechanics to describe them is fabulous for practical purposes, but those same physicists will put absolutely no stake in this practice as a means of parsing the world into distinct categories of being.

• Classicality as absence of entanglement Owing to the discussion of ubiquitous uncontrolled entanglement in the pre-

190 vious chapter, it is not hard to see why this particular attempt at a deﬁni- tion of classicality will fail. Even if one chooses to understand absence of entanglement as applying to the level of observation, this does not provide a hard-and-fast line, as how one deﬁnes observation depends on many sub- jective features of the situation in which one is performing the observation. Clearly that will not do.

• Classicality as absence of interference This definition falls prey to issues akin to those arising for the case of missing entanglement described above. If decoherence accounts for the suppression of interference effects and measurement approximately destroys interference in most systems, it certainly cannot serve as a divisor between classical and quantum objects or behavior. One must also remember that decoherence suppresses interference effects in a single basis—the einselected basis—whose stability is a function of the environment’s dynamical reaction to the system.

Thus this particular definition fails in both directions, for even objects one would want to name classical are on this definition only failing to exhibit interference in the preferred basis, but might have exhibited interference effects had the environment been comprised differently or the observation carried out in a different experimental arrangement, and so forth. In other words, the fact that any system—large or small, high or low energy, etc.—does or does not give rise to interference effects is a consequence of the system’s interaction with its particular environment and the nature of their interaction,

191 as well as being contingent upon the basis chosen for measurement.

• Classicality from quantum Darwinism Quantum Darwinism is a way of codifying the replication of information throughout an environment after decoherence has occurred. The greater the degree to which information is replicated in the environment, the more “classical” the system in that environment is considered to be. In chapter 2 we discusses various limits for scattering-induced decoherence (section 2.3.3), noting that one’s ability to obtain information about the CS from the environment is directly connected to decoherence of the CS; while one limit required only a single (quantum) interaction to obtain full position- determining information about the CS, the opposite limit required multiple scattering events to obtain the same information. In the former limiting case, it is likely that many more than a single interaction will take place with respect to the CS (and this probability increases as the size of the CS grows toward the macroscopic end of things), and a great deal of information about the CS will be found replicated all over the environment.

This definition, too, suffers from a sliding line in that what belongs to the quantum or the classical description is largely dependent on what one defines to be the system and environment to begin with. As such, quantum Dar- winism will not be able to provide the thorough-going distinction between a classical and a quantum world. Quantum Darwinism really amounts to nothing more than requiring maximally localized correlations by looking at

192 information stored in the environment. Since this is a definition in terms of correlations, it is in effect defining classicality as the absence of nonlocal (quantum) correlations. Once again this will do no good, the crucial point being that uncontrolled nonlocal correlations are ubiquitous.

4.4.3 Caveat: Classical spacetime

Though I cannot give a full discussion of the role of decoherence in quantum field theory or quantum electrodynamics here, there is one aspect of this area of physics that might cause tension for a physicist who otherwise would accept quantum fundamentalism without pause. That aspect has to do with the fact that quantum field theory and quantum electrodynamics are both fields that, though quantum theoretic themselves, are nevertheless embedded within a classical spacetime manifold. One might wonder if there is something to this seemingly necessary invocation of classicality, but one need not wonder long. If the role of decoherence in quantum theory is a relatively new arena of inquiry, then investigations of decoherence within field theories (e.g., quantum electrodynamics and perhaps even quantum chromodynamics) and in cosmology is barely out of the gate. This is so with good reason: complications abound in this arena because no quantized theory of gravity exists as yet. Nevertheless, already in the mid-1980s Joos and Zeh had realized the explanatory utility of decoherence with respect to spacetime (Joos and Zeh 1985). More recently, Claus Kiefer has done much to encourage theoretical work along these lines, in part motivated by the (assumed shared) desire to explain this seeming

193 reliance upon classicality.7 Very brieﬂy, Kiefer’s answer as to the reliance upon classical spacetime is in the same vein as many arguments from decoherence: whatever aspect of the classical entity (here spacetime) that appears fundamental can in fact be given a deeper explanation in terms of decoherence. But ﬁrst: Kiefer describes the crucial role of classical spacetime by stating (Giulini et al. 1996, pp. 146–147):

...[T]he assumed presence of an external time parameter is crucial for the standard framework of quantum theory: Matrix elements are evaluated at a given time, and the whole notion of unitary development would break down if such a time parameter were absent.

But he continues a few pages later (ibid., p. 148):

Again, the general notion of decoherence turns out to be a viable concept. The essential idea is based on the fact that gravity couples [read: becomes entangled] universally to all forms of energy. Gravity is then “measured” by matter, and a general superposition of gravitational quantum states is decohered, i.e., becomes “localised in conﬁguration space” after matter degrees of freedom (which are here considered as playing the irrelevant part) are integrated out. (Emphasis original)

Following the example of Joos (1987b), Kiefer asks his reader to consider a homogeneous gravitational field whose state is a superposition of two different field strengths (pp. 148–149). If we send a particle through this field, the particle will act as a measuring device on the field with respect to (or in the basis of) field strength. This is due to the fact that the particle’s trajectory will depend on the spacetime metric and therefore will commute with field strength; this results

7See Kiefer’s contribution to Giulini et al. (1996), pp. 137–156; also see the list of references provided in 1.5.

194 in entanglement in that basis. This entanglement, as we well know, will lead to decoherence of the superposed field strengths, and voilà:we have the appearance of a classical spacetime—that is, a metric whose field strength is measured in a non-superposed state. Of course the story is more complicated than this, but the results are, to the best of our current knowledge, consistent with quantum fundamentalism. And that will suffice for present purposes. To conclude this section on attempts at maintaining quantum fundamentalism while simultaneously demanding something deeper from “classicality” than explanatory depth or practical utility, it appears that no such definition will succeed. Thus no definition of classicality that might potentially weaken the nontological thesis has survived criticism. The weak reading of classicality to which we now turn is not going to pose a threat to the nontological thesis via attacks or worries regarding quantum fundamentalism. However, it is trivially easy to slip from a weak definition into something slightly more ambiguous that does commit the sort of blunders I’ve accused others of in the above sections. By way of a caution, then, we will take some time to explore two recent accounts that have received significant attention in the philosophy of physics community and that focus not on metaphysics but on explanatory strength deriving from semiclassical and classical methods.

195 4.5 Batterman, Bokulich & the obstinance of classical structures

Prior to his 2002 book, The Devil in the Details, Batterman authored a suite of papers (Batterman 1993, 1995, 2002) in which he argues that much of what gets “studied under the rubric of ‘quantum chaos’ is parasitic upon classical dynamics” (Batterman 1993, p. 50). The possible point of conﬂict for present purposes lies in how one interprets the word “parasitic”; if the term comes to mean something in a metaphysical way, than such an interpretation will run afoul of the physics for reasons already given. Batterman walks a ﬁne line in developing what he calls the “third theory” approach, embracing as he does a position that in places seems to go beyond questions of mere explanation into the ambiguous realm of interpretation. This is seen in statements like the following (Batterman 2002, p. 110):

Once again, their [quantum mechanical] features are all contained in the Schr¨odingerequation—at least in the asymptotics of its combined long-time and semiclassical limits—yet, their interpretation requires reference to classical mechanics.

When Batterman gets down to proving his point, he does so by restricting his considerations to “roughly the quantum equivalent to... conservative classical Hamiltonian systems” (Batterman 1993, p. 52)—in other words, finite- dimensional, closed, isolated systems. Within these restrictions he then stresses “a finite and bound quantum system governed by the Schrödingerequation cannot exhibit the sort of sensitive dependence on its initial state characteristic of the classical chaotic systems” (all emphasized in original). This would be unprob- lematic but for the fact that no such system exists. This serves as a reminder

196 that Batterman’s statements must be interpreted within a carefully deﬁned, non- metaphysical framework if they are to remain valid. Other moments for potential misinterpretation of Batterman arise when he maintains in Batterman (1993), contra the foundational work of physicist Michael Berry and the latter’s conclusion of quantum fundamentality, that there is “really a third theory involved here [in the semiclassical regime]; and what is interesting and worthy of further philosophical investigation are the relations between it and both the classical and quantum theories” (Batterman 1993, p. 54). This third theory of semiclassical mechanics is supposed to occupy the zone between quantum and classical mechanics (ibid, p. 63), but again it is purportedly not meant to do so metaphysically. The recent work of Bokulich (cf. Bokulich 2008a,b) makes claims very similar to those of Batterman regarding the insuﬃciency of pluralism, reduction or emergence to characterize the quantum-to-classical transition. Drawing on four examples in semiclassical mechanics, Bokulich argues (again, in much resonance with Batterman’s themes) that the semiclassical explanatory level is necessary to understand certain phenomena not adequately explained by classical mechanics nor (against expectation) by the more fundamental quantum theory. Bokulich invokes four case studies—the spectrum of the helium atom, the anomalous behavior of diamagnetic Rydberg atoms, wave function scarring and quantum dots—to demonstrate three things central to her thesis: (i) a variety of quantum phenomena exist for which quantum mechanics is nevertheless not the most appropriate theoretical framework but semiclassical mechanics is (ii) semiclassical mechanics

197 is comprised of “a thorough hybridization of classical quantum ideas” and (iii) the classical structures remaining in semiclassical mechanics are not mere calculational tools, but are “manifesting themselves in surprising ways in quantum experiments” (Bokulich 2008b, pp. 2–3; emphasis original). However, it turns out that quantum mechanics (especially with the aid of decoherence) is explanatorily rich enough to deal with the four quantum phenomena Bokulich presents in her case studies. Thus even were one to read too much metaphysics into Bokulich’s account, one would still encounter serious diﬃculties if adopting the same examples as invoked for Bokulich’s purposes. Semiclassical mechanics, surely Batterman and Bokulich would agree, only appears to depend upon classical notions like Newtonian trajectories. What we see instead is that these obstinate classical structures are in fact quantum mechanical.8 In the four case studies presented by Bokulich to support her view, each relies on the ﬁction of classical trajectories. However, in all cases the systems exhibiting the apparently classical behavior are within certain environments, and therefore upon interaction will become entangled, leading to decoherence. As described

8Note, once again, that I am in no wise attempting in my explanations to argue that the physicist who solved or explained these phenomena in terms of semiclassical mechanics were wrong, per se. Nor am I arguing that semiclassical methods should not be considered useful tools. However, as both Bokulich and Batterman might be read as indicating that semiclassical methods are important beyond heuristics and beyond calculational tools, I diverge from them on this point. Indeed, semiclassical methods on my view cannot be anything but useful tools, as no phenomena exist that are not through-and-through quantum mechanical. One must be careful in this domain of inquiry, as in the next chapter’s, not to confuse (as so many authors have and continue to do) the import of claims: are they epistemic? Ontic? Methodological? Such attention (or lack thereof) to this point has resulted in something of a meta-intertheory confusion: is the thesis regarding quantum-classical relations itself meant to belong to a theory of metaphysics or epistemology—or is it not meant to be philosophical at all? Likewise, is the tenor of the arguments in support of these conclusions and the perceived consequences meant to inform scientists’ methodology? Philosophers’ explanations?

198 in preceding chapters, decoherence suppresses interference between elements of the phase space of the quantum system and thereby effectively constrains the system’s motions to those of a well localized wave packet following an approximately Newtonian trajectory. More succinctly: in each of the four case studies presented, significant explanatory progress was made based on the inclusion of classical trajectories in the description of the phenomena; however, we know that the appearance of classical trajectories is due to the stability of certain component states over superposition states of those components via the decohering effect of the environment, which serves to randomize the phase relations among components states in the superposition. The semiclassical explanation is given a deeper explanation from decoherence processes: the physicists who relied on classical behavior to explain certain phenomena were justified in doing so in virtue of quantum behavior.9 A review of Bokulich’s book written by Belot and Jansson (2010) presents a critique of both Bokulich and Batterman that summarizes quite nicely the points I wish to stress:

Bokulich and others see explanations that draw on semiclassical considerations as involving elements of classical physics as well as of quantum physics. For those who accept this point of view, it is natural to look for an account of explanation under which explanations can ap-

9In support of this argument, one need only look to recent work done on these phenomena in light of decoherence. For example, Rydberg atoms (unusually large atoms in strong magnetic ﬁelds) have been studied using decoherence models by Keeler et al. (2010) and Maioli et al. (2005), while investigations on quantum dots and decoherence can be found in Golubev et al. (2010) and Nakamura and Harayama (2004). On decoherence and chaos more generally, see Braun (2001), Buric (2010), Habib et al. (1998), Namiki et al. (1996), Wang et al. (2010) and Zurek (1998).

199 peal to structure that is present in less than fundamental theories but absent in fundamental theories. But there is an alternative way of thinking of semiclassical mechanics: as the study of approximate solutions to the equations of quantum mechanics in some asymptotic regime (large energy levels, etc.) So understood, this project lies squarely within quantum mechanics: starting with the formalism of quantum mechanics one proves theorems about approximate solutions—theorems that happen to involve some of the mathematical apparatus of classical mechanics. But this need not tempt us to think that there is physics in our explanations that is not quantum physics. (p. 82)

While the topic of useful fictions has gained some popularity in recent philosophical literature, the importance of the explanatory work done by such fictions in semiclassical mechanics does not change what I argue to be the case—namely, that decoherence in particular and quantum mechanics more generally can provide sufficient explanations for these phenomena, and for this reason great care must be exercised when describing concepts like useful fictions. Other instances requiring great care include interpretation that one might easily infer from the concluding chapter of Bokulich (2008b). Here Bokulich articulates her stance (which she names “Interstructuralism”) and stresses that her thesis is not meant to be anti-quantum fundamentalist with respect to the metaphysics. And yet, she want to impress upon her readers that there is a power—an obstinacy—to classical descriptions that reaches beyond mere explanatory necessity; just how it is meant to supersede explanation and yet remain non- metaphysical is not easy to discern, but in as much as semiclassicality has power beyond explanation on Bokulich’s view, it runs the risk of entering metaphysically fuzzy terrain.

200 To summarize my analysis of both Batterman and Bokulich’s views, it is not that I take issue with their claims about the explanatory strength of classical concepts or the utility of semiclassical methods for understanding; it is only to the extent that both authors seem impelled to move beyond explanation and imbue their respective stances with something deeper—and the extent to which interpretations of their work easily encourage this troublesome view of classicality—it is only to this extent that worries arise regarding the nontological thesis and quantum fundamentalism. Slippage between epistemic and ontic claims is not uncommon; we’d do well to take care when so much is at stake.

4.6 Closing the (non-vicious) epistemic circle

In the first chapter I briefly introduced the notion of the epistemic circle, or loop, involving classical mechanics and quantum mechanics. The concept of the epistemic circle is from Shimony 1989, and it is meant to describe the fascinating historical fact that it was only by implementing classical concepts and the correspondence principle between classical mechanics and quantum mechanics that the latter was articulated, yet now we find that it is in virtue of quantum mechanics that the full truth underlying classical mechanics (i.e., that it is only apparent and that this appearance is contingent upon quantum processes) comes to light. Decoherence is the crucial link needed to close this loop, and it seems to do so beautifully. However, there are many authors who continue to express in writing and in word their convictions that classical concepts remain an indispensable part of the

201 world, and that this epistemic loop is viciously circular. This reticence to let go of classical ideas can also be seen lurking behind (or even explicitly motivating) the various attempts at deﬁning distinct classical behavior, regimes, objects, etc., described in the earlier parts of this chapter. In particular, a few authors have taken refuge in the correspondence principle, demanding that this principle (whose intended function was primarily heuristic) nevertheless remains the point on which all questions about the classical-to-quantum transition must settle. Related to these convictions are notions, still expressed in philosophy and physics (though it is becoming less so), that puzzles like nature’s apparent preference for speciﬁc variables or for classical observables must be explained in terms of some selection principle, or that superselection rules cannot be given deeper explanations. To begin with, consider the just warning on this point given in Bitbol (2009), p. 354 (emphasis original):

Being a pragmatic precondition for is not tantamount to Being full stop. Having been used as an indispensible [sic] starting point of an epistemic process is not equivalent to having more ontological weight than the end produce of this very epistemic process. One should realize that choosing a starting point has no ontological implication at all.

The historical fact of the matter is that being creatures of a particular size and with particular faculties of perception and interpretation and so forth, we came to know our world in a classical way before we understood it in a quantum mechanical way. The ﬁrst class of variables considered by scientists were, naturally, those that described the familiar, macroscopic world. But now that we have available to us a deeper understanding of the microphysics, we must recognize that our ability

202 to describe the world as “classical” at all was contingent on the methods by which we measured it. Such concepts as were successfully used for millennia to describe objects at certain scales are, we now know, concepts whose very existence is parasitic upon quantum interactions. There is simply no need, in light of decoherence especially, to puzzle any longer about the strange fact that nature seems to present itself more easily in certain ways over others (e.g., in a position basis rather than superposition of positions). Let us consider the question of observables for a moment, and on what grounds certain operators have come to be known as observables (that is, as belonging to the special class of operators corresponding to “measurable” physical properties). The same story applies here: why certain operators lend themselves more readily to physical interpretation over others is a contingent fact based on the nature of environmental interactions, and not due to some hidden law of nature that selects or prefers such operators. As discussed in previous chapters, the robustness of a particular basis—its stability with respect to uncontrollable environmental interaction—is determined by the dynamics of the interaction and the eﬀects of decoherence within that basis (i.e., to approximately diagonalize the einselected basis).10 Because certain bases (e.g., superposed bases) are extremely unstable

10Diagonalization of a basis means that the vectors in the system’s Hilbert space, when ori- ented according to that basis, are rendered approximately orthogonal. The degree of orthogonal- ity of the vectors in a system’s Hilbert space (remembering that these vectors represent possible states the system could occupy) in physical terms translates as the degree of distinguishability between vectors/states in that space. Perfect distinguishability corresponds to vectors that are orthogonal with respect to one another in the Hilbert space of the system, which in turn means that all the vectors are pure states. There is no interference between two vectors in a perfectly orthogonal space because the phase between them must be zero, since their relative angle is 90 degrees. In other words, in a perfectly diagonalized or orthogonal space, none of the vectors

203 under evolution within an environment, the probability of observing a system in such bases is extremely small. More technically, we set a certain class of linear algebraic operators called Hermitian operators to correspond to “classical variables” because they posses a speciﬁc mathematical property (self-adjointness) that guarantees that their action on a quantum system will return physical quantities—properties that can be interpreted and are distinguishable. By associating this class of operators with historically and contingently preferred variables like position, momentum, energy and time we adopted a way of doing quantum mechanics that determined which variable could be interpreted as physical or “real,” and which could not. To further demonstrate this point, consider the existence of current research programs in quantum mechanics that make use of non-Hermitian operators—that is, programs that take to be “observable” certain quantities that do not correspond to traditional variables. Though the experimental nature of the physics is much more diﬃcult, researchers have realized that entire regimes of quantum phenomena and systems at the microscale have remained unexplored simply because they were deemed unphysical long ago. Now that we understand the role of operators can be interpreted more broadly, entirely new arenas of investigation are open to us.11 Nature, so it seems, does not care to divide itself according to how it is per-

(representing possible states of the system) overlap with one another, and thus the interference between diﬀerent states is damped, and interference phenomena, upon measurement of this system, are not observable. 11An earlier example of this work includes Hatano (1999). More recent theoretical and experimental work with non-Hermitian operators can be found in Longhi (2010), Rotter (2009), Tolkunov and Privman (2004) and Znojil (2009).

204 ceived by creatures at our scale. The world is impervious to whether we choose Hermitian or non-Hermitian operators to act on quantum systems, how we divide particular subspaces into subsystems, or how we parse the universe’s degrees of freedom. While it is clear that we must divide the world somehow in order to make sense of it (recall the introductory discussion of chapter 3), what becomes even more clear through decoherence is the utter disregard the world has for our choices in these respects. The historical iteration between the development of quantum mechanics in terms of linear algebra and the experimental conﬁrmation of particular measurement outcomes in particular bases is complex. Nevertheless, with all of the history at our feet, we can now see why it was that certain operators appeared to us to be more convenient, and hence why certain operators came to be considered classical as opposed to others that were less convenient to measure. The nature of operators, and their applicability to particular systems to be measured, is entirely derivative. There are no truly “selected” or “favored” observables. There are no mysterious selection rules to be discovered, explaining why macroscopic objects tend to be measurable with greatest ease in the position basis and not some other basis: quantum dynamics is suﬃcient.

4.7 Conclusion

In this chapter I demonstrated that there can be no metaphysically robust meaning to deﬁnitions of classicality as not-quantum, and that even those (like Batterman and Bokulich) who would refrain from giving their views of classicality

205 metaphysical import are easily misconstrued as doing just that. In some cases a misunderstanding of the process of decoherence or invalid interpretations of quantum mechanics have been the reason that attempts to define classicality have gone amiss; such was the case in the accounts of Ballentine, Wiebe and Emerson, who all three claimed that decoherence was unnecessary for recovering “classical behavior.” In the final section, I briefly discussed an issue that has caused much confusion regarding the transition from quantum to classical, especially with respect to questions of explanatory heavy-lifting in, e.g., the seeming emergence of certain operators as well suited for most measurement processes. Again, the “classicality” of certain observables can be explained as arising from the dynamics of decoherence, and in this sense, too, the idea of classicality can make no contribution to our understanding of the way the world is. Furthermore, for those with worries about the sort of circularity invoked when one discusses the correspondence principle or the relation between classical and quantum theories more generally, we see that this circularity is not vicious, but rests on various contingent features. At some point, axioms must be postulated; my argument is that these axioms should be those of quantum mechanics, in terms of which all classical mechanics, behavior, states—even variables—can now be understood. There is no classical world. But more generally, there are no levels. All things are quantum mechanical, and decoherence not only plays an important role in explaining why the world appears at all classical or semiclassical, but furthermore it gives us the dynamical story of how such apparently classical or

206 semiclassical states of affairs come to be out of a fundamentally quantum world. This chapter has focused on the why aspect of this explanation, but again I stress the points made in particular in chapter 3: it is the specific details of the models of decoherence and the great success of these models that grants us the means and the confidence to assert that it is this dynamical story that explains the appearance of classicality and semiclassicality. I wish to emphasize as a final point that the purported explanations provided by decoherence not only cut metaphysical ice, but do so without invoking particular interpretations of quantum mechanics or getting caught up in the weeds of the measurement problem. This is by my lights the principle virtue of decoherence understood as a straightforward physical process, and as a process captured increasingly accurately by not-so-toy models.

207 CHAPTER 5

NONREDUCTIVE QUANTUM MONISM

5.1 Introduction

I have claimed—and hopefully to some extent already demonstrated—that good metaphysical work can be done by focusing on the process of decoherence and exploring its models in a detailed, philosophically-sensitive way sans the excess baggage that arrives in the form of the measurement problem or debates over interpretations of quantum mechanics. The slight awkwardness encountered at this point is that a majority of the extant philosophy literature on decoherence I naturally wish to analyze with respect to my thesis is written from within a particular interpretation of quantum mechanics or is an attempt to address the measurement problem in one of its many guises. This puts my thesis at cross- purposes with the aims of most others. For this reason, the remaining chapters focus less on philosophical accounts that make use of decoherence, and more on general philosophical accounts that are nevertheless aﬀected by what decoherence reveals. The question of interlevel relations has generated rather diﬀerent debates in philosophy of science versus philosophy more generally speaking. In the former,

208 the question has for nearly a century been largely a two-party debate between reduction and emergence (this century’s iteration of the debate being attributable to the rise of British Emergentism and reactions to it), though a third independent party headed by Nancy Cartwright (and to some extent, John Dupré) has recommended pluralism as an alternative view. But quite apart from which views one considers among the principle contenders, the goal of the debate has been the same—to understand how various entities (be they physical phenomena or “meta-entities” like laws, theories or the special sciences) interact with one another. Often the mode of analysis in these debates concerns arguments regarding what constitutes fundamentality, if and how causality or temporal ordering (diachronic vs. synchronic) plays a relational role, which view affords greater explanatory utility, and the like. Finding candidate exemplars of one’s interlevel relation of choice within the science itself also plays a substantial role in such debates. Philosophers of science concerned specifically with ontological interlevel relations (as opposed to theoretical or nomological interlevel relations, whose relata are “meta-entities”) want to know, for example, if the world consists of one level (or category, or kind) of phenomena or many; if the latter, then one wishes to know how levels (categories, kinds) stand in relation to one another. If the former, then a different suite of questions arises, more on which will be said presently. Metaphysicians who analyze interlevel relations typically do so within the context of debates on mereology and parthood, the constitution and composition of material beings, or the relationship between a property and the thing instantiat-

209 ing it. Because the language and goals of such debates are somewhat at odds with the language and goals of the philosophy of science interlevel relations debate, I tackle the latter here and address the former in the next chapter. Recall that the lesson of the previous chapter was to establish the non-existence of the classical world through quantum fundamentalism, which follows from entanglement and subsequent widespread decoherence. The chapter was also in part a ﬁrst attempt at understanding the scope of the nontological thesis to extend beyond just the denial of an autonomous classical realm (which is, after all, a rather mundane fact for most physicists), and to impress upon the reader that the force of the nontological thesis carries through to the core: there exist no levels tout court. Not only is there no metaphysically meaningful classical-quantum divide— there are no ontologically robust divisi to be discovered and deﬁned among any candidate “levels” at all. Such a level-less, division-less ontology ought to steer us toward a species of monism—that is, a quantum monism understood as follows: there is one thing, and it is a quantum thing (a sort of holism-monism), or as a one-category ontology: there are many things, but they are all of the quantum sort. A way to make sense of the former would be to take one’s cues from Spinoza and reify the only truly unitary wave function—that of the universe writ large—though there may be less speculative ways to endorse such holistic monism.1 Another valid way to understand quantum monism might be to allow that

1The topic of quantum holism has been approached various ways throughout the last few decades. A few of the important attempts at this approach include Esfeld (2001), Esfeld (2004), Healey (1991), Howard (1989) and Teller (1986).

210 there are many things—lots of stuﬀ—but only one way of being, i.e., quantumly. Perhaps this is the most fruitful way to go, as it leaves open the question of how one individuates subsystems from other systems, or how one divides the true quantumness of the world into analyzable, manipulable, understandable units. For these reasons (and more that will become evident shortly), it will be most helpful to assume a reading of quantum monism that is not of the holistic variety. As I am deeply convinced that metaphysics is (and perhaps always will be) underdetermined by our physics, the fact that there are several ways one might understand the quantum monistic position that seems to follow from the nontological thesis is in no way a detriment to that position. In fact, I count the richness and metaphysical creativity it encourages as a particular virtue of the account. I did, however, name it the nontological thesis on purpose. If one accepts it, then as a consequence certain views regarding interlevel relations cannot subsist without also resulting in serious problems. That is to say, certain ways of constructing the world and interpreting the relations therein can no longer be considered viable candidates for metaphysical views, given quantum monism on any plausible reading. Focusing on interlevel relations as manifest in the bipartite deliberation over reductionism vs. emergentism (with an independent section on supervenience), I ask whether these relational views can still be considered accurate, appropriate, or informative in a level-less world. As we shall see, neither reduction nor emergence seem to latch onto the physical world as described by decoherence in an interesting or useful way. Therefore, I suggest that these terms be discarded in favor of some better description of the world—perhaps one that

211 carries with it less extraneous philosophical baggage and that satisfies the original goals of the debate: to illumine the underlying relations between entities and domains of phenomena, and to explain their implications for ontology. By way of proviso, there is a matter of house-keeping to attend to before continuing with the body of the chapter. I must clarify that in speaking of things (object, entities, etc.) I mean to restrict myself to empirical things. I am concerned in this thesis primarily with electrons, tables, planets and the like. (I might call them integrable objects but for the unfortunate false connection this encourages between the physicist’s view of chaotic systems as non-integrable systems). Though metaphysicians will no doubt chide me for being too narrow at this juncture, I do not here introduce the canonical metaphysical topics of persons or minds. Thus when I say that “all being is quantum being” or “all stuff is quantum stuff,” this is to be read with the silent modifier within the domain of physical (or material, or empirical) entities. I am a substance dualist and so do not endorse an ontology of all and only quantum being—I believe there are immaterial ways of being, too. A materialist who takes the nontological thesis seriously may be forced to concede, for example, that his being a man and not a Barcalounger is merely a difference in quantum being, if that; so be it. I do not feel compelled to explore this perspective at present. Besides (and this is the main point), as C. U. Moulines has written, “The issue of analyzing reduction and emergence in the realm of things directly or indirectly connected to our sense experience is rich enough to deserve a treatment on its own” (Moulines 2006, p. 315). Indeed. Yet another proviso is in order before we begin in earnest, and it concerns how

212 I will deﬁne both reduction and emergence. This is a perennial problem. For as many authors as write on interlevel relations, there exist equally many versions of what the terms mean, and what they take as relata. In his contribution to a special issues of Synthese dedicated to emergence and reduction (2006, Volume 151), Moulines characterizes exactly the sense in which I shall understand reduction and emergence in what follows. He writes (p. 314):

The concepts of reduction and emergence clearly appear to have primarily an ontological meaning. They appear to refer to kinds of beings and/or kinds of relationships between kinds of beings. They are ontological categories. It might well be that the terms “reduction” and “emergence” also have other meanings or connotations, which are not ontological in any reasonable sense and which might be interesting for some reason or other. However, they are not primary ones... We are interested in ﬁnding out what it may mean to say that a particular kind of entity... is reducible to another, prima facie very diﬀerent kind of entity...; also, we are interested in the meaning of a phrase like “this kind of being... emerges from that other kind...”

In what follows, I look at typical accounts of reduction and emergence qua ontological theses. I conclude that although it might be possible to salvage some weakened account of reduction or possibly also of emergence, what decoherence and the nontological thesis highlight is that neither of these relational terms ﬁt very well the world as our best physics describes it. In fact, these terms have tended to introduce more complications than they are worth. Though I will not proclaim outright that reduction and emergence are dead, for the reasons listed immediately above and elsewhere in this thesis, I will venture to say that they are henceforth outmoded.

213 5.2 Reduction

It is common for accounts of reduction to begin with a discussion of Nagel’s syntactic approach to intertheory reduction (Nagel 1961, 2008). Of course, Nagel’s account has since been judged on the whole too simplistic in its assumptions regarding the construction of bridge principles and its characterization of the laws and scientific theories themselves. Hence Nagelian reduction has largely been forsaken. I already separate myself from much of this literature by restricting focus to ontology. Approaching the topic of reduction from within the framework of quantum monism is in a sense antecedent to claims about intertheoretic and perhaps even nomological reduction. Yet much (if not all) current discussion in philosophy of physics—whether the dimension of the philosophical question be within ontology or elsewhere—still takes as a starting point the question of intertheoretic relations. I briefly address my concerns with this approach before continuing. Presumably, the thinking underlying much of the present debate on interlevel relations involves an inference of the following sort: our scientific theories hand the world to us in a way that appears divided, and these theories enjoy high degrees of empirical verification. Hence, if the way the world is is in accordance with our best theories, and our best theories are leveled, the world must also be leveled. This argument not only gets one levels, but in addition one can look to the theories themselves to provide definitions or parameters for the levels and fill them with content in accordance with the respective domains of inquiry of the theories.

214 This mode of reasoning has the air of truth to it, in large part owing to its deference to physical theories—always a meritorious feature in philosophy of science arguments. But recall that one of the strategies of the previous chapter was to caution against the deep metaphysical errors one exposes oneself to when one allows arguments from pragmatism, fecundity, appearance or methodology to stipulate one’s ontology. Intertheoretic reduction cannot generate an appropriate ontological perspective, and this should be brought to mind whenever conversation within this arena suggests the conferral of ontic status to the claims made therein. A monistic world wherein all material being is done quantumly will not lend itself readily to a usual sense of reduction wherein the predicate “is reducible to” requires a subject and an object—two relata at the very least. This is immediately problematic for the Spinozean monist, for there are no ontological relata or relations on such a view. We have already bracketed this form of monism as the less helpful of the two available options, and now we can see why: relata-less reduction is a non-starter. Moulines—whose article I quoted at the end of the prior section—suggests that ontological reduction and emergence might be understood as questions about kinds as opposed to particulars (Moulines 2006, p. 315). According to him, applying relational terms to particulars is allowed, albeit indirectly, as a consequence of the necessarily prior attribution of particulars to their respective kinds. This take on reduction (and emergence) is also a nonstarter under quantum monism, understood in either way. Even if we wished to make a reduction claim in terms of particulars, those particulars—whether speciﬁc laws, behaviors or the like—must

215 belong to the same kind. There are no two kinds between which to prop the reduction relation. What of other sorts of reduction within a one-category ontology, then? There is some sense in which we might say there exist relata here, but whether or not these relata are capable of bearing the “reduces to” relation will be answered in the negative in the following chapter. Typical accounts of reduction require qualitative, not merely quantitative, distinguishability between the reducer and the reducee: the relata must differ in terms of fundamentality, or complexity, or explanatory power, or some combination of these. It would be hard to defend a variety of reduction that did not rely on such differences between relata, at least. Fundamentality as a comparative notion is nonsensical in a monistic framework, and so this mode of grounding reduction can already be dismissed. No feature of an entirely quantum world can legitimately be said to be more or less fundamental than another. Neither is explanatory power of particular interest (at least initially) for an ontic project such as this; furthermore, we have seen from our discussion of both Batterman and Bokulich in the previous chapter that relying on this method to define relata can get one into hot water rather quickly with the metaphysics, as it lends itself more easily to mischaracterization. Complexity falls short of the mark, too. Though explanatory reduction is often successfully couched in terms of complexity relations, we at least have cause to doubt that this sort of reduction has ontic grit owing to the fact that complexity of a given system can be seen, in light of decoherence, to be a contingent property of that system. As discussed in previous chapters, one thing decoherence ought

216 to cure us of is the notion that the world has its own principled ideas about how it should and can be subdivided. Instead, through the specific character of dynamical interplay among our chosen system of interest and its environment we learn that the world could have been otherwise. One can slice the world (consisting solely in quantum stuff) in myriad ways; the divisi we thought were handed to us by nature turn out to have no ontological significance. This is in effect what we do when we model decoherence: we impose on the world a division between some subsystem which we designate the system of interest and its environment, and through various mathematical means are able to treat these two systems like different things. This despite the fact that they are of a kind—in the deep sense that the interactions among their various degrees of freedom do not respect whatever divisions we impose upon them. The success of decoherence models, ironically enough, confirm the world’s indifference in this respect. Furthermore, all will concede that in order to analyze the world in any respect such divisions must be made. But to draw ontological conclusions from this pragmatic necessity is to misunderstand the very lesson of decoherence. Owing to ubiquitous entanglement and the quantum correlations between and among the universe’s stuff, any parsing of nature on our behalf can only ever be approximate, and for pragmatic purposes. It is not representative of what we know to be the case beyond appearance. Recall that validity of the Schrödinger equation is only maintained at the level of the entangled system as a whole. In order to understand and in some artificial sense isolate the dynamics of a subset of the entangled whole, we are forced to be clever with our linear algebra and

217 our engineering of controlled settings. But our ability to treat a subsystem as approximately independent from the rest must not obscure the fact that the two subsystems are not really two, but are an inextricably woven one. The only way we can discuss the dynamics of a subset is by cheating—by employing useful (nay— necessary) fictions about the independence of one subsystem from any other. As things now stand, the prospects do not look good for finding some appropriate qualitative difference between relata in our quantum-monistic world that might substantiate a general reduction claim. Let us look at reduction from a slightly different vantage point, then: is there a metaphysically important sense in which properties can be said to reduce to the quantum stuff in the world? The answer to this question is negative, and quite obviously so according to most philosopher of physics. The reason comes from quantum mechanics itself, so there’s no dodging it: an entangled system has properties (one of them being “is entangled”) that do not reduce. There is no sense in which it is correct to say that the quantum correlations manifest between (say) two previously-separable-but-now-entangled systems are reliant upon the constituents of the now-entangled system—if it is even appropriate to speak of “constituents” of an entangled “composite” to begin with. Even our language here is somewhat misleading, as it fails to capture precisely what is involved in the entanglement relation. What it does not involve is clear, and that is reduction. This prominent instance of the failure of ontological reduction, noted (among others) by Esfeld (2001, 2004), Howard (2003), Silberstein and McGeever (1999) and Teller (1986), is not the only example from physics one might cite in order

218 to argue against reductionism. That ontological reduction fails as an interlevel relation is also exemplified by the well-known problem of ergodicity—a meaningful thermodynamical property at the level of multi-particle systems that does not exist at the level of the individual molecules constituting the ergodic system. A last attempt at defining reduction generally might be made via part-whole reduction. Indeed, many treatments of ontological reductionism veer in this specific direction. I will have more to say about mereology in the next chapter; here it will suffice to say that things don’t look good for part and whole talk, and hence for part-whole reduction. One can immediately see that the example of the entangled system will run equally well as an argument against this species of reduction. But perhaps the types of reduction considered so far are too amorphous to count towards a solid refutation of this interlevel relation. Let us then supplement the above work by considering the highly-influential account of reduction proffered by Jaegwon Kim, and see what conclusions can be drawn with respect to it.

5.2.1 Kim’s functional reduction

In his 1998 and 1999, Kim argues for a particular sort of reduction called “functional reduction” that holds between properties, and is supposed to have the advantage over Nagelian reduction of not requiring bridge principles.2 Kim’s

2More precisely, Kim says “[Functional reduction] contrasts with Nagelian reduction via bridge laws in which [the higher-level property] only nomologically correlates with [the lower- level property in a particular structure] but remains distinct from it, and similarly for [the lower-level property in a diﬀerent structure], and so on” (Kim 1998, p. 110).

219 step-wise model for functional reduction is summarized nicely in Rueger (2006, p. 336) as follows:

1. Functionalize a higher-level property M in terms of a functional or causal role,

2. Find a “mechanism” P (the realizer of M)—that is, a property based itself on lower-level properties and that ﬁlls the causal role, and

3. Find a theory at the lower level explaining how P is able to ﬁll the causal role constitutive of M.

Kim uses the example of the transparency of water as a property to be reduced to lower-level properties under this algorithm by first rewriting the property “is transparent” in causal or functional terms, understanding the property as “the capacity of a substance to transmit light beams intact.”3 Step 2 then asks us to find a mechanism that realizes this functional property. In other words, we want to find a lower-level property P that gives a substance this capacity to transmit light beams intact, rendering “is transparent” semantically equivalent to “having the property P .” P , of course, will be microphysical—let’s even assume it is quantum mechanical, to be helpful. At this point ,it appears we have from Kim’s schema a type of reduction that tells us something of substance: it explains the relationship between an observed property (transparency) and how that observed property can be said to reduce,

3Kim (1998), pp. 24–26 and 98–100. See also Rueger (2006) pp. 336–337 and Batterman (2002) pp. 68–71.

220 without the mess of bridge principles, to the underlying microphysics. All of this is so far in accordance with quantum monism, is it not? And yet is it not an instance of successful ontological interlevel reduction? Prima facie things appear positive for Kim’s account. They appear even more positive if we consider a criticism of Rueger’s that actually strengthens Kim’s account for our purposes: Rueger points out that, strictly speaking, Kim’s functional reduction fails as a characterization of interlevel relations because it is in fact a type of intralevel reduction. The property P —which is the realizer of M (the property to be reduced)—is still a macroproperty, albeit expressed in microphysical terms. Thus both M and P are macroproperties, occupying the same ontological level (Rueger 2006, p 337). If Rueger’s analysis is correct, then Kim’s functional reduction will pass as acceptable under quantum monism at precisely the place most other accounts fail, as it gives us a sort of reduction that can take place within a single level. Whether or not Kim’s functional reduction qua intralevel reduction produces something metaphysically interesting will be answered when the next chapter analyzes the possibility of intralevel relations more generally. It is important at this point to be wary of an issue dealt with in the previous chapter. The temptation in reading Kim’s and others’s accounts to begin sliding towards an understanding of reduction as merely explanatory instead of ontic is strong, for (as mentioned already) explanatory reduction is an unquestionably useful and interesting relation in many cases. However, resisting this temptation and restricting our discussion to ontological reduction places Kim’s functional reductionism in a diﬀerent, weaker light regardless of one’s reading it as an inter-

221 or intralevel account. Kim gets into trouble when we attempt to read his macro/micro categorization of properties in a metaphysical and not merely epistemic or explanatorily way. For one, it has been argued at some length already that properties like “micro” and “macro” cannot hold metaphysical water. We can explain why such properties appear to belong to different size or energy classes (or whatever Kim has in mind when distinguishing various property levels from one another), but that is all the farther we can go. Perhaps explanatory strength is all the farther Kim intended to go with his definitions of macroproperties and so forth. Perhaps his point is simply that the underlying micro-realizers are sufficient in the sense that they contain all the attributes, or “powers” (though I shudder to use this oft-employed yet deeply mysterious term) found at the higher level, while the higher-level manifests only a subset of such “powers” found in its realizer(s). Kistler’s response to Rueger in effect argues this point on Kim’s behalf (Kistler, 2006). But this is no solution—it is a concession that the most we can (and should expect to) derive from functional reductionism is microphysical explanation. Kim’s account is furthermore subject to failure in virtue of entanglement, which by now we recognize to be the rule and not the exception for physical systems. Even if we attempt to understand the entanglement relation diachronically and without making the macro/micro distinction, and while remaining on a single ontic level, problems arise. Consider once again the example of the K-meson described in chapter 2. The property of entanglement in this case only refers to the resultant system (Klong or Kshort); although we know diachronically that the

222 entangled system is in some sense caused by previously defined separable K-meson and K-meson antiparticle “subsystems,” there is no sense in which, after interaction, entanglement can be reduced to those subsystems. In other words, the property of emergence, while in some sense brought into existence by certain “inputs” (two subsystems), nevertheless cannot be said to relate in any metaphysical way other than temporally to these inputs. And so our reduction scheme cannot get off the ground. To summarize what we’ve discovered regarding reduction so far, it appears that no matter how one chooses to define the relata, there is no strong sense in which reduction maps onto the world. One need only look to the example of an entangled system to see as much, though I have tried to push reduction a bit farther along by considering Kim’s more nuanced account. If some still remain unconvinced of the fairness of the preceding investigation, I stress once again the weightiness of two specific considerations. First, one must certainly concede that the consequences of decoherence and the resulting impossibility of metaphysically dividing nature into levels makes the notion of ontological reduction meaningless— or at the very least, renders it a weaker relation than is surely worth fighting for. It is unclear what we stand to gain with respect to the initial goals of this debate on such terms. Secondly, we have on hand physical examples, by no means representative of infrequent or rare situations, that illustrate the failure of ontic reduction. If the cases of ergodicity and entanglement are not enough, add to the arsenal the strange behavior of K mesons as described in chapter 2. There we saw that the existence

223 of the strange creatures Kshort and Klong is the consequence of a superposition forming between the K meson and its antiparticle state—a truly weird, thoroughly quantum mechanical state of aﬀairs that eludes usual relational descriptions, not least of which reduction. As Batterman so nicely put it: “perhaps reduction as it has been traditionally understood is just not the right sort of relation to be worrying about” (Batterman 2002, p. 61). I heartily agree.

5.3 Supervenience

A quick and easy description of supervenience will be sufficient for a demonstration of its failure by the physics. Roughly, one might characterize supervenience as “no change at one level without change at another.” This is a tricky definition to satisfy in a level-less ontology. Howard points out again that one need only go as far as an entangled pair to demonstrate the failure of supervenience (Howard 2003, pp. 9–13). Questions of the ontological status of the subsystems aside, it is clear that the entangled system’s properties fail to supervene on the properties of the subsystems that went into its creation. This is trivially true at the mathematical level, and empirically verified many times over by each violation of Bell’s inequalities for correlations of entangled pairs. Instead, the entanglement cascade (entanglement begetting decoherence begetting further entanglement) cannot be stopped in principle, in which case the failure of supervenience to successfully characterize interlevel relations is thorough-going. What if we take a slightly different definition of supervenience? Chalmers suggests we define it thus: “B-properties supervene on A-properties if no two

224 possible situations are identical with respect to their A-properties while differing in their B-properties” (Chalmers 2008, p. 412). Though for Chalmers (and most others interested in supervenience) only A-properties tend to be associated with physical properties, it is of course assumed, given the current context, that we will understand both A- and B-properties to be physical properties. It is a little harder to say whether entanglement will fail to supervene under this definition. To ease matters, Chalmers introduces two different ways one might understand supervenience thus defined as local or global. Local supervenience holds if A-properties of an individual determine the B-properties of that individual (ibid., pp. 412–413). Local supervenience will fail in cases where the B-property is “somehow context-dependent”—that is, when some object’s having the B-property depends “not only on the object’s physical constitution but also on its environment and its history” (ibid., pp. 412–413). Chalmers likely did not anticipate his choice of the word “environment” would suit present purposes so well (or so ironically), but it does help us see quite quickly why, given decoherence, local supervenience cannot hold in our exemple of an entangled system. Supposedly it is the entangled system whose properties ought to supervene on the more basic (read: temporally prior, not fundamentally prior, as we know already not to be the case) A-properties—here, the subsystems that, upon interaction, become entangled. Thus local supervenience will fail in any case where a system’s having some B-property (nonlocal correlations) depends on something over and above the physical constitution of the system (the subsystems themselves, which do not exhibit nonlocal correlations). This is really all that must

225 be said with respect to supervenience.

5.4 Emergence

Reduction and supervenience had the superficial convenience of being defin- able in terms of certain semantic or other formal relations. The same cannot be said of the roiling sea of proposed emergence relations. A first-pass list of properties that might signify a genuine case of ontological emergence includes such potentially ambiguous terms as novelty, irreducibility, unpredictability, un- explainability, transcendence, over-and-aboveness, and the like. Batterman calls “naked emergence” (2002, p. 118) this: when “an emergent property of a whole somehow ‘transcends’ the properties of the parts.” Yet another way to generally characterize emergence might be to say an emergent “thing” is one that is novel at some level analysis yet not derivable from lower levels. Obviously, the level- talk will be hard to escape in emergence; nevertheless, might some sense of this interlevel relation remain relevant? We can calm things a bit by excluding some accounts of emergence from the start. Given the arguments in the previous section, it is trivial to say that regardless of how one chooses to understand emergence, in order to succeed in this context it must entail not-reduction, or at a minimum be agnostic with respect to reduction, since we already know that that relation will not suffice to capture what seems to be going on “out there.” Thus we can divide the myriad senses of emergence that remain in play into two classes, somewhat following the taxonomy given by Howard (2003): definitions of emergence including (possibly among other

226 things) the straight-out denial of ontic reduction, or definitions of emergence that are silent on the question (and perhaps only explicitly deny weaker varieties of reduction). In our screening out any sense of emergence that doesn’t properly ignore or deny ontic reduction, we additionally want to ensure that part-whole reduction is ignored or denied as well, on much the same grounds (though, again, I refer the reader to the detailed discussion of the failure of mereology that takes place in chapter 6). In other words, we not only want to eliminate candidate relations that might in any way overlap with ontological reduction, but we will furthermore keep clear of emergence-like relations that rely on mereological distinctions. There is one final filter we must apply to our search for a passable emergence- like relation, and that is the ontology filter—i.e., we will only consider definitions of ontological emergence as opposed to, say, epistemological emergence. Silberstein and McGeever, in their excellent paper (Silberstein and McGeever 1999), provide a way to distinguish between epistemological emergence and ontological emergence. Epistemological emergence, they state, is an instance of emergence in which it is “merely an artifact of a particular model or formalism generated by macroscopic analysis, functional description or some other kind of ‘higher-level’ description or explanation” (ibid., p. 182). Or perhaps a better way to say this is as follows:

The ﬁrst problem [in the re-emergence of emergence talk] is whether any speciﬁc claim is merely an epistemological one regarding the in- eliminable nature (emergence) of some ‘higher-level’ description or explanation, or whether it is a robust ontological claim about the emergence of some novel feature of reality. If the former, then we should ask why the higher-level description or explanation is in-eliminable.

227 We should also ask, given that ontological reductionism is true of the system in question, how it is exactly that the higher-level description manages to have explanatory and/or predictive value. If the latter, what exactly is being alleged to emerged from what? For example, is it properties, property instances, entities, new laws or dynamics that are being allege to emerge? We must also ask how the emergence of the new feature is allege to occur, and what relations it bears to that from which it emerged. Claims for emergence are often unclear or confused about their answers to these questions. (Ibid., p. 185)

We might re-characterize (and thereby generalize) the given definition of ontological emergence by removing all references to causality, in this manner: ontologically emergent properties (and note—we are taking our relata to be properties here, in order to remain within the bounds of quantum monism) are properties belonging to a system (considered as a whole) that are consistent with, but do not reduce to or derive from, the properties of any of the parts or sub-set of the parts less than the whole. Thus Silberstein and McGeever want to call ontological emergence the failure of part-whole reductionism. Perhaps red flags are already being raised with respect to the language of parts and wholes in this definition. There isn’t really any way to talk about ontological emergence without invoking mereology—that is, without talking about some system in terms of its parts, and it itself as a whole. But this is further problematic in the sense that ontological emergence so defined entails the failure of part-whole reductionism, which is a notion also crucial to many mereological accounts. Part-whole reductionism, as Silberstein and McGeever summarize, is simply the notion that all wholes are entirely reducible to their parts, and those parts are reducible to their parts, until one gets to the basic parts

228 or “simples.” Our by-now familiar example of entanglement already answers to this. One attempt to understand emergence that might save us from parthood problems is found in the work of Batterman already described in chapter 4 (Batterman 2002). Near the end of the book, Batterman claims that one of the case studies he has used to demonstrate his “third theory” view—the case of universal properties in rainbows—is a case of emergence that avoids the mereological aspect of the received view of emergence (ibid., p. 113). This is achieved by adopting a definition of emergence that changes in what sense the emergent entity is considered to be novel. There is much Batterman says that fits well within our ontological framework and within quantum monism specifically. For instance, in explicating his account of emergence, he notes that typical philosophical accounts are couched in terms of a hierarchy of levels, but that “ ‘Level’ talk here [in his example of caustic optics and rainbows] seems completely inappropriate” (ibid., p. 116). He suggests instead that the necessary distinction between emergent properties and their base be couched in terms of scale. These seems satisfactory, for when we bring our entangled system back under consideration, we know given the nontological thesis that level talk won’t suffice for our purposes either, whereas scale might do the trick. There is surely some ontological grit to the association of the emergent property “entanglement” as existing at a different scale than the subsystem base from which it emerged, isn’t there? The problem one encounters with this type of emergence (and definitions of

229 emergence that involve mereological distinctions) is only brought to light by decoherence, and it is as follows. While it seems correct to analyze an entangled system in terms of emergent properties or with respect to different scales, our ability to use this characterization meaningfully rests on the assumption that there is a way of stopping our analysis at some point, roping off a particular system, and then applying the property “is emergent” to the relevant feature(s). Decoher- ence shows us that the world pays no heed to our choosing to stop at any point and apply a description of some sort or other. If we try to place weight on our ability to isolate a part of the world in order to label it emergent, we are fighting against the very phenomenon that gave us grounds to claim emergence in the first place—widespread entanglement. If, as the physics seems to show, there is no way to stop the cascade of entangled systems, then at what level of analysis do we meaningfully apply the term “emergence”? To what properties does it correspond, if those properties appearing to belong to an entangled system (e.g., the correlation between measured pairs of entangled spin particles) only in virtue of decoherence? Thus the interlevel relation of emergence, whether one understands it as the denial of part-whole reduction àla Silberstein and McGeever or in terms of scale àla Batterman, doesn’t seem to do much good work in the debate, either.

5.5 Conclusion

We began by recognizing that the consequence of the nontological thesis (understood in both ways: as the nonexistence of the classical world and the nonexistence of any levels) was a sort of quantum monism. The two most prominent

230 candidates for describing interlevel relations—reduction and emergence—both fail to capture the known dynamics structure of this quantum-monisitc world. Super- venience isn’t going to win the day, either. Thus we are left with a one-category ontology comprised of quanta whose relations to one another, in the sense that there are “one anothers” to have relations with—are not adequately characterized by reduction or emergence. Hence nonreductive quantum monism. Now we turn to an exploration of nonreductive quantum monism with respect to more traditional metaphysics topics: parts and wholes relations (mereology) and the ontic status of properties. I argue that decoherence makes it metaphysically vacuous to talk about intralevel relations as well as interlevel ones.

231 CHAPTER 6

WHY WE CAN’T HAVE NICE THINGS

We shall seek to construct a metaphysics of matter which shall make the gulf between physics and perception as small, and the inferences involved in the causal theory of perception as little dubious, as possible. We do not want the percept to appear mysteriously at the end of a causal chain composed of events of a totally diﬀerent nature; if we can construct a theory of the physical world which makes its events continuous with perception, we have improved the metaphysical status of physics, even if we cannot prove more than that our theory is possible.

Russell, The Analysis of Matter (p. 275)

6.1 Introduction

Authors Ladyman, Ross, Spurrett and Collier begin their recent book Every Thing Must Go with an admittedly polemical chapter that is, in eﬀect, a wrist- slapping of a good many metaphysical views of the 20th century due to their misuse of science (Ladyman et al. 2007). They chide modern and contemporary philosophers for embracing views like atomism or locality in some fundamental way, despite the well-known conﬂict between such beliefs and our best physical theories. The authors divide current received views about the fundamental

232 makeup of the world into two categories, neither of which is in accord with modern physics: (i) atomistic ontologies based on simples (partless parts) and (ii) “gunk” ontologies built from inﬁnitely divisible parts (ibid., p. 20). Ladyman et al. also disparage what they call metaphysical anthropology—metaphysical theses constructed entirely in ignorance of modern physics and supported on (familiar metaphysical) grounds of intuition, simplicity, fecundity and the like. None of these canonical metaphysical virtues are guaranteed to map onto the way the world is independent of human experience. In contrast, the authors state (ibid., p. 5) “We, however, are interested in objective truth rather than philosophical anthropology,” and continue to say that they take particular issue with metaphysicians who claim the former motivation yet carry out the latter by ignoring or misunderstanding the physics. I will not rehearse their (perhaps overly strong) grievances with current analytic metaphysics here, as my aim is not to argue for scientism or against analytic metaphysics writ large. Nevertheless, for both philosophers of science and metaphysicians, the lessons of decoherence will alter the metaphysical terrain. We have already seen in what ways old interlevel relational terms like emergence, reduction and supervenience prove inadequate for describing physical systems in the wild. The purpose of this chapter is to begin thinking about how to construct a world that is relationally described by nonreductive quantum monism: what do the primary ontological constituents of such a world look like? Or rather, in keeping with the underdetermination of metaphysics by physics, what can the world’s basic constituents not be? And given the results of these inquiries, what are the

233 consequences for live debates about (for example) compositional mereology, the property/object distinction or the nature of properties? Before getting into these questions it is worth pausing to consider just what relationship is envisioned between metaphysics and the more philosophy of science- flavored work done in the previous chapter. There are some camps of the latter subfield that might wonder where metaphysics comes into it, if it should come in at all. But I understand the relationship between the two fields to be mutually informative. Instead of focusing primarily on metaphysical accounts that are inherently antithetical to philosophy of science (in the vein of Ladyman et al.), I choose to understand the basic project of ontology as one that stands to gain from collaboration between metaphysicians and philosophers of science. A metaphysician who is not only sympathetic to but in fact heralds such a consilient view is L. A. Paul. Paul looks at the relationship between physics and philosophy in the following light in her manuscript “Building the World from Fundamental Constituents” (p. 9):1

The separate-but-connected relationship between science and metaphysics means that even though metaphysicians have their own job to do, they need to pay attention to the science. Metaphysics is constrained by physics, but not determined by physics.

This precisely characterizes the way I envision the interaction between metaphysics and science, with philosophers of science standing at the nexus and with all sides conceding epistemic humility due to underdetermination. That said, while the previous chapter discussed the nontological thesis from a philosophy of science

1Obtained from the author with permission. Manuscript dated 3 June 2010.

234 perspective focusing on interlevel relations, the present chapter will address the slightly diﬀerent question of intralevel relations, and will do so in a way more germane to metaphysicians. Before proceeding with this task, however, some clariﬁcation regarding terminology and a slight reorientation regarding method must be addressed.

6.1.1 A note about terminology

The first item of note concerns the position of nonreductive quantum monism arrived at in the previous chapter. Though the nonreductive aspect of this position translates across disciplines rather easily (as do the conclusions regarding the inadequacies of ontic reduction, emergence and supervience), the “monism” aspect of this position must undergo slight reinterpretation. In the context of philosophy of science, terms like monism, or what it means to have a one-category ontology, are already understood to refer primarily to the realm of empirical objects, or at least the realm of entities central to scientific theories. Metaphysicians approach the categorical hierarchy of their ontologies in myriad ways. Depending on the account, emphases are made on grounds of distinctions like more versus less fundamental, abstract versus concrete, particular versus universal, properties as tropes or modes or particulars versus properties as universals, and so on. In the previous chapter I remarked that for as many authors as wrote on the topic of emergence and reduction, there were just as many (if not more) methods of parsing the conceptual space and defining one’s terms. So too ontological categories in contemporary analytic metaphysics. I will adopt one

235 particular understanding to work with in what follows, and I choose it on the basis that I take it to be the most illuminating regarding the quest to understand the metaphysical consequences of the nontological thesis. Perhaps the nontological thesis has ramiﬁcations for other methods of slicing the landscape, but plenty can be said by employing just the particular scheme I adopt. In keeping with the focus of this thesis on entities qua empirical or physical objects, I will bring the nontological thesis to bear on two classes of category-ﬁllers: quantitative and qualitative “things.” Examples of usual metaphysical objects that might be classed as quantitative include property tropes or modes, physical particulars, material parts/wholes or spatiotemporal parts/wholes, and concrete objects/individuals. Examples of usual metaphysical objects falling into the qualitative class are properties understood as universals or universals more generally, and abstract parts/wholes. Quiddities and haecceities might even be said to inhabit this latter class. Understood in this manner, nonreductive quantum monism (as it has begun to be established) applies solely to the metaphysical class of quantitative entities. I do not extend the thesis to include things like mental states or ectoplasm in my characterization of the world as quantum-monistic for the same reasons I chose to engage primarily with empirical entities: there is work aplenty to be done in this corner before venturing into the realm of philosophy of mind or more ambitious metaphysical categorical schema. Only in so constraining the purview of this chapter will the largely philosophy-of-science issues dealt with thus far translate into tractable metaphysical language.

236 6.1.2 A note about method

Much of the motivation for accepting or denying a particular metaphysical view and the merits upon which the view stands or falls are distinctly metaphysical. This is not surprising. However, it is somewhat orthogonal to the usual criteria applied to theses in philosophy of physics (which are often science-related, unsurprisingly). Since this thesis as a whole was written under the conviction that the relationship between metaphysics (or philosophy more generally) and science is as described in the above quote from Paul, I have started from the physics (presented as philosophically neutrally yet sensitively as possible) and take it as the ultimate constraint on possible philosophical positions. As such, there is a very deep sense in which I don’t care whether or not certain philosophical virtues are satisﬁed or vices exacerbated by the proposed view, constructed as it was on physical foundations. For example, I am not concerned in what follows with conceptual simplicity or intuition-matching (or even perception-matching, to some extent). Perhaps some will decide that this is too uncouth an approach to metaphysical argumentation. However, since this method of doing philosophy takes as its input all the relevant empirical evidence, there is no denying the strength such a view ought to have when compared to, say, a metaphysical thesis constructed purely on traditional metaphysical values. At the same time, a metaphysical position that takes as its input various empirical results from science can verge on the positivistic, and I hope to avoid this by remaining committed to metaphysical underdetermination by the physics. Indeed, much of the physics revealed to us through studies of

237 decoherence is strange enough to elicit multiple metaphysical readings of a new and unintuitive sort while at the same time decidedly eliminating other, perhaps more traditional, metaphysical readings. But sheer strangeness in the physics is nothing to shirk; indeed, I count it a virtue of decoherence that what it seems to say is the case (or, rather, what it seems to say is not the case) in the physical realm is too far removed from intuition to have been borne of anyone’s head. There is no pre-analyticity here, and this gives it the ring of truth to my ears. One prominent criterion for adjudicating the validity of a metaphysical thesis that has been a particular favorite scapegoat for the anti-metaphysicians (many of whom are philosophers of science) is the logical strength or cohesiveness of a view: when properly explicated in formal logical terms, does the view get us all the things we should want or expect from our metaphysics? As the anti- metaphysicians have argued, one’s ability to rewrite a view in the language of logic and deduce consequences therefrom takes for granted the historical fact that our articulation of diﬀerent logical schemata was contingent upon an entirely classical worldview. This worldview substantiated the false impression that the external world conformed to our perception of it as containing well deﬁned individuals with local properties. At the very least, based on the physics I have presented in this thesis there ought to be reasonable cause to doubt the appropriateness of such intuition-based notions in contemporary ontology if one wishes to construct an ontology that maps onto the world as our best physical theory understands it, and not merely as it strikes us from our particular vantage point as humans (the previously mentioned

238 approach that Ladyman et al. have baptized “metaphysical anthropology”). These two notes about transitioning from the philosophy of science to a more metaphysics-centered discussion being accomplished, the remainder of the chapter is structured as follows. I begin by focusing solely on the quantitative aspects of a potential ontology, and ask what the nontological thesis disallows or allows along these lines. I explore this question by introducing a thought experiment and a real experiment, both of which aim at explicating the unintuitive consequences of a decohered world. The answer comes in the form of a corollary to the nontological thesis that has considerable implications regarding the possibility of (quantitative) compositional mereology. I argue that several popular mereological schemes must be eliminated in light of the physics. The next section adopts a more optimistic stance towards mereology by considering the potential for qualitative composition. In particular, I focus on the traditional view of properties qua abstract entities and/or universals, and investigate (using yet another thought experiment) the dynamics of property observation. I conclude that one possible attempt at a qualitative single-category ontology described by L. A. Paul is potentially consistent with the physics presented in this thesis, and suggest further development along these lines—especially concerning the property/object debate.

6.2 Why we can’t have nice things

What we have seen from the physics so far is this: though the ontic status of interference terms among superpositions in a system’s Hilbert space remains

239 open, the system can nevertheless not be interpreted to exist in a definite state, but only an approximately (and apparently) definite state, with all the usual properties attributable to non-superposed states. Recall the example of the Wigner representation of quantum Brownian motion presented in chapter 3. This particular representation models the central system in phase space as two major peaks representing two vector states of the system that are (nearly) perfectly out of phase with respect to one another. The interference terms—the mathematical characterization of phase relations existing between these two major components of the superposition (the perfectly out of phase states)—are suppressed upon interaction with even a weakly interacting environment, and this is the process of decoherence. And though the suppression in time of these interference terms appears completely destructive, we know that the interference terms always exist and in fact (by Poincarérecurrence) will at some future time resume their initial values. They do not disappear, and we are not ever left with a truly “classical” situation. The question of just how or in what sense these interference terms remain in existence is a mystery, and any attempt to move beyond this point immediately ushers one into the arena of metaphysical supposition—into interpretations of quantum mechanics. For instance, one might adopt a robust view of existence with respect to these possible states and believe that the system really does occupy all these states simultaneously; the fact that we simply do not observe the system to “really” occupy them is because of contingencies like the method of observation, the composition of the environment, or what bit of the quantum-monistic world

240 we’ve deﬁned to be the initial system of interest. No material thing exists in a truly distinct, individual, isolated way. While it might not be the metaphysical consequence of widespread decoherence that (for instance) the chair over there actually exists as a smeared-out ensemble of all possible states, it is necessarily the case that the chair cannot be said to be a nice thing that really, actually, according to our best physics, exists in one spatial- temporal location or with deﬁnite-valued degrees of freedom. How a thing is a thing—how that thing exists—is a great deal more mysterious than we imagined it to be.2 Perhaps a worry that arises at this point concerns the ontic status attributable to unmeasurable things: is it really fair to postulate existence if we can never measure the things as existing, and instead must rely on the strength of such claims as extracted from those aspects of the theory which are testable? Though I grant it feels strange to adopt existence claims indirectly (so to speak), to respond to this unease by forbidding inferences from empirical consequences strikes me as too harsh. It is no new thing in physics to deal with increasingly unobservable or unmeasurable entities; though it is indeed a place to be extra cautious with respect to metaphysical or other philosophical claims, the sheer volume of empirical data

2One possible way of characterizing this strange sense in which things still are things in certain limited contexts is to follow the suggestions given in section 4 of Brading and Skiles (ming). There the authors suggest that precise deﬁnitions of thinghood committing one to individuality (among other usual criteria susceptible to criticism from physics) can be avoided by letting countability be a suﬃcient condition. Though this view strikes me as more plausible on many counts than most other recent treatments of haecceities, it still cannot avoid the particular problems arising from decoherence: in what sense can countability be considered a metaphysically salutary notion if our ability to count is itself moored to spatial-temporal features of physical objects that, by decoherence, are revealed to be merely apparent?

241 undergirding the models of decoherence as applied to physical, chemical, and biological systems lends strength to this case. Furthermore, there are measurable manifestations of decoherence and entanglement, some of which were mentioned in previous chapters. They include phenomena like superfluidity and the behavior of Josephson Junctions during superconduction, or the quantum behavior of massive Weber bars. If one still wants to deny that there’s something metaphysically interesting and strange going on here, then the onus is on the dissenter to explain the success of decoherence models in some other way, arguing against the nature of what is usually considered ample evidence from the perspective of the physicists testing such theories. Finally, taking issue with measurements that are merely “indirect” is a naive reaction (with positivist flavor) that sends one down a slippery slope of defining what constitutes indirect versus direct evidence. Not only do these worries strike me as outmoded, I cannot fully treat such issues here without departing significantly from the principle aim of this work. Though prima facie it is hard to see that there is any real metaphysical space to work with between what might be allowed and what is definitely not allowed given the physics, my worry here is not to catalogue all remaining possibilities; I leave this project to posterity. It is enough for present purposes to examine more fully the consequences of what is not metaphysically allowed (yet another sense in which the project is nontological). I believe that there are a few concrete things to be said about possible metaphysical projects in light of all this. As a first matter of business, I want to state what I consider to be a corollary to the nontological thesis. We have established that, ontologically speaking, there

242 are no levels. There is no metaphysically robust way to divide the physical world into levels. Below I oﬀer several arguments that this thesis leads inevitably to the following corollary: ontologically speaking, there are no quantitative parts or wholes. If we cannot divide the world in one way, it seems to follow we cannot divide the world at all. Let us begin to investigate the meaning of this corollary by running a thought experiment.

6.2.1 A thought experiment to illustrate this point

Assume a world identical to this one but for the size of metaphysicians, who’ve been shrunk to a size so small that their faculties are capable of witnessing, in real-time, the processes of decoherence for objects considered medium-sized dry goods in our usual world. In this parallel world, the tiny metaphysicians are placed in an advanced physics lab wherein there is an apparatus (say) that can bring into existence instantaneously an 8-ounce diner-style coffee mug, much like the one presently sitting on my desk. In the real world (where the coffee cup already exists and is sitting on my desk), metaphysicians might assign my cup the property of being singly spatio- temporally located. Specifically, it has the property at time t0 of having its center of mass being located at coordinates (x0, y0, z0) with respect to some origin. Then, given the dimensions of the cup, we should be able to say that it has the property at t0 of occupying some volume of three-dimensional space. But what do we mean when we say the cup occupies a certain volume of space at a particular time? The mereologist would say something along the lines of all the parts of the cup being

243 contained within that volume, and no parts of the cup not being contained within that volume. But we’ve already run into two problems. The first is that, even if we could at time t0 pretend to separate those subatomic particles that compose the cup from those that do not, immediately after time t0 those particles would become entangled with their environment (non-cup parts), and would no longer be describable in local terms. The second problem is, again, connected to the size and energy scale at which we just happen to be doing metaphysics. Let us go to our possible world where metaphysicians are the size of sub-atomic particles, and are thus capable of observing decoherence in real-time. Even if we start by assuming that the coffee cup will pop into existence at time t0 (and therefore that it will be legitimate at that time to describe the coffee cup as an isolated system, as it has not previously interacted with any other system), what do we observe (say, in the position basis) as the systems constituting the cup evolve? We will see that the particles are smeared out, and that the volume of space in which our “cup” is meant to exist is smeared out beyond the borders of that volume. But as the particles interact with one another and with the external environment, the superposition of position states that we may have observed at times immediately after the cup’s appearance will fade until our observation is of the cup standing in relation to one another such that the cup appears to be spatially-temporally located with values (x0, y0, z0, t0). In this short thought experiment I have assumed a particular physical interpretation of the interference terms (namely, that they appear to our tiny metaphysi-

244 cians to be smeared out). To be perfectly clear, the physics does not stipulate just what the world would really look like at such a scale (maybe states ﬂicker in and out, or pulse, or something truly weird) except that it cannot appear localized in position. What ’s the important realization here? The result is that we realize it is in large part an artifact of the size of humans and the time scale on which we are able to observe physical processes that accounts for the appearance of things that are nice; we know this is not the full truth, however.

6.2.2 A real experiment to illustrate this point

Consider one version of many famous Bell-type experiments in which an entangled pair (usually electrons or photons) is created and the correlations between various properties of the entangled subsystems are subsequently deduced from repeated measurements on values like spin orientation or polarization. These experiments are part of the canon of empirical veriﬁcations of quantum mechanics, for in classical scenarios one expects certain statistical correlation between measured values and quantum mechanics predicts certain other statistical correlations, and it is the latter that have been repeatedly observed. For present purposes, let us consider a pair of entangled photons such as are created instantaneously via the process of spontaneous parametric down- conversion a process initiated by a parent photon passing through a nonlinear crystal lattice and “splitting” into two maximally entangled daughter photons. The daughter photons are then sent in opposite directions by a clever arrangement of mirrors, and are ultimately measured (with respect to, say, their polarization

245 so in the polarization basis) by photo-detectors situated some distance apart from each other and from the origin of the daughter photons. The technology of the photo-detectors and the laser emitting the parent photon are such that we can tune the laser to very low pulse setting (so it emits single photons at a time) and then can say with some confidence, and using data measured at identical times from both detectors, that we are in fact measuring a certain property—the polarization—of each daughter photon. Such experiments have been done on many occasions since the 1980s; more recently, these Bell-type experiments have even been performed successfully in undergraduate laboratories.3 A metaphysician’s analysis of such an experiment might be as follows. She might say, “Here we have apparently taken independent readings measuring a particular quantitative property, and have with confidence assigned the property (say, of being linearly polarized at an angle of 45 degrees off-axis in the positive direction) to one of an entangled pair. Given what the previous chapter tells us about the nature of entangled systems, we shouldn’t be able to assign individuality or other, less metaphysical properties like polarization, to part of an entangled whole. What’s going on here? I know that the wave function describing the two photons cannot be separated in terms of one photon or the other. Yet, can’t I still do my metaphysics of quantitative parts and wholes based on the fact that in this experiment we can nevertheless evaluate the entangled system in terms of its parts?”

3See Branning et al. (2009), Carlson et al. (2006), Dehlinger and Mitchell (2002a), Dehlinger and Mitchell (2002b), Galvez et al. (2005) and Thorn et al. (2004).

246 So it would seem. But my response to the metaphysician is to say that it merely seems this way to her in virtue of decoherence—not even epistemic limitations of our apparatus or anything like that, but based on the way the particles interact with their environment. The first thing to note that the only reason such experiments work is that the timeframe in which one can collected polarization data after the entangled pair is created is extremely small and highly controlled precisely so that decoherence effects will not destroy the relationship “between” the daughter photons. In other words, decoherence is actively avoided as long as possible in order to achieve measurement outcomes. Furthermore, these experiments only work if the apparatus is shielded heavily from its environment; the daughter photons must be protected from stray photons in the rest of the laboratory, or from the laser itself, and so forth. Thus the metaphysician must first realize that she is witnessing an intensely controlled, highly unnatural creation and subsequent measurement of entangled pairs that has been isolated almost entirely effectively from the environment and therefore prevented from being decohered in as much as is possible. We are already a long way from physics in the rough, and this ought to factor into the metaphysical conclusions we draw. All this aside, decoherence will occur whether or not we try to prevent it; as described in earlier chapters, no matter how well we can isolate our system from environmental noise, decoherence will occur within the shielding due to the un- avoidable presence of the apparatus itself or the intrinsic dynamics of the daughter photons. The photons appear to have individual polarization values because they are entangled in that basis, thus superpositions of polarization are unstable states

247 under evolution, and thus the probability of our measuring the pair to exhibit such a state—what that might even look like—is unclear. As for the apparent individuality of the photons we are tempted to believe in owing to the spatial separation of the two detectors and the fact that they give simultaneous readings whose values correspond to the predictions of quantum mechanics (both serving as empirical evidence that it is the same pair of entangled photons being measured at that time)—this, too, is due to our choosing to measure the entangled system in a particular basis and not others. If we could somehow measure the photons in, say, the position basis, then our measurements would yield highly nonclassical state of affairs of the photons being spread out or smeared initially after their creation, but as they fly toward the detectors, they become entangled with the apparatus itself, which leads to decoherence in the position basis, and eventually we would no longer observe smearing in the position basis. Even in this case we know that the superposition of position terms are still there in the description of the entangled photons, but that the effect of the photon interaction with the rest of the apparatus is to diminish those values, which translates into a prodigiously low probability of ever measuring our photons as smeared out in the position basis. I give one more example, and this time at a different scale, to persuade the metaphysician that her intuitions regarding mereological analysis seem ill suited to describe the dynamics given to us by decoherence models. This is the macroscopic case describe in chapter 4 of Hyperion, the moon of Saturn that ought to be tumbling chaotically but that is never observed in a nonclassical states. Again, were our metaphysician to inhabit a world in which Hyperion could be consid-

248 ered truly isolated, and in which the physical object Hyperion could be said to supervene on its physical parts—in such a counterfactual world, it would not be a mistake to say that the spatial position at time t = t0 of a particular crater on the moon is (x0, y0, z0, ), or that Hyperion’s baked-potato-like shape supervenes on the relationship among the elements composing it, or any other quantifiable part assignable to Hyperion that somehow depends on Hyperion’s existing within a particular region of space at a time. However, we know that this is not the sort of world we inhabit; were we able to feasibly measure Hyperion in a basis of superposition of position, for instance, we would observe the moon (and its “parts”) to exist in a highly nonclassical, nonlocal way. Again, how it would actually appear to us is not described by quantum theory, except to say that it could not be described as existing in a well defined region of spacetime. Consequently, how any of Hyperion’s properties relate or are grounded in the physical nature of Hyperion or its “parts” also appears ill posed. Even in a normal basis like the positions basis we know that it is due to decoherence that weird superpositions of states describing the co-location of Hyperion, for example, exist in Hyperion’s Hilbert space; yet, we will never measure Hyperion in such states because the suppression due to decoherence is so effective that we would need to measure Hyperion continually for many lifetimes of the universe in order to witness it occupying such a state. It is due to decoherence suppressing these strange non-separable, co-located states of Hyperion in most bases we might choose to measure it in that gives rise to the appearance of Hyperion as an object that can be described in terms of its parts. But this is not the physical truth of

249 how Hyperion exists, so to speak. It is only an extremely effective approximation that ignores a great deal of information we have about this system. One might still play the mereology game and describe Hyperion or a pair of entangled photons in terms of their “parts,” but at a cost; I am of the opinion that this cost is so great as to render a metaphysics thus carried out to be an inaccurate description of the world. It would be a metaphysics of approximations in which one’s ability to do the metaphysics in terms of those approximations—that is, to make those approximations in the first place—is owing to the very aspect of the world one takes for granted in order to get this metaphysical story off the ground.

6.3 Consequences for quantitative mereology

Mereology is the attempt to understand parthood, and the relation of parts to wholes. As such, it obviously assumes the correctness of describing the world in terms of parts and wholes. However, as we have seen it is entanglement and decoherence—both uniquely quantum processes—that give rise to the stability in systems that render them (apparently) describable in terms of quantitative parts and wholes. Recall the discussion of the conceptual nature of decoherence oﬀered in chapter 2 (before meddling with formalism as a way to decompose the world into something analyzable). It was there discussed just how widespread decoherence processes are—that tubiquitous and uncontrollable entanglement ex laboratorium beget decoherence, which in turn creates further entanglement, and so on. In this way, the CS initially studied with respect to its own environment becomes a new

250 system, CS-E, which in turn becomes entangled with respect to its environment, ad infinitum. The result is a cascade—an inverse Matryoshka doll effect, if you will—of entangled, and subsequently decohered, systems and environments. This means there are no quantitative parts and wholes divisions to be made at the level of ontology, at least not for any whole less than the universe entire. Even were a part/whole relation to come into existence via spontaneous creation of an antiparticle (itself further divisible into quarks) in some remote region of the universe, the particle in question would immediately become entangled with its environment, be it interstellar dust, quasi-vacuum fluctuations or even the all-pervasive cosmic microwave background radiation. These interactions are sufficient to induce decoherence, and to do so exceedingly rapidly compared to other relevant physical processes. One of the most popular (quantitative) mereological views currently on offer is that of spatiotemporal mereological composition. This view holds that the world at its most fundamental level is made of spacetime points, and that all there is in the world supervenes onto some well defined set of spacetime points. This sort of ontology is the kind Ladyman et al. classify as atomistic, in the sense that no part (occupying its respective spatiotemporal region) overlaps with any other. The smallest bits on such a view are still supposedly those given by our latest physics, yet are still assumed to supervene locally. This is a rather naive view, for one might at least want to be able to talk about vague objects in one’s ontology—things like clouds, whose parts are fuzzy or indeterminate. But even without complicating the picture with clouds and the like, this view contradicts

251 the physics. The problem decoherence and quantum mechanics more generally poses for quantitative compositional mereology is deeper than the fuzziness or indeterminacy or vagueness issues mereologists acknowledge when they try to understand and systematize parthood relations for objects like clouds or properties like bald- ness. On many views, fuzziness, indeterminacy and/or vagueness are de dicto, not de re. To put it in more philosophy-of-science friendly terms, the vagueness mereologists have in mind when trying to ascribe certain water vapor molecules to the cloud’s constituent parts is (at least as far as I am aware) not the same sort of vagueness I am claiming comes from quantum mechanics. In the instance of the cloud, for example, the spatiotemporal compositional mereologist will assume that there exists a fact of the matter about the spatiotemporal location of all the water vapor molecules in the sky and it is some issue with our language—with the word “cloud” failing to refer appropriately—that causes the mereological fuzziness. What is being claimed here is that even if one wanted to precisify one’s language in order to be rid of semantic vagueness (e.g., by deﬁning clouds as objects with a particular density of water vapor molecules or something along those lines), and even if one were able to evaluate the density of all the water vapor molecules in the sky and so discern whether or not a particular unit volume is part of a cloud or not, the fact of the matter is that each and every water vapor molecule is entangled to its environment, and therefore the correct description of the molecule necessarily involves nonlocal physical features. This is a sort of vagueness that runs to the core, for no matter how detailed the semantics or how precise the

252 deﬁnition of what constitutes a physical part, in virtue of naming anything a part one has already fallen prey to the illusion that there exist local descriptions of physical entities. We know that this is false. Let us examine in more detail some of the tenets of a broader class of mereological views—one that is not restricted to the admittedly naive spatiotemporal compositional view, but one that nevertheless makes little or no progress for mereology.

6.3.1 Compositional mereology

One of today’s leading metaphysicians, Ted Sider, recently published a paper simply called “Parthood” (Sider, 2007). It only takes Sider a few sentences before espousing a contradiction with modern physics: he assumes the very sort of spatiotemporal locality of parthood just discussed: he states that at the very heart of mereology is the assumption that a given object o can be considered to be exactly located in some spacetime region R. One of the aims of this paper is to elucidate the strength of the composition as identity relation. This mereological relation states that wholes are nothing over and above their parts, and it is central to many mereological investigations. If applied to quantitative or physical understandings of parthood, we know already from the example of entangled pairs that this relation fails: an entangled system is over and above its parts, having distinct properties and physical interpretation. Since all uncontrolled systems are entangled (or become so incredibly rapidly), composition fails as an identity relation for physical systems.

253 In this same paper, Sider goes on to construct a list of various theses that “ﬂow from, and help to articulate, a picture of parthood: of parts as specially characterizing a thing, of the whole being nothing more than the parts” (ibid., p. 69). This list includes, among other theses, the following proposals rendered problematic by nonlocal physical features of the world or the failure of composition as identity for physical parts.

1. Inheritance of location. If x is part of y, then y is located wherever x is located. Let one of the daughter photons from the Bell-type experiment described above serve as x (although we have already conceded much according to some philosophers of physics in allowing energy quanta to be called proper parts of an entangled system). If this photon is a part of an entangled pair, then the entangled pair is located wherever that photon is located. This is clearly inappropriate. It is nonsense to say that the entangled system is located wherever one of the daughter photons is located. The entangled system is not local, in that its description includes (highly) nonlocal, superposed position states which cannot be said to not exist. Neither can the entangled system be said to supervene on any subregion of spacetime, because the entangled system is itself part of an entangled system with its own environment. Widespread, uncontrollable, nonlocal quantum interactions render any candidate x—call it a “central system” if you wish— -indescribable in terms of any local spatial-temporal region. Neither can any candidate y—the entangled system—be said to supervene on any subregion of spacetime, because the entangled system is itself part of an entangled

254 system with its own environment.

2. Transitivity: Parthood is transitive (e.g., if x is part of y and y is part of z, then x is part of z). There are two points to be made here. The ﬁrst is that, as in the above case, nonlocal dynamics make it inappropriate to speak of subregions or subspaces as being truly isolated from one another. Second, even were we to understand x, y and z as generally as possible by assigning them appropriately related regions of the universe’s Hilbert space, the lesson from chapter 4 included noting the imperviousness of the physical world to the way we slice degrees of freedom, Hilbert spaces, or any other general mathematical description. There is no absolute way to clearly establish transitivity among possible candidate parts. In other words, it is trivially true that for any x it will be a some entangled system y which in turn will be part of another entangled system z—the inverse Matryoshka Doll eﬀect described above. Such a degree of triviality makes transitivity of quantitative parthood metaphysically vacuous in the sense that no understanding about parthood or its relation to the whole will be gained under this thesis, except the very general fact that (as the physics describes) the only true whole is that of the universe entire. And whether or not the universe entire is a legitimate “whole” is yet another consideration.

3. Uniqueness of composition: No proper parts have more than one mereological sum. For any two “proper parts” of an entangled system, the mereological description of that system will be entirely dependent upon the environment of that entangled system. For instance, that the handle of my

255 coffee cup is a proper part of the coffee cup by compositional mereology is understood to be an illusion owing to the appearance of the handle-part of my coffee cup occupying a subregion of the volume of space occupied by the whole coffee cup (and not vice versa, which accounts for the “properness” of the parthood relation). The physical state of affairs is, as should by now be clear, quite a bit more complicated than this, starting with the fact that it is strictly false that the handle (and the whole cup, for that matter) occupy well defined spatiotemporal regions.

4. Weak supplementation: If x is part of y and x 6= y, then y has a part that does not overlap x. This final item is rendered moot with respect to the physics by its appeal to “overlap,” which is a spatial-temporal relation dependent upon locality. And, as we have seen, nonlocality is the way of the world. One might instead consider weak supplementation as a way of defining parthood, in which case the world fails to obey once again. If one photon is to be defined as “part” (in the classical mereological sense of parthood) of the entangled photon pair, then it should be the case that if the two entities are not equivalent (and they are not, demonstrably so), the entangled pair has a part that does not overlap with the one photon. But the description of the entangled system is non-separable, meaning that there is no sense in which the two photons do not overlap. Thus we cannot consider them parts of an entangled whole. This argument can be run ad infinitum, as all systems are part of some entangled system.

Another widely held thesis of composition mereology is asymmetry, which

256 states that for distinct parts x and y, it cannot be the case that x is part of y and y a part of x. This is related to the deﬁnition of proper parthood, and runs into problems with quantum theory. Even if we grant that parts can be considered distinct in a non-contingent, approximate way (which we already know to be false—that anything appears distinct at all is due to decoherence!), this thesis implies a particular ordering of the physical universe that fails to describe the real world. The same can be said for all of quantitative compositional mereology: the ordering it tries to impose on the universe technically does not exist (or isn’t best characterized in terms of parts and wholes talk), and to add insult to injury, the very fact that the universe appears describable in terms of compositional mereological relations in the ﬁrst place is contingent upon the particular quantum interactions that occur at the level of observation among systems and their environments.

6.3.2 Traditional bundle theory

Traditional bundle theory is one of the most widely endorsed single-category ontologies. Bundle theorists achieve their one-category status by denying the existence of a categorical distinction between properties and objects. In essence, there are no such things as qualitative features in this ontology, and many of the issues found with compositional mereology will apply—even to heightened degree—in bundle theory. One example is as follows. In the Hilbert space describing the entangled photon pair, it seems to be the case that (due to the setup of the experiment and other

257 factors) there exist individuated parts in virtue of our ability to assign distinct polarization values to each, or by describing the experiment as having measured one photon “here at this detector” and the other “over there at that detector.” However, as Ladyman and Ross point out (Ladyman et al. 2007, pp. 132–135), quanta belonging to entangled pairs (a universal situation, so I’ve argued) share identical extrinsic and intrinsic properties, and this causes deep problems for bundle theorists. For instance, it is unclear how one gets parts at all on a such a view without also forsaking usual deﬁnitions of individuality and identity. Bundles of tropes (property instances) don’t seem structurally rich enough to capture what’s going on in the world.

6.3.3 Mereological nihilism

I take mereological nihilism to be a position that denies the existence of wholes. One way to see this as a metaphysical view of “parts all the way down.” Such an ontology is usually associated with “gunk,” a term ﬁrst employed by Lewis (1991) to describe a view opposite to that of atomism. There is an interesting sense in which one might recover the nonreductive aspect of the world described in the previous chapter via mereological nihilism: no matter at what size or energy scale the smallest current building block as given to us by current physics exists, it cannot be considered a part-less part. There are no simples or (philosophical) atoms in a gunk ontology, and this captures to some degree the idea that all physical systems are necessarily a “part” of some further entangled system, which is in turn a part of a further entangled system, etc. The

258 benefit of mereological nihilism is that it allows us to have parts in a slightly different way than other views—one that captures the force of the entanglement cascade. Yet this same position viewed in another light exposes all the more forcefully the limitations of mereological approaches to ontology. This is accomplished by noting that no matter which direction one chooses to interpret gunk (as “parts all the way up” or “parts all the way down”), the fact remains that gunk ontologies rely on parthood relations between and among physical systems, and as such assume it is correct to attribute local properties to candidate parts. Many of the criticisms for compositional mereology thus remain salient here. To summarize this section, investigating different approaches to quantitative mereology in light of the nontological thesis confirms the corollary stated above, denying the existence of parts and wholes. Decoherence renders mereological approaches highly problematic, though mereological nihilism held some virtue in its ability to recognize the sense in which our physical universe is multiply (nay, infinitely) parted, and part-able. Recall the two types of ontologies Ladyman and Ross et al. class all science-ignorant metaphysics into: the first class includes simples-based or atomistic ontologies, such as undergird compositional mereology and perhaps some varieties of bundle theory. The second class is that of gunk ontologies, that while an improvement still fail to adequately describe the physical world. If the only true whole is the universe, we aren’t likely to make any interesting metaphysical discoveries by focusing on part-whole relations—for this whole is not only untestable qua whole, but is also too far removed from the sorts

259 of things we do want to construct a metaphysics around to be worth the worry.

6.4 Consequences for qualitative mereology?

If we can’t have nice things, and quantitative mereology (or much of it) seems premised upon the existence of nice things, then maybe qualitative mereology will be a more fruitful way to approach ontology. Since the only ontology I am aware of that is strictly qualitative is the proposed property bundle theory of L.A. Paul, I focus on the issue of properties in what follows. If the NT-corollary is right, then subsystems of the universal Hilbert space cannot be said to be separate from their environments in a way that makes property-assignment straightforward. To illustrate this point, I describe another thought experiment concerning the dynamics of property perception. After that, I brieﬂy discuss Paul’s proposal, and argue that though it strikes me as somewhat hard to swallow, it has the enormous merits of not only being consistent with the physics (as far as I understand the philosophical implications of the thesis), but also seeks to dissolve the property/object distinction. Though I do not have the space to fully consider what this distinction might look like (if it stands at all) in light of decoherence, consider the following to be a ﬁrst-pass discussion along these lines.

6.4.1 One more thought experiment

Let us say that our metaphysician is out walking on a ﬁne summer’s day, and stops to gaze at the alternating patterns of light and shadow as the sun shines through the green leaves of a big oak tree. The metaphysician considers one

260 leaf in particular, and asks: if I want to say that this leaf has the property of being green—in what sense does the property exist? That is, what is its ontic status? Let us translate this situation into more physics-savvy language for our metaphysician by saying that the leaf contains chemicals undergoing the process of photosynthesis, which has the physical consequences of absorbing certain parts of the visible spectrum while emitting others (in particular, photons of wavelengths approximately 500 nanometers—a wavelength that corresponds to green light in the visible spectrum). Can’t we then say that the property of greenness belongs to the leaf? If we say it belongs the the leaf as a whole, in virtue of the chemical processes occurring in the leaf, we are making approximation that are only allowed because of decoherence. The leaf itself is a huge object (in terms of usual quantum objects), and as such it is, we can very safely assume, already decohered in various bases with respect to its environment. As such, the metaphysician’s observation of the leaf as green and thereby attributing the property of greenness to it is only possible when we understand that the observation of greenness is the eﬀect of our eyes absorbing photons of a particular wavelength and with a particular energy, and subsequently of the physiological mechanisms within our eyes translating this particular received energy into electrical pulses that are read by our brain as “greenness.” One need not mess with the complication of the property of greenness as it exists in our minds or brains or whatever we need only go so far as our metaphysician’s pupil taking in a packet of energy as a discrete (and therefore analyzable) packet of energy.

261 Our metaphysician’s physiological ability to absorb the light is already a result of quantum interactions, and in a way represents a loss of information about the way the leaf really is the moment the metaphysician’s eye receives the energy, owing to decoherence: because a given photon becomes entangled with its environment from the time of its emission via photosynthesis to its absorption as energy in the eye (and furthermore, we know that its entanglement will occur as soon as the photon is emitted, and extremely rapidly) and that thereafter we can describe the dynamics of the situation in terms of a particular decoherence model. Since the environment is a natural one, it is safe to map its dynamics onto a field of infinitely many delocalized bosonic field modes (an oscillator environment), and since a photon is best described as a discrete system (since it is a microscopic particle, and thus it is best to analyze it in the energy basis, and the energy basis for this sort of photon is going to be discrete)the spin-oscillator model tells us not only that decoherence will happen in the energy basis, thus effectively suppressing superpositions of energy states (the result of which being the overwhelmingly probability that the metaphysician’s eye will “see” the photon as a definite energy eigenstate). But the model also tells us that this suppression of weird energy states will happen on a scale significantly faster than the time in which it can be observed by a human eye after its emission from the leaf. Thus, although the real description of the photon involves lots of strange states in many bases, our metaphysician will nevertheless always measure it to be in a definite energy eigenstate—one that corresponds to greenness. In other words: all said and done, according to our best physics, the leaf has

262 the property of appearing approximately green to the metaphysician because all the strange other energy states that are a superposition of, e.g., one wavelength of greenness to another wavelength of greenness, still exist but would not be measured by our eyes given many lifetimes of the universe in which to observe it. In what sense does the leaf have or contain or engender, etc. , the property of greenness, owing to our understanding that the leaf’s appearing to be green in the ﬁrst place is due to a world of contingencies? Contingencies allowing not just for the perception of a particular property as instantiated, nor of merely the content of that property, but our ability to perceive the leaf as a thing somehow related to the property of greenness, too. To avoid this problem, one can choose to build one’s ontology by interpreting all properties as qualitative or by dissolving the property/object distinction all together. With respect to the former, one can retain the intuitively ingrained property/object distinction, but at the price of understanding all properties to be merely qualitative, including properties like mass, angular momentum, position and the like. This will strike many, I venture, as too high a price, leaving the alternative: dissolving the property/object distinction in favor of a purely qualitative mereology. This is the position adopted by L.A. Paul in a recent manuscript (Paul (Paul)). Though it strikes me as very strange to consider the world to be nothing but bundles of qualitative properties and property fusions, this position at least avoids some of the criticisms decoherence brings against quantitative ontologies. Whether or not Paul and others wishing to adopt a strictly qualitative ontology can explain how their world is entirely qualitative while avoiding the temptation

263 to imbue the world with parts, wholes, slices or levels remains to be seen.

6.5 Conclusion

Continuing to pretend we can have isolated or closed systems is to turn one’s back on what I consider many of the goals that metaphysics began with, and to content oneself with a brand of phenomenologically-driven metaphysics. To metaphysician friends I say: do so if you wish. But then you must no longer claim that your metaphysics maps onto the world as we know it truly to be. Even though decoherence leaves (at this point at least) a great puzzle what the world is truly like, the science is clear enough with respect to the way the world is not. And the way the world is not ought to play no role in metaphysical views that are meant to extend beyond the epistemic. I have argued in previous chapters that the dynamical story of decoherence in various system-environment situations allows us to explain the appearance of a classical, macroscopic world. It turns out that things like tables, chairs and coﬀee cups have been described in terms of parts and wholes in the past because of their underlying quantum mechanical nature. The truth as stated in the corollary to the nontological thesis is that no such thing as quantitative parts and wholes exist, and this has deep implications for quantitative compositional mereology, bundle theories and gunk ontologies. Furthermore, the lessons of decoherence have interesting consequences regarding not only the ontic status of properties, but property/object relations generally. I have introduced a thought experiment and one metaphysical thesis—that of Paul’s one-category property bundles theory—

264 by way of starting down this new path of metaphysical inquiry, but by no means have I said all that can be said regarding properties and their role in physically consistent metaphysical views. The deep reason why we believe we have gotten away with, and continue to get away with, talking about coﬀee cups and tables in terms of particles-arranged- table-wise is owing to the comparative instability of superposed states to subsist upon interaction with a table-environment compared to the robust non-superposed states that do remain stable during evolution. It is an artifact of the world we live in being entangled and decohered through-and-through, coupled with the size and speed with which we “measure” or observe systems, that we used to believe we could have nice things. Now we know just why this isn’t so.

265 CHAPTER 7

CONCLUDING REMARKS

Quantum decoherence is a dynamical process resulting from the entanglement of a system with its environment and the subsequent delocalization of that system’s phase relations into the environment. This delocalization forces the quantum system into a state that is apparently classical by prodigiously suppressing features that typically give rise to so-called quantum behavior. Thus it has been frequently proposed by physicists and philosophers alike that decoherence explains the dynamical transition from quantum behavior to classical behavior. Statements like this assume the existence of distinct realms, however, and the present thesis was an exploration of the metaphysical consequences of quantum decoherence motivated by the question of the quantum-to-classical transition and interlevel relations: if there are true “classical” and “quantum” domains or levels, what are the relations between them? And if there are no such levels, what follows? I have argued that there can be no in-principle levels, and what follows is a position tentatively named nonreductive quantum monism, where all the being in the physical world is done quantum-ly. This name is meant to reﬂect the failure

266 of usual interlevel relations to characterize the world in accordance with quantum mechanics and decoherence processes. Among the relations considered were reduction (including Kim’s functional-reduction schema), supervenience (including Chalmers’s local supervenience) and emergence (defined in turn as both the denial of part-whole reduction and the existence of novel features at one scale not present at some other scale). In chapter 6 it was argued that nonreductive quantum monism (with appropriate modifiers attached to this position depending on whether one wishes to read it as a philosophy of science position or as a view within contemporary metaphysics) entails the nonexistence of parts and wholes, which has sweeping ramifications for quantitative compositional mereology and similar views. The possibility of qualitative compositional mereology was discussed, though if this account turns out to ground properties (qualitative though they be) in anything physical, it will run afoul of the same problems condemning quantitative objects. Interesting consequences regarding the property/object are likewise taken from the corollary to the nontological thesis. In particular, if the property itself is quantitative or is grounded in something quantitative, it is difficult to see how such property-object relations can avoid imposing definiteness divisi in the ontology. Whether or not there is a story to be told about properties that avoids running into contradiction with the conclusions in this thesis is yet to be seen. This particular question strikes me as one of potential interest to philosophers of physics and metaphysicians alike, wherein the specific modes of inquiry brought to the table by these respective fields stand to enrich the larger discussion.

267 Another possible venue for fruitful exchange between philosophers of physics and metaphysicians that falls out of investigations of decoherence involves current debates in philosophy of mind. For example, it would be interesting to consider certain accounts in the philosophy of mind in which mental states, or perhaps consciousness, are explained in terms of supervenience relations with a physical base (i.e., a certain brain state). As we have seen, if either end of the supervenience relation requires defining an independent or local or well-defined or closed physical region, the conclusions in this thesis contradict this possibility. Of course, such an analysis deserves much more careful consideration than I have just provided; I merely use it as an example of an arena where extensions of the conclusions in this thesis may be of interest. I have counted as one of the merits of the above investigations its principled avoidance of the question of the interpretation of quantum mechanics and the measurement problem variously construed. Not only does this lend the present philosophical investigations a novelty (at least to my knowledge, compared to other works on the philosophy of decoherence), but it leaves open these perennial debates to engage with precisely those question I have intentionally bracketed. Nevertheless, I maintain that reference to these usual questions is not necessary— that good philosophical work can be done without adopting a specific interpretational framework and without recourse to the measurement problem. The proof of this claim is in the pudding, so to speak, but the fact that I have managed to arrive at some interesting results from such grounds is surely testimony to this methodological point of departure from normalcy.

268 Another novel feature of the above work was a presentation of the philosophical assumptions and consequences of the four canonical models applied to system- environment interactions. To my knowledge, no serious philosophical work based on decoherence has been done that takes seriously these four models, and my thesis sought to remedy this situation. The motivation for this position is simply the recognition that one’s philosophical conclusions, should that philosophy take as its input scientiﬁcally informed accounts of the world, ought to rely signiﬁcantly on the details of the science as it is currently being done in the laboratory. I have attempted to do just this. In the end, I conclude that given the knowledge of quantum decoherence to be derived from philosophically neutral investigations of the basic principles, concepts and models of this physical process, one can claim with respect to ontology that there exist no levels. This claim—called the “nontological thesis”—exposes as ill posed questions regarding the transition from the quantum regime to the classical regime and reveals the inappropriateness of interlevel relations (like reduction, supervenience and emergence) operating within metaphysical frameworks. The nontological thesis had further important consequences regarding intralevel relations: not only have we seen there are there no meaningful ways to carve the world into levels, but there appear to be no metaphysically robust ways of understanding the world as containing parts and wholes, either. These conclusions drastically alter the philosophical terrain—not just in theoretical and experimental physics or in the philosophy of physics, but in traditional metaphysics as well. Though the target audience of the preceding thesis

269 was therefore much broader than is typical for a work of this sort, this was an inevitable consequence of the subject matter. In addition, it is my hope that in so constructing this study of decoherence, members of various disciplines and with varying research interests will feel compelled to respond to, expand upon, or perhaps even challenge the oftentimes provocative claims I have made. I invite one and all to explore such questions with me.

270 BIBLIOGRAPHY

(2011, March). Antarctic Connection: Vostok Station. http://www.antarcticconnection.com/ antarctic/stations/vostok.shtml. Re- trieved 16 March 2011. Adler, S. L. (2003). Why decoherence has not solved the measurement problem: A response to P.W. Anderson. Studies in the History and Philosophy of Modern Physics 34B, 135–142. Albert, D. and B. Loewer (1990). Some alleged solutions to the measurement problem. Synthese 86, 87–98. Allori, V. and N. Zanghi (2001). On the classical limit of quantum mechanics. quant-ph/0112009v1. Ambegaokar, V., U. Eckern, and G. Schoen (1982). Quantum dynamics of tunneling between superconductors. Physical Review Letters 48 (25), 1745–1748. Anderson, P. (2001a). Reply to Cartwright. Studies in the History and Philosophy of Modern Physics 32, 499–500. Anderson, P. (2001b). Science: A ‘dappled world’ or a ‘seamless web’? Studies in the History and Philosophy of Modern Physics 32, 487–494. Anglin, J., J. Paz, and W. Zurek (1997, June). Deconstructing decoherence. Physical Review A 55 (6), 4041–4053. Arndt, M., O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger (1999). Wave-particle duality of C60 molecules. Nature 401, 680–682. Arndt, M., O. Nairz, and A. Zeilinger (2002). Interferometry with macromolecules: Quantum paradigms tested in the mesoscopic world. In R. Bertlmann and A. Zeilinger (Eds.), Quantum [Un]Speakables: From Bell to Quantum Informa- tion, pp. 333–351. Berlin: Springer.

271 Asadian, A., M. Tiersch, G. Guerreschi, J. Cai, S. Popescu, and H. J. Briegel (2010, July). Motional eﬀects on the eﬃciency of excitation transfer. New Journal of Physics 12, 075019.

Bacciagaluppi, G. (2007). The role of decoherence in quantum mechanics. The Stanford Encyclopedia of Philosophy. http://plato.standford.edu/entries/qm- decoherence. Retrieved 8 February 2009.

Bacciagaluppi, G. and M. Hemmo (1996, Sept). Modal interpretations, decoherence and measurements. Studies in the History and Philosophy of Modern Physics 27 (3), 239–277.

Ballentine, L. (1970). The statistical interpretation of quantum mechanics. Re- views of Modern Physics 42 (4), 358–381.

Ballentine, L., Y. Yang, and J. Zibin (1994). Inadequacy of Ehrenfest’s theorem to characterize the classical regime. Physical Review A 50, 2854–2859.

Banks, T. (2008, 22 Sep). Locality and the classical limit of quantum mechanics. arXiv:0809.3764.

Banks, T., W. Fischler, and S. Paban (2002). Recurrent nightmares?: Measure- ment theory in de Sitter space. Journal of High Energy Physics 12 (2002), 062(19).

Batterman, R. (1993). Quantum chaos and semiclassical mechanics. In PSA 1992: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Volume 2, pp. 50–65.

Batterman, R. (1995). Theories between theories: asymptotic limiting intertheoretic relations. Synthese 103, 171–201.

Batterman, R. (2002). The Devil in the Details: Asymptotic Reasoning in Expla- nation, Reduction and Emergence. Oxford: Oxford University Press.

Belot, G. and L. Jansson (2010, Jan). Book review: Alisa Bokulich, Reexamining the Quantum-Classical Relation: Beyond reductionism and pluralism. Studies in the History and Philosophy of Modern Physics 41 (1), 81–83.

Bene, G. and D. Dieks (2002). A perspectival version of the modal interpretation of quantum mechanics and the origin of macroscopic behavior. Foundations of Physics 32, 645–672.

272 Bernu, J., S. Deleglise, C. Sayrin, S. Kuhr, I. Dotsenko, M. Brune, J.-M. Raimond, and S. Haroche (2008, Oct). Freezing coherent field growth in a cavity by the quantum zeno effect. Physical Review Letters 101, 180402(4). Berry, M. (1994). Asymptotics, singularities and the reduction of theories. In D. Prawitz, B. Skyrms, and D. Westerstahl (Eds.), Logic, Methodology and Philosophy of Science IX. Amsterdam: Elsevier. Berry, M. (2001). Chaos and the semiclassical limit of quantum mechanics (is the moon there when somebody looks?). In R. Russell, P. Clayton, K. Wegter- McNelly, and J. Polkinghorne (Eds.), Quantum Mechanics: Scientific Perspec- tives on Divine Action. Vatican Observatory: CTNS Publications. Bertet, P., A. Auffeves, S. Maioli, T. Osnaghi, T. Meunier, M. Brune, J.-M. Raimond, and S. Haroche (2002). Direct measurement of the Wigner function of a one-photon Fock state in a cavity. Physical Review Letters 89, 200402. Bitbol, M. (2009). Decoherence and the constitution of objectivity. In M. Bitbol, P. Kerszberg, and J. Petitot (Eds.), Constituting Objectivity: Transcendental Perspectives on Modern Physics, The Western Ontario Series in Philosophy of Science, pp. 347–357. Springer. Bittner, E. R. and P. J. Rossky (1995, Nov). Quantum decoherence in mixed quantum-classical systems: Nonadiabatic processes. Journal of Chemical Physics 103 (18), 8130–8143. Bittner, E. R. and P. J. Rossky (1997). Decoherent histories and non-adiabatic quantum molecular dynamics simulations. Journal of Chemical Physics 107, 8611. Blanchard, P., D. Giulini, E. Joos, C. Kiefer, and I.-O. Stamatescu (Eds.) (2000). Decoherence: Theoretical, Experimental, and Conceptual Problems, Berlin. Pro- ceedings of the Bielefeld Workshop, Nov. 1998: Springer. Blencowe, M. (2004). Quantum electromechanical systems. Physics Reports 395, 159–222. Bokulich, A. (2008a). Can classical structures explain quantum phenomena? British Journal for the Philosophy of Science 59 (2), 217–235. Bokulich, A. (2008b). Reexamining the Quantum-Classical Relation: Beyond Re- ductionism and Pluralism. Cambridge University Press.

273 Born, M., W. Heisenberg, and P. Jordan (1926). Zur Quantenmechanik II. Zeitschrift f¨urPhysik 35, 557–615.

Brading, K. and A. Skiles (forthcoming). Underdetermination as a path to ontic structual realism. In E. Landry and D. Rickles (Eds.), Structure, Object, and Causality: Proceedings of the Banﬀ Workshop on Structural Realism, University of Western Ontario Series in Philosophy of Science. Spinger.

Branning, D., S. Bahndari, and M. Beck (2009). Low-cost coincidence- counting electronics for undergraduate quantum optics. American Journal of Physics 77 (7), 667–670.

Braun, D. (2001). Dissipative quantum chaos and decoherence, Volume 172 of Springer tracts in modern physics. Berlin; New York: Springer.

Breuer, H.-P. and F. Petruccione (2002). The Theory of Open Quantum Systems. Oxford: Oxford University Press.

Brezger, B., L. Hackerm¨uller, S. Uttenthaler, J. Petschinka, M. Arndt, and A. Zeilinger (2002). Matter-wave interferometer for large molecules. Physi- cal Review Letters 88, 100404.

Briegel, H. J. and S. Popescu (2009, Oct). Entanglement and intra-molecular cooling in biological systems? A quantum thermodynamic perspective. arxiv:0806.4552.

Brune, M., E. Hagley, J. Dreyer, X. Maˆıtre, A. Maali, C. Wunderlich, J.-M. Raimond, and S. Haroche (1996). Observing the progression decoherence of the meter in a quantaum measurement. Physical Review Letters 77, 4887.

Brune, M., S. Haroche, J.-M. Raimond, L. Davidovich, and N. Zagury (1992). Manipulation of photons in a cavity by dispersive atom-ﬁeld coupling: Quan- tum non-demonlition measurements and generation of Schr¨odingercat states. Physical Review A 45, 5193.

Buric, N. (2010). Decoherence and classical behavior with a chaotic macroscopic environment. Physics Letters A 374 (40), 4090–4098.

Cai, J., J. Guerreschi, and H. J. Briegel (2010). Quantum control and entanglement in a chemical compass. Physical Review Letters 104 (22), 220502(4).

274 Caldeira, A. and A. Leggett (1983). Path integral approach to quantum Brownian motion. Physica 121A, 587–616.

Caldeira, A. and A. Leggett (1985, Feb). Inﬂuence of damping on quantum interference: An exactly soluble model. Physical Review A 31 (2), 1059–1066.

Camilleri, K. (2009, Dec). A history of entanglement: Decoherence and the interpretation problem. Studies in the History and Philosophy of Modern Physics 40 (4), 290–302.

Carlson, J. A., M. D. Olmstead, and M. Beck (2006). Quantum mysteries tested: An experiment implementing Hardy’s test of local realism. American Journal of Physics 74 (3), 180–186.

Cartwright, N. (1983). How the Laws of Physics Lie. New York: Clarendon Press.

Cartwright, N. (1994a). Fundamentalism vs. the patchwork of laws. Proceedings of the Aristotelian Society, New Series 94, 279–292.

Cartwright, N. (1994b). The metaphysics of the disuniﬁed world. Proceedings of the Biennial Meeting of the Philosophy of Science Association 1994, 357–364.

Cartwright, N. (1999). The Dappled World. Cambridge: Cambridge University Press.

Castagnino, M., S. Fortin, and O. Lombardi (2010, Jan). Suppression of decoherence in a generalization of the spin-bath model. Journal of Physics A: Mathematical and Theoretical 43, 065304(20).

Chakravarty, S. (1982). Quantum ﬂuctuations in the tunneling between superconductors. Physical Review Letters 49, 681–684.

Chakravarty, S. and A. J. Leggett (1984). Dynamics of the two-state system with ohmic dissipation. Physical Review Letters 52, 5–8.

Chalmers, D. (2008). Supervenience. In M. A. Bedau and P. Humphreys (Eds.), Emergence, Chapter 23, pp. 411–425. Cambridge, MA: MIT Press.

Cheng, Y.-C. and G. Fleming (2009). Dynamics of light harvesting in photosynthesis. Annual Review of Physical Chemistry 60, 241–262.

275 Collini, E., C. Y. Wong, K. E. Wilk, P. M. Curmi, P. Brumer, and G. D. Scholes (2010, 4 Feb). Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature. Nature 463, 644–648.

Cucchietti, F., J. Paz, and W. Zurek (2005). Decoherence from spin environments. Physical Review A 72 (5), 052113(8).

Davidovich, L., M. Brune, J.-M. Raimond, and S. Haroche (1996). Mesoscopic quantum coherences in cavity QED: Preparation and decoherence monitoring schemes. Physical Review A 53, 1295.

Dehlinger, D. and M. W. Mitchell (2002a). Entangled photon apparatus for the undergraduate laboratory. American Journal of Physics 70 (9), 898–902.

Dehlinger, D. and M. W. Mitchell (2002b). Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory. American Journal of Physics 70 (9), 903–910.

Del Buono, G., T. Cohen, and P. J. Rossky (1994). Eﬀects of long-range interactions on the dynamics of ions in aqueous solution. Journal of Molecular Liquids 60, 221. d’Espagnat, B. (1976). Conceptual Foundations of Quantum Mechanics (2nd ed.). Addison Wesley.

Dobrovitski, V., H. De Raedt, M. Katsnelson, and B. Harmon (2003). Quan- tum oscillations without quantum coherence. Physical Review Letters 90 (21), 210401(4).

Duplantier, B., J.-M. Raimond, and V. Rivasseau (Eds.) (2007). Quantum Deco- herence, Volume 48 of Poincar´eSeminar 2005. Birkh¨auserVerlag.

Emerson, J. (2001). Chaos and quantum-classical correspondence for coupled spins. Ph. D. thesis, Simon Fraser University.

Emerson, J. and L. Ballentine (2001). Quantum-classical correspondence for the equilibrium distributions of two interacting spins. Physical Review E 64 (2), 026217(11).

Engel, G., T. Calhoun, E. Read, T.-K. Ahn, T. Mancˇal,Y.-C. Cheng, R. Blanken- ship, and G. Fleming (2007). Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature 446, 782–784.

276 Esfeld, M. (2001). Holism in Philosophy of Mind and Philosophy of Physics. Dordrecht: Kluwer. Esfeld, M. (2004). Quantum entanglement and a metaphysics of relations. Studies in the History and Philosophy of Modern Physics 35B, 601–617. Fa-Qiang, W., Z. Zhi-Ming, and L. Rui-Sheng (2009). Decoherence of two-qubit system in a non-Markovian squeezed reservoir. Chinese Physics B 18, 597. Feynman, R. and F. L. Vernon (1963, Oct). The theory of a general quantum system interacting with a linear dissipative system. Annals of Physics 24, 118– 173. Fodor, J. (1974). Special science (or: The disnuity of science as a working hypothesis). Synthese 28, 97–115. Fodor, J. (1997). Special sciences: Still autonomous after all these years. Philo- sophical Perspectives 11: Mind, Causation and World, 149–163. Galvez, E. J., C. H. Holbrow, M. J. Pysher, J. W. Martin, N. Courtemanche, L. Heilig, and J. Spencer (2005). Interference with correlated photons: Five quantum mechanics experiments for undergraduates. American Journal of Physics 73 (2), 127–140. Gauger, E., E. Rieper, J. Morton, S. Benjamin, and V. Vedral (2011). Sustained quantum coherence and entanglement in the avian compass. Physical Review Letters 106, 040503. Gell-Mann, M. and A. Pais (1955). Behavior of neutral particles under charge conjugation. Physical Review 97 (5), 1387–1389. Ghirardi, G., A. Rimini, and T. Weber (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review D 34 (2), 470 ff. Giraud, A. and J. Serreau (2010). Decoherence and thermalization of pure quantum state in quantum field theory. Physical Review Letters 104, 230405. Giulini, D., E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu, and H. Zeh (1996). Decoherence and the Appearance of a Classical World in Quantum Theory (2nd ed.). Springer-Verlag. Giulini, D., C. Kiefer, and H. Zeh (1995, April). Symmetries, superselection rules, and decoherence. Physics Letters A 199 (5-6), 291–298.

277 Goggin, M., B. Sundaram, and P. Milonni (1990). Quantum logistic map. Physical Review A 41, 5705–5708. Golubev, D., A. Semenov, and A. Zaikin (2010). Weak localization, Aharonov- Bohm oscillations, and decoherence in arrays of quantum dots. Low Temperature Physics 36 (10), 933–950. Habib, S., Y. Kluger, E. Mottola, and J. Paz (1996). Dissipation and decoherence in mean field theory. Physical Review Letters 76 (25), 4660–4663. Habib, S., K. Shizume, and W. Zurek (1998). Decoherence, chaos, and the correspondence principle. Physical Review Letters 80, 4361–4365. Hackermüller,L., K. Hornberger, B. Brezger, A. Zeilinger, and M. Arndt (2004, Feb). Decoherence of matter waves by thermal emission of radiation. Na- ture 427, 711–714. Hackermüller, L., S. Uttenthaler, K. Hornberger, E. Reiger, B. Brezger, A. Zeilinger, and M. Arndt (2003). Wave nature of biomolecules and fluo- rofullerenes. Physical Review Letters 91, 090408. Halliwell, J. (1995). A review of the decoherent histories approach to quantum mechanics. Annals of the New York Academy of Sciences 755, 726–740. Han, S., Y. Yu, X. Chu, S.-I. Chu, and Z. Wang (2001). Time-resolved measurement of dissipation-induced decoherence in a Josephson Junction. Sci- ence 293 (5534), 1457–1459. Hartmann, L., W. Dür,and H. J. Briegel (2006). Steady state entanglement in open and noisy quantum systems at high temperature. Physical Review A 74, 052304. Hartmann, S., C. Hoefer, and L. Bovens (Eds.) (2008). Nancy Cartwright’s Phi- losophy of Science. New York: Routledge. Hatano, N. (1999). Non-Hermitian quantum mechanics and localization in physical systems. In Y. Ono and K. Fujikawa (Eds.), Quantum Coherence and Decoherence, North-Holland Delta Series, pp. 319–322. Amsterdam: Elsevier. Healey, R. (1991). Holism and nonseparability. Journal of Philosophy 88, 393–421. Heaney, L., J. Anders, and V. Vedral (2007, July). Spatial entanglement of a free bosonic field. arXiv:quant-ph/0607069v3.

278 Hines, A. P. and P. Stamp (2008). Decoherence in quantum walks and quantum computers. Canadian Journal of Physics 86, 541–548.

Hoefer, C. (2008). For fundamentalism. In S. Hartmann, C. Hoefer, and L. Bovens (Eds.), Nancy Cartwright’s Philosophy of Science, pp. 307–321. New York: Routledge.

Hornberger, K., S. Uttenthaler, B. Brezger, L. Hackerm¨uller,M. Arndt, and A. Zeilinger (2003, April). Collisional decoherence observed in matter wave interferometry. Physical Review Letters 90 (16), 160401(4).

Howard, D. (1989). Holism, separability, and the metaphysical implications of the Bell experiments. In J. Cushing and E. McMullin (Eds.), Philosophical Conse- quences of Quantum Theory, pp. 224–253. Notre Dame, Indiana: University of Notre Dame Press.

Howard, D. (2003). Reduction and emergence in the physical sciences: Some lessons from the particle physics-condensed matter physics debate. Manuscript available at http://www.nd.edu/ dhoward1/Papers.html.

Hwang, H. and P. J. Rossky (2004). Electronic decoherence induced by intramolec- ular vibrational motions in a betaine dye molecule. Journal of Physical Chem- istry B 108, 6723–6732.

Jacobs, K. (2009). A Monte Carlo method for modeling thermal damping: Beyond the Brownian motion master equation. Europhysics Letters 85, 40002.

Jang, S., M. Newton, and R. Silbey (2007). Multichromophoric F¨orsterresonance energy transfer from B800 to B850 in the light harvesting complex 2: evidence for subtle energetic optimization by purple bacteria. Journal of Physical Chem- istry B 111, 6807–6814.

Janssen, H. (2008). Reconstructing reality: Environment-induced decoherence, the measurement problem, and the emergence of deﬁniteness in quantum mechanics. philsci-archive.pitt.edu.

Joos, E. (1987a). Comment on “Uniﬁed dynamics for microscopic and macroscopic systems”. Physical Review D 36 (10), 3285–3286.

Joos, E. (1987b). Time arrow in quantum theory. Journal of Non-Equilibrium Thermodynamics 12, 27–44.

279 Joos, E. and H. Zeh (1985). The emergence of classical properties through interaction with the environment. Zeitschrift f¨urPhysik B 59, 223–243.

Keeler, M., W. Setzer, and W. Martin (2010). Rydberg-electron decoherence in experimentally obtained recurrence spectra. Physical Review A 82 (5), 053406(9).

Kiefer, C. (1992). Decoherence in quantum electrodynamics and quantum cosmology. Physical Review D 46, 1658.

Kiefer, C. (1993). Topology, decoherence, and semiclassical gravity. Physical Review D 47, 5414–5421.

Kiefer, C. (1996). Decoherence of black holes by Hawking radiation. Physical Review D 53, 7050–7061.

Kim, J. (1998). Mind in a physical world. Cambridge, MA: MIT Press.

Kistler, M. (2006). Reduction and emergence in the physical sciences: Reply to Rueger. Synthese 151, 347–354.

Kokorowski, D. A., A. D. Cronin, T. D. Robers, and D. E. Pritchard (2001, March). From single- to multiple-photon decoherence in an atom interferometer. Physical Review Letters 86 (11), 2191–2195.

Koksma, J., T. Prokopec, and M. G. Schmidt (2010). Decoherence in an interacting quantum ﬁeld theory: The vacuum case. Physical Review D 81, 065030.

Konetchnyi, A., M. Mensky, and V. Namiot (1993, June). Physical model for monitoring the position of a quantum particle. Physics Letters A 177 (4-5), 283–289.

K¨ubler,O. and H. Zeh (1973). Dynamics of quantum correlations. Annalen der Physik 76, 405–418.

Ladyman, J., D. Ross, D. Spurrett, and J. Collier (2007). Every Thing Must Go. Oxford: Oxford University Press.

Lan, B. and R. Fox (1991). Quantum-classical correspondence and quantum chaos in the periodically kicked pendulum. Physical Review A 43 (2), 646–655.

Landau, L. (1996, Feb). Macroscopic observation of a quantum particle in a slowly varying potential: On the classical limit of quantum theory. Annals of Physics 246 (1), 190–227.

280 Landsman, N. (1995, April). Observation and superselection in quantum mechanics. Studies in the History and Philosophy of Modern Physics 26 (1), 45–73.

Lee, H., Y.-C. Cheng, and G. Fleming (2007). Coherence dynamics in photosynthesis: protein protection of exitonic coherence. Science 316, 1462.

Lee, T. and C. Yang (1956). Question of parity conservation in weak interactions. Physical Review 194 (1), 254–258.

Leggett, A., S. Chakravarty, A. Dorsey, M. Fisher, A. Garg, and W. Zwerger (1987, Jan). Dynamics of the dissipative two-state system. Reviews of Modern Physics 59 (1), 1–85.

Leggett, A. and A. Garg (1985). Quantum mechanics versus macroscopic realism: Is the ﬂux there when nobody looks? Physical Review Letters 54 (9), 857–860.

Lewis, D. K. (1991). Parts of Classes. Blackwell.

Liboﬀ, R. (1984). The correspondence principle revisited. Physics Today 37, 50–55.

Lombardi, O., S. Fortin, and M. Castagnino (forthcoming). The problem of iden- tifying the system and the environment in the phenomenon of decoherence.

Lombardi, O., S. Fortin, M. Castagnino, and J. S. Ardenghi (2010). Compat- ibility between environment-induced decoherence and the modal-Hamiltonian interpretation of quantum mechanics.

Lombardo, F. and F. Mazzitelli (1996). Coarse graining and decoherence in quantum ﬁeld theory. Physical Review D 53 (4), 2001–2011.

Longhi, S. (2010). Optical realization of relativistic non-Hermitian quantum mechanics. Physical Review Letters 105 (1), 013903(4).

Maioli, P., T. Meunier, S. Gleyzes, A. Auﬀeves, G. Nogues, M. Brune, J.-M. Raimond, and S. Haroche (2005). Non-destructive Rydberg atom counting with mesoscopic ﬁelds in a cavity. Physical Review Letters 94, 113601.

Mavromatos, N. and S. Sarkar (2007). Probing models of quantum decoherence in particle physics and cosmology. Journal of Physics Conference Series 67, 012011.

281 Mensky, M. B. (2000). Quantum Measurements and Decoherence: Models and Phenomenology. Dordrecht, Boston & London: Kluwer Academic Publishers.

Messiah, A. (1965). Quantum Mechanics, Volume I and II. Amsterdam: North- Holland Publishing.

Mohseni, M., P. Rebentrost, S. Lloyd, and A. Aspuru-Guzik (2008). Environment- assisted quantum walks in photosynthetic energy transfer. Journal of Chemical Physics 129, 174106.

Monroe, C., D. Meekhof, B. E. King, and D. J. Wineland (1996). A ‘Schr¨odinger cat’ superposition state of an atom. Science 272, 1131.

Motakabbir, K., J. Schnitker, and P. J. Rossky (1992). A comparison of classical and quantum analyses of electron localization sites in liquid water. Journal of Chemical Physics 97, 2055.

Moulines, C. U. (2006). Ontology, reduction, emergence: A general frame. Syn- these 151, 313–323.

Myatt, C. J., B. E. King, Q. Turchette, C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe, and D. J. Wineland (2000, 20 Jan). Decoherence of quantum superpositions through coupling to engineered reservoirs. Nature 403, 269–273.

Nagel, E. (1961). The Structure of Science. London: Routledge and Kegan Paul.

Nagel, E. (2008). Issues in the logic of reductive explanations. In M. A. Bedau and P. Humphreys (Eds.), Emergence, Chapter 19, pp. 359–373. Cambridge, MA: MIT Press.

Nakamura, K. and T. Harayama (2004). Quantum Chaos and Quantum Dots. Oxford: Oxford University Press.

Namiki, M., I. Ohba, K.-i. Maeda, and Y. Aizawa (Eds.) (1996). Quantum Physics, Chaos Theory, and Cosmology. Woodbury, NY: AIP Press.

Namiki, M., S. Pascazio, and H. Nakazato (1997). Decoherence and Quantum Measurements. Singapore: World Scientiﬁc.

Paul, L. Parthood and individuation in a one-category ontology. Unpublished manuscript obtained from author.

282 Paul, L. (2010, 3 June). Building the world from fundamental constituents. Un- published manuscript obtained from author.

Paz, J. and W. Zurek (1999). Quantum limit of decoherence: Environment induced superselection of energy eigenstates. Physical Review Letters 82, 5181– 5185.

Prezhdo, O. and P. J. Rossky (1998). Relationship between quantum decoherence times and solvation dynamics in condense phase chemical systems. Physical Review Letters 81, 5294.

Rae, A. I. (2009). Everett and the Born rule. Studies in the History and Philosophy of Modern Physics 40 (3), 243–250.

Raimond, J.-M., M. Brune, and S. Haroche (2001). Manipulating quantum entanglement with atoms and photons in a cavity. Reviews of Modern Physics 73, 565.

Raimond, J.-M. and S. Haroche (2005). Monitoring the decoherence of mesoscopic quantum superpositions in a cavity. S´eminarPoincar´e2, 25–64.

Richter, K. (2000). Semiclassical Theory of Mesoscopic Quantum Systems. New York: Springer-Verlag.

Rossky, P. J. (1998). Non-adiabatic quantum dynamics simulation using classical baths. In B. Berne, G. Ciccotti, and D. Coker (Eds.), Classical and Quantum Dynamics in Condensed Phase Simulations. Singapore: World Scientiﬁc.

Rotter, I. (2009). A non-Hermitian Hamilton operator and the physics of open quantum systems. Journal of Physics A: Mathematical and Theoretical 42 (15), 153001.

Rueger, A. (2006). Functional reduction and emergence in the physical sciences. Synthese 151, 335–346.

Russell, B. (1954). The Analysis of Matter (Reprinted ed.). New York: Dover.

Saunders, S. (2004). Derivation of the Born rule from operational assumptions. Proceedings of the Royal Society of London A 460, 1771–1788.

Schlosshauer, M. (2007). Decoherence and the Quantum-to-Classical Transition (2nd ed.). Berlin: Springer.

283 Schlosshauer, M. (2008). Classicality, the ensemble interpretation, and decoherence: Resolving the Hyperion dispute. Foundations of Physics 38, 796–803.

Schlosshauer, M., A. P. Hines, and G. J. Milburn (2008). Decoherence and dissipation of a quantum harmonic oscillator coupled to two-level systems. Physical Review A 77 (2), 022111(7).

Schr¨odinger, E. (1935). Die gegenw¨artigSituation in der Quantenmechanik. Naturwissenschaften 23, 807–812, 823–828, 844–849.

Schr¨odinger,E. (1935, 1936). Discussion of probability relations between separated systems. Proceedings of the Cambridge Philosophical Society 31, 32, 555–563, 446–452.

Schwartz, B., E. R. Bittner, O. Prezhdo, and P. J. Rossky (1996). Quantum decoherence and the isotope eﬀect in condensed phase nonadiabatic molecular dynamics simulations. Journal of Chemical Physics 104, 5942.

Senthilkumar, K., F. Grozema, C. Guerra, F. Bickelhaupt, F. Lewis, Y. Berlin, M. Ratner, and L. Siebbeles (2005). Absolute rates of hole transfer in DNA. Journal of the American Chemical Society 127 (42), 14894–14903.

Shimony, A. (1989). Search for a worldview which can accommodate our knowledge of microphysics. In J. Cushing and E. McMullin (Eds.), Philosophical Consequences of Quantum Theory, pp. 25–37. Notre Dame, Indiana: Univer- sity of Notre Dame Press.

Sider, T. (2007). Parthood. Philosophical Review 116 (1), 51–91.

Silberstein, M. and J. McGeever (1999). The search for ontological emergence. The Philosophical Quarterly 49 (195), 182–200.

Squires, E. J. (1990). On an alleged “proof” of the quantum probability law. Physics Letters A 145 (2-3), 67–68.

Stamp, P. (2006). The decoherence puzzle. Studies in the History and Philosophy of Modern Physics 37, 467–497.

Teller, P. (1986). Relational holism and quantum mechanics. British Journal for the Philosophy of Science 37, 71–81.

284 Thorn, J. J., M. S. Neel, V. W. Donato, G. S. Bergreen, R. E. Davies, and M. Beck (2004). Observing the quantum behavior of light in an undergraduate laboratory. American Journal of Physics 72 (9), 1210–1219.

Tolkunov, D. and V. Privman (2004). Short-time decoherence for general system- environment interactions. Physical Review A 69 (6), 062309(7).

van Grondelle, R. and V. Novoderezhkin (2006). Energy transfer in photosynthesis: experimental insights and quantitative models. Physical Chemistry Chem- ical Physics 8, 793–807.

Venkatesan, A., K. J. Lulla, M. J. Patton, A. D. Armour, C. J. Mellor, and J. R. Owers-Bradley (2010). Dissipation due to tunneling two-level systems in gold nanomechanical resonators. Physical Review B 81, 073410.

Voss, R. F. and R. A. Webb (1981). Macroscopic quantum tunneling in 1- micrometer Nb Josephson Junctions. Physical Review Letters 47 (4), 265–268.

Wallace, D. (2007). The quantum measurement problem: State of play. quant- ph/0712.0149v1.

Wallace, D. (2010). Decoherence and ontology. In S. Saunders, J. Barrett, A. Kent, and D. Wallace (Eds.), Many Worlds? Everett, Quantum Theory, & Reality, pp. 53–72. Oxford: Oxford University Press.

Wang, W., L. Wang, and X. Yi (2010). Lyapunov control on quantum open systems in decoherence-free subspaces. Physical Review A 82 (3), 034308(4).

Wei-Ci, L., W. Fa-Qiang, and L. Rui-Sheng (2010). Concurrence evolution of two qubits coupled with one-mode cavity separately. Chinese Physics B 19, 094204.

Wesenberg, J. H., Z. Kurucz, and K. M. lmer (2011). Dynamics of the collective modes of an inhomogeneous spin ensemble in a cavity. Physical Review A 83, 023826.

Wiebe, N. and L. Ballentine (2005). Quantum mechanics of Hyperion. Physical Review A 72, 022109(15).

Wilde, M. M., J. M. McCracken, and A. Mizel (2010, May). Could light harvesting complexes exhibit non-classical eﬀects at room temperature? Proceedings of the Royal Society of London A 466 (2117), 1347–1363.

285 Wisdom, J., S. Peale, and F. Mignard (1984). The chaotic rotation of Hyperion. Icarus 58, 137–152.

Yu, Y., S. Han, X. Chu, S.-I. Chu, and Z. Wang (2002). Coherent temporal oscillations of macroscopic quantum states in a Josephson Junction. Science 296, 889–892.

Zeh, H. (1970). On the interpretation of measurement in quantum theory. Foun- dations of Physics 1, 69–76.

Zeh, H. (1973). Toward a quantum theory of observation. Foundations of Physics 3, 109–116.

Zeh, H. (2002). Ch 2: Basic concepts and their interpretation. quant- ph/9506020v3.

Zeh, H. (2007). Roots and fruits of decoherence. In B. Duplantier, J.-M. Rai- mond, and V. Rivasseau (Eds.), Quantum Decoherence, pp. 151–176. Birkh¨auser Verlag.

Zilly, M., O. Ujs´aghy, and D. E. Wolf (2009). Statistical model for the eﬀects of dephasing on transport properties of large samples. The European Physical Journal B 68 (2), 237–246.

Zilly, M., O. Ujs´aghy, and D. E. Wolf (2010). Conductance of DNA molecules: Eﬀects of decoherence and bonding. Physical Review B 82, 125125(7).

Znojil, M. (2009). Scattering theory using smeared non-Hermitian potentials. Physical Review D 80 (4), 045009(12).

Zurek, W. (1981). Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse? Physical Review D 24, 1516–1525.

Zurek, W. (1982). Environment-induced superselection rules. Physical Review D 26, 1862–1880.

Zurek, W. (1991). Decoherence and the transition from quantum to classical. Physics Today 44, 36–44.

Zurek, W. (1993). Preferred states, predictability, classicality and the environment-induced decoherence. Progress of Theoretical Physics 89, 281–312.

286 Zurek, W. (1998). Decoherence, chaos, quantum-classical correspondence, and the algorithmic arrow of time. Physica Scripta T76, 186.

Zurek, W. (2003). Decoherence, einselection, and the quantum origins of the classical. Reviews of Modern Physics 75, 715–775.

Zurek, W. and J. Paz (1997). Why we don’t need quantum planetary dynamics: Decoherence and the correspondence principle for chaotic systems. In B. Hu and D. Feng (Eds.), Quantum Classical Correspondence: Proceedings of the 4th Drexel Symposium on Quantum Nonintegrability, 1994, Cambridge, MA, pp. 367–379. Drexel University: International Press.

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