Topics in the Foundations of Quantum Theory and Relativity

Transcription

1 THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Topics in the Foundations of Quantum Theory and Relativity HANS WESTMAN Department of Theoretical Physics Chalmers University of Technology and Göteborg University Göteborg, Sweden 2004

2 Topics in the Foundations of Quantum Theory and Relativity Hans Westman ISBN c Hans Westman, Department of Theoretical Physics Astronomy and Astrophysics group Chalmers University of Technology and Göteborg University SE Göteborg Sweden Telephone +46(0) Front cover: A numerical simulation of a simple quantum system illustrating how quantum probabilities emerges through a relaxation process in the deterministic hidden variable theory of de Broglie and Bohm. Back cover: An illustration of the meaning of violations of Bell inequalities in deterministic hidden variable theories. The amount those inequalities are violated puts a lower bound on the dark region. The four circles represent the nonlocal transition sets,,, and. Chalmersbibliotekets reproservice Göteborg, Sweden, 2004

3 Abstract In this thesis we focus on foundational issues in both quantum theory and general relativity. As regards quantum theory, we discuss the measurement problem and how it is solved in the hidden variable theory of de Broglie and Bohm. We discuss how quantum probabilities arises in that theory through a relaxation process. This relaxation process is explicitly demonstrated by aid of a numerical simulation. Then we discuss quantum non-locality both from a deterministic hidden variable point of view as well as for the standard quantum theory. As regards general relativity, we discuss the status of the equivalence principle in quantum field theory in curved spacetime. Then we discuss the the issue of general covariance, Einstein s hole problem, its solution, and the nature of observables in general relativity. i

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5 Acknowledgments My five years as a PhD-student would not have been the same were it not for numerous friends, family, mentors, and an open-minded supervisor. The first big thanks goes to Marek Abramowicz for letting me work freely on subjects that I find interesting. It is difficult for me to find words to express my gratitude to the two mentors that guided me through the jungle of the foundations of physics. A plant can only grow in good soil. Dearest mentors, Antony and Sebastiano, I have learned from you more than you can possibly imagine. Nothing would have been the same (briefly) if it were not for my true friend and spacetime lover Rickard Jonsson. Thanks for helping me out in numerous situations. May the gravitational force be with you! I would like to thank Nikola Markovic, Ulf Torkelsson, and Andreas Bäck for introducing me to the Fortran 90 programming language. Special thanks goes to Andreas Bäck for numerous coca-cola breaks at 3 o clock, interesting discussions on statistical physics, and for our many laughs. May they prolong your life! Heartfelt thanks goes to my father for helping me out with numerous things during the stressful time of writing this thesis. I also thank my mother and father for their musical genes. Playing the piano is a wonderful way to relax the brain from all thinking of physics. I would also like to thank my brothers Olle and Johan, one for a good jogging company, and the other for inspiring me with interesting music. I want to thank Behrooz Razaznejad and Kristian Dimitrievski, my friends during my student years. Much of my interest in the foundations of physics were fueled by our early discussions. Finally, I would like to thank all the people in the string theory group, especially Martin Cederwall and the PhD students for several interesting discussions. October 2004 Hans Westman iii

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7 Appended Papers Paper I Particle Detectors, Geodesic Motion, and the Equivalence Principle S. Sonego, H. Westman - gr-qc/ , Class. Quant. Grav. 21 (2004) Paper II Dynamical Origin of Quantum Probabilities A. Valentini and H. Westman - quant-ph/ , accepted for publication in Proc. Roy. Soc. Paper III Nonlocality, Contextuality and Transition Sets H. Westman - To be submitted to Annals of Physics. Paper IV Events as Point-Coincidences H. Westman and S. Sonego - in preparation for submission to Class. Quant. Grav. Papers not Appended Generalizing Optical Geometry R. Jonsson, H. Westman To be submitted to Class. Quant. Grav. v

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9 CONTENTS 1 Introduction 1 2 The Measurement Problem Precise argument Different reactions Pilot Wave Theory The pilot wave dynamics Solving the measurement problem The infamous impossibility proofs The Origin of Quantum Probabilities The subquantum -theorem Proof that Understanding quantum equilibrium The nesting theorem The approach of Dürr, Goldstein and Zanghì A Numerical Study Relaxation in a two-dimensional box Numerical technique Two ways of significantly decreasing computation time Program listing Quantum Non-Locality Non-locality in deterministic hidden variable theories The EPRB gedanken experiment The idea of hidden variables Nonlocal transition sets The meaning of violations of Bell inequalities Signal locality The non-locality of quantum theory The EPR argument vii

10 viii CONTENTS The issue of counter-factual definiteness Bell s notion of local causality Particle Detectors, Geodesic Motion and the Equivalence Principle, Introduction Mathematical preliminaries Normal modes The Dewitt monopole detector Discussion On General Covariance General invariance as a mathematical symmetry The Klein-Gordon field Einstein s vacuum field equations Einstein s hole problem Solution to the hole problem General invariance vs Lorentz invariance The role of the manifold in general relativity The manifold structure and quantum non-locality Conclusions and Outlook 73

11 Introduction This thesis is a study of foundational issues in both quantum theory and relativity. Despite the intense effort of physicists, a quantum theory of gravity is largely missing. As well as technical problems, such as the infinities in perturbative attempts to quantize gravity, there are also conceptual problems like the problem of time and the problem of observables. Quantum theory, even taken separately, poses difficult conceptual problems. Despite almost 80 years since the Schrödinger equations were first published, there is still no consensus of how to make sense of quantum theory. More acutely, as noted by John Bell, standard formulations of quantum theory is intrinsically vague and need undefined notions such as macroscopic, classical apparatus, etc.. This is related to the notorious measurement problem which we will discuss along with several proposals of how to solve it, or ignore it. As a specific answer to the measurement problem we will discuss the hidden variable theory of de Broglie and Bohm. This deterministic theory which reproduces all predictions of quantum theory, has existed ever since 1928, but curiously has mostly been ignored. We shall see how it solves the measurement problem, and perhaps more interestingly, how quantum probabilities can be explained (not postulated) by it. In this thesis a numerical simulation is presented that shows in detail how the quantum probabilities emerge for a simple system. We also discuss hidden variable theories in general. In particular we provide a clear graphical illustration of both the meaning of violations of Bell inequalities and how an effective locality emerges as a contingent feature of the quantum statistics. We will also discuss the irreducible non-locality inherent in quantum theory, as demonstrated by Bell. We will then turn to general relativity. First we discuss the status of the equivalence principle in curved spacetime. It is shown that there is in general no correlation between the response of a particle detector and its motion being a geodesic. 1

12 2 1. Introduction Then we proceed to discuss the issue of general covariance. Einstein s theory is invariant with respect to all differentiable coordinate transformations. This leads to an apparent problem, which is Einstein s hole problem. The solution of the problem, as Einstein came to realize, is that the coordinates in a general relativity are very different from those used in, for example, special relativity. While the coordinates in special relativity can be interpreted as readings of physical objects such as clocks and rulers, the coordinates in general relativity are mere parameters devoid of any such physical meaning. The parameters are needed only in order to write down field equations and to construct solutions. However, after such solutions are obtained, the parameters can, and should be, completely eliminated. One is left with gauge independent quantities that summarizes relations between field values alone. We end by a discussion of the role of the manifold in general relativity.

13 The Measurement Problem They somehow believe that the quantum theory provides a description of reality, and even a complete description; this interpretation is, however, refuted most elegantly by your system of radioactive atom + Geiger counter.... A. Einstein in a letter to E. Schrödinger The measurement problem dates back to 1927 (see ref. [35], p. 27) when Einstein realized that the superposition principle (the linearity of the Schrödinger equation) generates difficulties for recovering classical behavior of macroscopic systems. Some eight years later Schrödinger, after a congenial correspondence with Einstein, published his notorious article with the malicious cat experiment. It was conceived, by both Einstein and Schrödinger, as a reductio ad absurdum type argument aimed at demonstrating the incompleteness of the quantum mechanical formalism. The argument is very general with basically only two assumptions: The Schrödinger equation is universally valid: Everything can be described by it, even macroscopic measuring devices. The wavefunction, as given by the linear Schrödinger equation, provides a complete description of an individual system. The completeness assumption amounts to this: There is nothing further to say about the state of affairs of an individual system than what is encoded in the wavefunction. In particular, one should not think of the electron in a hydrogen atom in its ground state as occupying a precise position in space. For example, pilot wave theory (see section 3), in which there actually is such a well defined position of the electron, is ruled out by the completeness assumption. The completeness assumption becomes deeply problematic when quantum mechanics is used to describe the interaction between a measurement apparatus and a 3

14 4 2. The Measurement Problem quantum system. For it is an immediate mathematical consequence of the linearity of the Schödinger equation that if the microscopic system is in a linear superposition then, after such an interaction, the apparatus will be in a linear superposition of macroscopically distinguishable configurations. We see only one configuration (the specific outcome) but the wavefunction (as calculated from the linear Schrödinger equation) makes no such commitment. This is the measurement problem that Bell (see ref. [1], p. 201) summarized so succinctly Either the wavefunction, as given by the Schrödinger equation, is not everything, or it is not right. 2.1 Precise argument We shall now turn to the precise argument in its modern form. We essentially follow Ballentine s treatment (see [37] p ). In quantum mechanics we are dealing with microscopic objects that cannot be seen by the naked eye. Therefore, since we cannot observe these systems directly, all we can do is to try to produce a reliable correlation between some property of the microscopic system and some property of a macroscopic object directly visible to human beings. The result of the measurement is inferred from the configuration of an appropriate macroscopic indicator variable, belonging to the measurement device (i.e a computer output, a meter needle, etc.). Definition: Measurement. Let (I) stand for the system we wish to measure and (II) the apparatus that records the result. A measurement in quantum mechanics is an interaction between a system and the experimental apparatus that produces a unique correlation between some dynamical variable of (I) and an appropriate indicator variable of (II). In order to make use of this definition we must treat the interaction between the system and apparatus in a fully quantum mechanical way. It is here we make use of the first assumption, that the Schrödinger equation is universally valid. The indicator variable of the experimental device may be the center of mass coordinate of a meter needle or some other thing. This argument is completely general and does not depend on such details. Nevertheless the example of the meter needle is good to have in mind. In order to distinguish between the final results, it must be assumed that the final wave-packet of the indicator variable (e.g the center of mass coordinate of the meter needle) is sufficiently localized in configuration space. If it is not sufficiently localized it would be impossible to read off the outcome since its configuration would then not be well-defined. 1 1 Here we are tacitly assuming that what we see is encoded in the wavefunction.

15 2.1. Precise argument 5 First, expand the state of the system (I) in the eigenvectors of the operator we wish to measure, i.e where we have. Further, let the system (II) be in a state where is the indicator variable (perhaps the center of mass coordinate of a meter needle) and the other degrees of freedom that are needed in order to give an exhaustive description of the experimental device (e.g quantum numbers of all the atoms etc.). Let us first consider the simple case where the system (I) is in an eigenstate of the observable. In this case we may apply the definition of a measurement. According to it we should apply an interaction that produces a unique correspondence between the value of the system (I) and the indicator variable of the system (II). How will such a interaction Hamiltonian look like? Well, the argument is so general that we do not need to go into specific details. 2 The only thing that is important is that a unique correspondence is produced by the interaction and that the evolution operator is linear. Let the system (II) be prepared in a state prior to the measurement. The state will then evolve to! " # " " &% $ (2.1) Here the unitary evolution has evolved the system (II) such that the value of the indicator variable is in unique correspondence with the value r. We have made the convenient restriction that the state of the system (I) is not changed by the interaction, but this is strictly speaking not necessary and may be dropped [37]. Further, we have allowed for possible changes in the system (II) by introducing a sum over $. If we think of the indicator variable as the center of mass coordinate of a meter needle, the above implies that it occupies a certain region in space. But of course we must be careful to design the interaction so that the possible results are macroscopically discernible. Hence the positions of the meter needle corresponding to different results of the experiment should be macroscopically distinguishable. Let ' us now consider a more general initial state of the system (I). The most general is of the unitary evolution, in order to deduce for some complex coefficients. Now we make use of the linearity ( *) " # " " $ (2.2) (2.3) 2 One example of such an interaction Hamiltonian would be +-,/.&02143 in the case where the indicator variable is the mass center coordinate of a meter needle.. is a coupling strength and 1 3 is the momentum conjugate to 5. The interaction is taken to be an impulse measurement, i.e during a very short time it is so strong that all other terms in the Hamiltonian may be neglected. This is called a von Neumann impulse measurement and is frequently used when one wishes to model quantum measurements theoretically.

16 6 2. The Measurement Problem Figure 2.1: The picture illustrates the ghostly superposition %/% % of macroscopically distinct configurations, i.e the outcomes labeled in the picture, of a meter needle depicted as the grey arrows. In reality we see only one outcome (depicted here by the black arrow) while the wavefunction in the ghostly superposition makes no such commitment. It is now readily seen that the system (II) is in a superposition of macroscopically distinct indicator states. If we let the indicator variable be the center of mass coordinate of a meter needle, we have just deduced that the final state is not localized around some definite point. Rather, it is a superposition of such well-localized states. But it is a fact of experience that we do not see this in a laboratory experiment. We get one outcome rather than a superposition of outcomes. It is now clear that there is is a clash between the linearity of the Schrödinger equation and the assumption that the wave function provides a complete description of reality. Indeed, there is more to say about the state of the affairs, namely the particular outcome. 3 In Fig. 2.1 the perplexing situation is illustrated. 2.2 Different reactions The measurement problem is the major crossroad when it comes to interpreting quantum theory. Here is a list of reactions that are commonly discussed in the literature. Needless to say, even within these views there are differences between its proponents. 3 It is important to realize just how general this argument is. Effectively we use only the linearity of the evolution operator. One can make a seemingly good objection though. We have assumed that we could prepare the system (II) in an exact state every time we carry out an experiment. But in practice it is impossible to prepare a complex system in exactly the same state each time. However, our inability to prepare the system (II) in exactly the same way every time can be taken into account by describing the system (II) as a mixed state. If one carries out the same calculation as above using density matrices (see [37] p ) we end up with the same contradiction: the indicator will have evolved into macroscopically distinct configurations.

17 2.2. Different reactions 7 Isn t this just a triviality? To the above argument it might be objected that, of course, quantum mechanics cannot make any commitment to any specific outcome. If quantum mechanics would uniquely single out any of these results it would be a deterministic theory. But quantum theory is a probabilistic theory that predicts probabilities of measurement outcomes and nothing more. So, trivially, it is incomplete in this sense. The wavefunction, in the superposition of macroscopically distinguishable configurations, does not encode all states of affairs, especially not the outcome. The reader may be interested to know that this was precisely the conclusion of Einstein and Schrödinger. They also thought this was the trivial solution. Quantum mechanics is incomplete in precisely this way. However, this is not the orthodox interpretation (see below). But perhaps there is a way to save the completeness assumption. Could it not be that quantum mechanics provides a complete description only for microscopic systems and somehow fails to do so when macroscopic systems, like classical measuring devices, are involved? In order to make sense of this idea it is an absolute necessity to make clear what macroscopic and microscopic means. After all a macroscopic system is just a collection of atoms. Thus, one needs to give precise answers to questions like: why should the wavefunction provide a complete description of one atom but not a collection of them? How many particles can at most be involved in order for the wavefunction to provide a complete description, & particles, or perhaps more? Perhaps the number of particles is not the relevant thing. Maybe a system needs to be heavy. But how heavy? One Planck mass, roughly the weight of a grain of sand? Bohr Bohr s solution(?) to the measurement problem was to censor any analysis of the interaction between a classical apparatus and quantum system (see ref. [9], p. 329). To him, these two formed an indivisible and unanalyzable whole. The interaction between macroscopic and microscopic objects is thus, according to Bohr, beyond scientific understanding, and in particular not through a purely quantum mechanical treatment. This is not a view without problems. Macroscopic objects are composed of atoms. So, put in a different way, this is what is being said: When one atom interacts with a collection of atoms one cannot analyze the situation scientifically. 4 Bell wrote wittily about the orthodox view (see ref. [1], p. 83) By resisting the impulse to analyze and localize, mental discomfort can be avoided. This is, as far as I understand it, the orthodox view... Many people are quite content with it. 4 Of course, Bohr was aware of this problem and tried to rectify it by somehow defining what a classical system is. One should also keep in mind that Bohr thought that classical concepts are autonomous from, and conceptually prior to quantum theory [54].

18 8 2. The Measurement Problem In any case, the quantum mechanical treatment of the measurement process has proven valuable in highlighting technically and conceptually important points, in particular the emergence of classical physics and the process of decoherence [19]. Decoherence? A not uncommon answer is that decoherence [19] solves the measurement problem. 5 It is true that decoherence provides a satisfactory explanation for why interference effects do not show up for macroscopic objects. But it does not deal with the perplexity that, according to the quantum dynamics, macroscopic objects will in general be in superpositions of macroscopically distinct configurations. In fact, the unitary evolution equation (2.3) is but an instance of a decoherence process that leads to a suppression of interference phenomena. Decoherence does not take away the superpositions of macroscopically distinct configurations. It predicts them. Therefore, decoherence cannot solve the measurement problem [54]. Does the wavefunction represent a maximal state of knowledge? Could it not be that the wavefunction encodes, not the actual state of affairs, but merely a state of maximal knowledge? When the observer knows about the result of the measurement he can safely collapse the wavefunction according to his increased knowledge (see e.g ref. [62], p ). But collapsing the wavefunction is not a passive act. It will destroy the possibility of interference for future experiments (see e.g ref. [49], p ). Therefore this view is problematic because we are not dealing with a probability but a probability amplitude. We are interested in matters of principle and therefore it is of no concern that such interference experiments are (because of decoherence) practically impossible to carry out. In any case, it is not always clear how one actually should apply the projection postulate (i.e the how to collapse the wavefunction) [38]. Furthermore, it is not clear how the word knowledge is used. Normally we understand that knowledge is about something. And that something should, for the sake of precision, be subject to an exact mathematical description. But clearly, this is missing here. We cannot say that the observer has acquired knowledge about the macroscopic settings of the measurement apparatus since the word macroscopic is not well-defined. Wavefunction collapse as a physical process A different view is that a collapse of the wavefunction is a physical process for which a mathematical precise recipe must be provided. The universal validity of the Schrödinger equation [47] is the questioned. Perhaps there are corrections that come into play when macroscopic systems are involved which brings about a wavefunction collapse yielding one of the states in the superposition. Such a proposal is 5 For example, in [53] Omnès writes about Schrödinger s cat Because of internal decoherence, it can be said that the cat is necessarily either alive or dead. It is not in a quantum superposition.

19 2.2. Different reactions 9 given by Ghirardi, Rimini, and Weber [23]. The present mathematically elegant formulation, in terms of states in Hilbert spaces and linear unitary evolution operators, surely speaks against any such arbitrary ad hoc modification. But if quantum mechanics and its formalism is phenomenological and just the limit of some other theory, such corrections would not be very strange. In Newtonian theory we are since long accustomed to post-newtonian corrections. These corrections are not viewed as arbitrary or ad hoc because we know there is a beautifully simple theory underlying Newtonian mechanics accounting precisely for those corrections. However, this approach is not without its share of problems. Infinite energy production is predicted in Lorentz invariant collapse theories. More worlds? Let us really take seriously the resulting superposition of macroscopically distinguishable configurations. Let us take the resulting superposition not as a problem, but as a factual description of the situation. This idea, taken to its logical conclusion, implies that there are actual parallel universes out there. This is the many-worlds interpretation. Below is a rough description of the idea together with some standard critique, as I have understood it. First of all there is not only one interpretation but many. However, there are essentially two different types of many-worlds interpretations: complete and incomplete ones. The incomplete interpretations [69, 70] postulate extra structure in addition to the wavefunction. In the complete interpretations [71, 72, 48] one would like to view quantum mechanics as a complete description without any need for extra baggage. As a response to a criticism of Bell (see [58] or [1], p. chapter 11) there has recently been a trend to take the complete many worlds interpretations more seriously. I shall therefore focus on these types here and the problems associated with them. If the wavefunction is everything, how do we identify in it the different worlds? To do that one needs to pick out some preferred basis to represent a state in Hilbert space. This is a technical problem and the decoherence program is believed to be able to solve it. See however [20] for technical and conceptual difficulties with mathematical attempts. Suppose however that such a solution to the preferred basis problem exist. Can we explain the relative frequencies, the statistics that quantum theory predicts, within this approach? To highlight this difficulty, consider a billion Stern-Gerlach apparatuses all measuring spin along a -axis. To each apparatus there is a corresponding electron almost in the spin state. Now let the electrons each pass through its corresponding apparatus. This results in a measurement with two possible outcomes ( or ) for each apparatus-electron pair, and a corresponding creation of macroscopically distinguishable configurations, that the many-worlds people call worlds. Since there are & apparatuses and two possible outcomes there will be worlds after the measurement. It is easily seen that the overwhelming majority of these worlds are compatible

20 10 2. The Measurement Problem with roughly a distribution of and outcomes, i.e roughly an equal amount of and outcomes. However, since the electrons were all prepared in an almost state quantum mechanics predicts probabilities and. Consequently, most of the outcomes should be and only very few should be. In order to deal with this situation one must then suppress the bad worlds somehow, and bring forth those that are more compatible with quantum statistics. One then introduces notions like measures of existence [48]. All worlds are not to be on equal footing. I am not sure what is being said here. Does this mean that some worlds exists less that others? But then, what does it mean to say that some worlds exists just a little bit? In the incomplete many-worlds interpretations, I do not see that there is, in principle, a problem with relative frequencies. I think any clean solution of the problem of relative frequencies will involve postulating extra structure and thereby accepting that the quantum mechanical description is incomplete. Hidden variables? Finally, perhaps one should accept that the wavefunction cannot provide a complete description of reality. This is the hidden variable program. The position of the meter needle depicted as a black arrow in Fig. 2.1 is not encoded in the wavefunction but is something separate. More generally, the world we see is not encoded in the wavefunction and therefore one should not be concerned about the wavefunction being in such strange superpositions. The measurement problem arises only when an image of the world is sought in the wavefunction. We shall now have a closer look at a particular hidden variable theory due to de Broglie and Bohm and see how it solves the measurement problem.

21 Pilot Wave Theory For twenty years people were saying that hidden variable theories were impossible. After Bohm did it, some of the same people said now it was trivial. They did a fantastic somersault. First they convinced themselves, in all sorts of ways, that it couldn t be done. And then it becomes trivial. J. Bell We shall now turn to a specific hidden variable theory that goes under the name pilot wave theory, or de Broglie-Bohm theory, or simply Bohmian mechanics. This theory was first introduced by de Broglie in 1928 [8] and later rediscovered 1952 by Bohm [5] in a slightly different form. In a sense, pilot wave theory can be viewed as a conceptually unambiguous and mathematically precise implementation of Bohr s idea of the conceptual priority of classical terms in quantum mechanics, although it should be seriously doubted that he would have liked it. In addition to the usual Schrödinger wavefunction defined on a classical configuration space, there is also a definite classical configuration whose evolution is choreographed in a precise mathematical way by the wavefunction. In accordance with Bohr s view the classical terms, i.e the classical configurations, are truly fundamental in this theory, and in this respect the theory displays a sort of quantum-classical dualism. Bohm did not merely rediscover de Broglie s 1928 pilot wave theory, he also demonstrated that pilot wave theory was capable of reproducing all of the predictions of quantum theory [5]. 1 By assuming that an ensemble of systems (all prepared with the same wavefunction) whose respective classical configurations are initially distributed according to the Born rule, one recovers the quantum 1 Pilot wave theory has been extended to relativistic field theory [11, 25]. See however [26] for a possible problem with the Grassmann field theory of fermions. 11

22 12 3. Pilot Wave Theory statistics for all possible future measurements on that system. The peculiarity is that in pilot wave theory the wavefunction has nothing a priori to do with probabilities, its role being to guide the configuration. As argued by Pauli [6] and Keller [7], this particular assumption can be regarded as physically dubious. Why should the wavefunction, whose role is to guide the configuration, also play the role as a probability? However, there is a similar issue in classical statistical mechanics. There is a welldefined way to compute probabilities in thermal equilibrium using the Gibbs statistical postulate ) (3.1) where is a normalization (also called the partition function) and is the temperature of the system. And as in pilot wave theory, the probability distribution involves an object (the Hamiltonian) that at the deeper deterministic level guides the configuration, which in the case of classical statistical mechanics is the Hamilton s equations. The statistics (3.1) can be understood in classical mechanics as arising in sufficiently complex situations. Therefore the above similarity between and the Hamiltonian (both guides the configuration) suggests that quantum probabilities could be understandable within pilot wave theory without any need for the extra assumption that Bohm made in his 1952 papers. This research field, the origin of quantum probabilities in pilot wave theory, was suggested by Bohm himself (see [5], p. 185) but crucial developments happened only in the early nineties [13, 24]. There are two conceptually different approaches, the first due to Valentini and the second due to Dürr, Goldstein, and Zanghì. My own part in this research field has been to study the emergence of quantum probabilities numerically in the first approach. The results are presented in Paper II. 3.1 The pilot wave dynamics The following is a very brief introduction to pilot wave theory. For a more detailed treatment see for example [1, 5, 9, 10] and the forthcoming book [17]. Pilot wave theory, being in a sense just a simple completion of quantum theory 2, shares with it the Schrödinger equation ( ) (3.2) 2 Historically this view is perhaps mistaken [18] since de Broglie s pilot wave ideas were the seeds of Schrödinger s wave mechanics. Thus, in one sense, quantum mechanics began as a deterministic theory and onle later were stripped of its corpuscle (i.e the classical configuration) by Schrödinger to yield the standard quantum theory.

23 3.1. The pilot wave dynamics 13 First let us present the idea in its full generality and then proceed to a concrete example. Let be the relevant configuration space (if we are dealing with particles it is the space of all particle configurations and if we are dealing with fields it is the space of all field configurations) and a specific configuration,. Whenever it is possible to write a continuity equation for (3.3) where is some vector defined on configuration space constructed out of the wavefunction 3, then the de Broglie-Bohm guiding equation is well-defined and reads Together with the Schrödinger equation (3.2), equation (3.4) constitutes a deterministic dynamics where a complete specification of initial data fully determines the future for all times. As a concrete example, consider a system of interacting particles with masses %/% % and configuration with potential. In this case the Schrödinger equation reads % %/% (3.4) % %/% (3.5) and the corresponding continuity equation is where " Im. Using equation (3.4) this implies the the guiding equation for the th particle Im % %/% (3.6) From equation (3.6) it is clear that the dynamics is explicitly nonlocal. The motion of the th particle is, for a general wavefunction, non-locally dependent on the positions of all the others. Pilot wave theory played a crucial role for Bell when he developed his critique of the impossibility theorems as well as his famous theorem demonstrating that any hidden variable theory must be nonlocal [73]. "!$#&% If one defines then the guiding equation can be written as $' (3.7) 3 There is an infinite amount of possible choices of ($) [22]. However, mathematical simplicity, e.g symmetries, often impose strong restrictions [24].

24 % Pilot Wave Theory One should however exercise some caution since the phase ' is in general multivalued. This happens, for example, for a hydrogen atom in the first excited state. But locally equation (3.7) is perfectly well-defined. By the definition of the guiding equation, if we consider an ensemble of such systems, prepared at some time, with the same wavefunction and an distribution of particle configurations taken to be then, for all future times %/% %, the distribution will be given by %/% % % %/% (3.8) % %/% (3.9) In analogy to classical statistical mechanics this distribution is called an equilibrium distribution. Any distribution with the above property is called equivariant [24]. However, since this is a deterministic theory with well-defined particle positions, there is no a priori reason for why it should have this particular distribution. It is certainly logically possible to imagine other distributions. Assuming that the ensemble is initially delivered with an equilibrium distribution, and using the guiding equation, the pilot wave theory yields the quantum statistics for all possible measurements on that system. Therefore, given this assumption, no experiment can discriminate between quantum theory and pilot wave theory. This result is a double-edged sword. On the one hand it is very nice to see that one reproduces all the experimentally verified predictions of quantum theory and that with a conceptually clean and mathematically unambiguous theory, but on the other hand one gets into trouble with Popper s idea of falsification. The details of the dynamics equation (3.4), one may argue, can not be falsified in equilibrium. 4 Since there are infinitely many possible choices of guiding equations [22] I do think this is a fair objection. Mathematical simplicity may suggest some forms of dynamics over others [24] but in the end no empirical discrimination will be possible. This situation is, however, different if we could have access (somehow) to ensembles not distributed according to the equilibrium distribution equation (3.9). In [16] it is suggested that particles that decouple soon after the big bang might be characterized by an anomalous statistics. 3.2 Solving the measurement problem That way seems to cheap to me A. Einstein 4 It might be, however, that pilot wave theory can be unambiguously applied to situations where it is difficult to see what quantum theory predicts. One example of this is quantum tunneling times [56]. Here it is hard to know what quantum theory really predicts (time is not an operator observable in quantum mechanics) while pilot wave theory seems to give a simple answer.

25 3.2. Solving the measurement problem 15 Pilot wave theory clearly disposes with the idea that the wavefunction provides a complete description of reality. Along with the wavefunction there is a classical configuration. For a believer in pilot wave theory it is of no concern that, after a measurement, the wavefunction is in a linear superposition of macroscopically distinct configurations, since the well-behaved classical configuration is not. And, according to pilot wave theory, it is the classical configuration we see, not the wavefunction. In Fig. 2.1 the black arrow is, in pilot wave theory, represented by a classical configuration guided by the wavefunction. The wavefunction in the ghostly superposition is not what we see. For a more detailed analysis of the measurement in pilot wave theory see [9, 11]. Pilot wave theory deals with the measurement problem surgically. It cleanly amputates the problematic part (the wavefunction evolving into linear superpositions of macroscopically distinct states) by postulating that what we see and experience has nothing to do with the wavefunction, but has everything to do with the position in configuration space occupied by the classical configuration. While it is true that this constitutes a logically consistent solution to the measurement problem one may have mixed feelings how satisfactory this solution is. What follows is a common metaphysical objection to the way pilot wave theory solves the measurement problem. The wavefunction is a dynamical entity that through decoherence will multiply into approximately autonomous branches. Those will presumably evolve in configuration space in a manner as to dynamically imitate the behavior of real worlds containing particles, dust, people, cars, etc. [58]. In pilot wave theory, the configuration will only end up in one of these branches. In this sense these worlds appear very real and it seems strange to me that those worlds should have nothing to to with the world we see. 5 Pilot wave theory might then be criticized in this way. I think this is a valuable critique that lends credit to the many-worlds interpretation. But rather than adopting the many worlds view (that has yet to be formulated consistently in my opinion) I have instead come to think of this as a serious problem of quantum theory itself. If I must accept parallel universes, that will per definition never be seen by me, in order to make sense of a theory, I prefer to take that as a reductio ad absurdum type argument. Interpretations naturally fall into two classes; (1) those interpreting the wavefunction as representing an ensemble of equally prepared systems and (2) those for which the wavefunction describes a property of an individual system (be it completely or incompletely). In the former class we find interpretations such as the statistical interpretation [37] or the epistemic interpretations (e.g Peierl s view [62]) and the latter finds its supporters among collapse proponents, Everettists and Bohmians. To me it seems that, in the end, neither of these classes of views will be correct. Quantum 5 One way to deal with this problem is to hope for a static wavefunction [57]. Such a static wavefunction is predicted by the Wheeler-de Witt equation. The time-evolving trajectory will then enter channels giving rise to time-dependent effective wavefunctions of subsystems. If this is doable (it is a technically difficult problem) that would eliminate the above critique.

26 16 3. Pilot Wave Theory theory somehow lies in between these two extremes, flirting with both and yet refuses commitment. I have therefore come to think that the measurement problem constitutes a real problem for quantum theory, not to be solved by interpreting the theory correctly but instead by replacing it by something better. My current view of pilot wave theory is that it is an advanced toy model. Completing quantum theory by adding a well-defined classical configuration constitutes a valuable trick that allows us to peek into the subquantum domain. This trick gives valuable hints as to how any such underlying theory must be constructed. Many features of pilot wave theory are in fact generic features of any theory underlying the statistics of quantum theory. By Bell s theorem ([1], chapter 2) such a theory must be nonlocal, and by the Bell-Kochen-Specker theorem ([1, 60, 52]) we know that concepts like momentum, energy, spin, etc. cannot be thought of as intrinsic properties of the system alone. Pilot wave theory also gives us very interesting hints concerning the origin of quantum probabilities. We shall turn to this subject in chapter The infamous impossibility proofs That the apparent indeterminism of quantum theory can be simulated deterministically is well known to every experimenter. It is now quite usual, in designing an experiment, to construct a Monte Carlo computer programme to simulate the expected behaviour. The running of the digital computer is quite deterministic even the so-called random numbers are determined in advance. J. Bell It is often claimed that no deterministic theory can reproduce the quantum statistics. The indeterminism of quantum theory is therefore thought of as a fundamental aspect of nature. To support this view, certain impossibility theorems are cited. Without exception, they all make assumptions that is not met by pilot wave hidden variable theory. Here we briefly discuss some examples. In 1932 von Neumann claimed to show that no deterministic theory could reproduce the statistics of quantum theory. von Neumann assumed that expectation values must be additive even for dispersion-free states. 6 Three years later, in 1935, a woman (Grete Hermann) pointed out a glaring deficiency in the argument [60]. Not very surprisingly she was completely ignored. Later in the early sixties Bell rediscovered ([1], chapter 1) Grete s critique of von Neumann s impossibiliy theorem. Nowadays (informed) scientists agree that the 6 A dispersion free state is characterized by having probability one or zero for the outcomes for all possible types of quantum measurements. In a sense, dispersion-free states are nothing but an extreme form of non-equilibrium.

27 3.3. The infamous impossibility proofs 17 von Neumann proof does not even get off the ground. The assumption that expectation values must be additive even for dispersion-free states implies, as clarified by Bell, a trivial contradiction in the case of spin measurements (see e.g [1], p ). It is by no means a logical necessity that expectation values must be additive. Pilot wave theory, for example, is simply not like that. It is enough that the additivity of expectation values is recovered for equilibrium distributions, and that is what happens in pilot wave theory. In the same paper Bell also provided another impossibility proof (later rediscovered by Kochen and Specker 7 ) that apparently makes a more convincing argument against hidden variables. Then Bell went on to criticize his own impossibility proof and noted that it can be dismissed on the same ground as von Neumann s. The Bell-Kochen-Specker impossibility proof assumes that two incompatible experiments must yield the same individual outcome for dispersion-free states. Clearly, neither that is a logical necessity, and it is violated by pilot wave theory. It is enough that one reproduces the quantum statistics (see Paper II for further details) for the equilibrium distribution. As stated by Bell himself, in evaluating those impossibility proofs, pilot wave theory was very valuable. It was simply a matter of understanding how that hidden variable theory circumvented the impossibility proofs. Remarkably, even today there are researchers who claim that hidden variable theories are not possible (see e.g [76]). Naturally, they have not cared to check their general reasoning against pilot wave theory. Then of course we have Bell s own non-locality theorem, which is perhaps the most cited. According to Wigner... the most convincing argument against the theory of hidden variables was presented by J. S. Bell (1964). However, Bell himself did not think of his theorem as establishing any argument against the possibility of hidden variable theories. He thought that his theorem established an incompatibility between local causality and quantum theory ([1], p. 172). The non-locality was not just established for hidden variable theories but also for the standard quantum theory. Bell stressed this repeatedly (see e.g ref. [3] and [1], p. 55, 106, 107, 110, 143). Conveniently, many researches miss (or ignore) this particular part of Bell s work when they talk about hidden variable theories and their supposed impossibility. 7 While Bell s proof makes use of a sort of continuity Kochen-Specker s theorem does without it.

28 18 3. Pilot Wave Theory

29 The Origin of Quantum Probabilities We shall in this section discuss in detail how the quantum probabilities emerges within the pilot wave theory. Given that an ensemble is initially delivered with an equilibrium distribution (3.8), the distribution will remain an equilibrium distribution for all times. We shall now try to relax that assumption and instead start with a more general distribution. It can then be argued, by aid of an -theorem [13], that almost all such initial distributions will relax to the equilibrium distribution on a coarse-grained level. What is actually demonstrated is that a certain class of distributions (characterized by the absence of microstructure) will initially evolve towards the equilibrium distribution on a coarse-grained level, while it is not established that it will actually reach it. Therefore numerical simulations are important. The main result of the numerical simulation reported in Paper II will be presented here. 4.1 The subquantum -theorem In classical statistical mechanics the Gibbs entropy is defined by ' % (4.1) where is a phase space volume element and is a phase space distribution. Since both and obey a Liouville relation 1 this quantity is constant in time. However, if one divides phase space into small cells one may define a coarsegrained distribution such that in each cell. Thus, is taken to be constant inside each cell but varying from one cell to another. Then it can be shown, 1 The boundaries of the volume element is evolved according to Hamilton s equations. 19

30 " " The Origin of Quantum Probabilities under the assumption that initially at ', 2, that the coarse-grained is indeed increasing in time. One should carefully distinguish this type of entropy, that is the Gibbs entropy, from Boltzmann s. They are conceptually very different. We shall comment on that later in section 4.3. In order to define a Gibbs entropy for pilot wave theory let us try to imitate the classical case as far as possible. Because of the fact that the particle motion is restricted to x " %, we will work in configuration space rather than in phase space. For the sake of simplicity we will work in a 3-dimensional configuration space. Generalization to arbitrary dimension is straightforward. First consider the expression (4.1). The important feature was that and separately obey the Liouville relation. However, the probability density in pilot wave theory is subject to the conservation law ' (4.2) which in turn implies that x % ". Since " % is in general not zero, this means that does not satisfy the Liouville property. However, we know that satisfies the continuity equation. This implies that the quantity is constant along trajectories since % % (4.3) Thus we may think of this as a quantum analog of the Liouville relation in classical statistical mechanics. To imitate the behavior we can not just consider the configuration space volume since. However, the quantity has this property. This is readily shown by recalling that % " is a measure of the increase of the volume element (for an infinitesimal lapse of time dt). Thus ' It is easy to see that Taken together equations (4.4) and (4.5 implies) ' ' ' ' (4.4) ' (4.5) (4.6) 2 This is an assumption of absence of microstructure of the initial probability distribution.