The Computable Universe: From Prespace Metaphysics to Discrete Quantum Mechanics Martin Leckey B.Sc. (Hons.), M.A. Submitted for the degree of Doctor of Philosophy Philosophy Department, Monash University October 1997 arXiv:1807.10826v1 [quant-ph] 30 Jun 2018 Contents Contents iii List of Figures vi Abstract vii Statement of Originality ix Acknowledgments xi 1 Introduction 1 2 Spaces, Causation, and Reality 7 2.1 Introduction . 7 2.2 Substantivalism, relationism, and the property view . 8 2.3 Spaces and perception . 8 2.4 Spaces and reality . 11 2.5 Spaces as property spaces . 12 2.6 Spaces and causation . 23 2.7 Space-time and causation . 26 2.8 Properties and laws . 26 2.9 Spaces and reduction . 28 2.10 Configuration space, phase space and logical space . 30 2.11 Quantum mechanics . 32 2.12 The wave function and reality . 36 2.13 Conclusion . 39 3 Prespace and Cellular Automata 43 3.1 Introduction . 43 3.2 Building the universe . 46 3.2.a UM1|Solipsistic dualism . 46 3.2.b UM2|Strong dualism . 47 iii Contents 3.2.c UM3|Materialism . 48 3.3 Cellular automata and discrete physics . 49 3.4 Functionalism . 52 3.5 Position and causation . 54 3.6 Mechanism . 56 3.7 Monism . 58 3.8 Spinoza . 58 3.9 Leibniz . 59 3.10 Kant . 60 3.11 Holism . 60 3.12 Relativity . 61 3.13 Computational heuristic principle . 62 3.14 Weak dualism . 64 3.15 Theism . 64 3.16 Conclusion . 67 4 Quantum Mechanics and Discrete Physics 69 4.1 Introduction . 69 4.2 The GRW model . 73 4.3 The CSL model . 75 4.4 Realism and holism . 77 4.5 The Critical Complexity theory of wave function collapse . 83 4.6 Discrete physics . 85 4.6.a Space and time discrete . 85 4.6.b All state quantities discrete valued . 85 4.6.c Discrete physics heuristic . 86 4.7 Discrete physics and wave function collapse . 86 4.7.a Single particle systems|the discrete representation . 87 4.7.b Single-particle wave functions and wave function collapse . 90 4.7.c Many-particle systems|the discrete representation . 92 4.7.d Many-particle systems and wave function collapse . 94 4.7.e Measurement . 96 4.8 The number of particles per wave function . 98 4.9 The collapse of the wave function|suggested form of collapse . 101 4.9.a Single particle systems . 102 4.9.b Many particle systems . 104 4.10 Transition from the \quantum realm" to the \classical realm" . 107 4.11 Comparison with other spontaneous localization theories . 110 4.12 Superconductivity . 113 4.13 The interpretation of the wave function|the ontology of CCQM . 115 4.14 Relativity and quantum field theory . 124 iv Contents 4.14.a Relativity and nonlocality . 124 4.14.b Quantum field theory . 126 5 Quantum Mechanics and Complexity 129 5.1 Relative volume as a measure of complexity . 129 5.1.a Physical complexity . 130 5.1.b Computational measures of complexity . 131 5.2 The computer simulation of quantum mechanics . 135 6 Entropy and the Arrow of Time 141 6.1 The wave function entropy . 141 6.2 The evolution of the wave function entropy with time . 146 6.3 The relationship between the wave function entropy and other entropy measures . 149 Bibliography 163 v List of Figures 2.1 Position space . 9 2.2 Velocity space . 10 3.1 \The Standard Model Lagrangian" (Austern 1991). 57 vi Abstract The central motivating idea behind the development of this work is the concept of prespace, a hypothetical structure that is postulated by some physicists to under- lie the fabric of space or space-time. I consider how such a structure could relate to space and space-time, and the rest of reality as we know it, and the implica- tions of the existence of this structure for quantum theory. Understanding how this structure could relate to space and to the rest of reality requires, I believe, that we consider how space itself relates to reality, and how other so-called \spaces" used in physics relate to reality. In chapter 2, I compare space and space-time to other spaces used in physics, such as configuration space, phase space and Hilbert space. I support what is known as the \property view" of space, opposing both the tra- ditional views of space and space-time, substantivalism and relationism. I argue that all these spaces are property spaces. After examining the relationships of these spaces to causality, I argue that configuration space has, due to its role in quantum mechanics, a special status in the microscopic world similar to the status of position space in the macroscopic world. In chapter 3, prespace itself is considered. One way of approaching this structure is through the comparison of the prespace structure with a computational system, in particular to a cellular automaton, in which space or space-time and all other physical quantities are broken down into discrete units. I suggest that one way open for a prespace metaphysics can be found if physics is made fully discrete in this way. I suggest as a heuristic principle that the physical laws of our world are such that the computational cost of implementing those laws on an arbitrary computational system is minimized, adapting a heuristic principle of this type proposed by Feynman (1982). In chapter 4, some of the ideas of the previous chapters are applied in an examination of the physics and metaphysics of quantum theory. I first discuss the \measurement problem" of quantum mechanics: this problem and its proposed solution are the primary subjects of chapter 4. It turns out that considering how quantum theory could be made fully discrete leads naturally to a suggestion of how standard linear quantum mechanics could be mod- ified to give rise to a solution to the measurement problem. The computational heuristic principle reinforces the same solution. I call the modified quantum me- chanics Critical Complexity Quantum Mechanics (CCQM). I compare CCQM with vii Abstract some of the other proposed solutions to the measurement problem, in particular the spontaneous localization model of Ghirardi et al. (1986). Finally, in chapters 5 and 6, I argue that the measure of complexity of quantum mechanical states I introduce in CCQM also provides a new definition of entropy for quantum mechanics, and sug- gests a solution to the problem of providing an objective foundation for statistical mechanics, thermodynamics, and the arrow of time. viii Statement of Originality This thesis contains no material which has been accepted for the award of any other degree and, to the best of my knowledge, contains no material previously published or written by another person, except where due reference is made in the text of the thesis. Martin J. Leckey ix Acknowledgments I would like to thank my principal supervisor John Bigelow for his assistance and encouragement during the course of my work on this thesis. His metaphysical realism (Bigelow 1988; Bigelow and Pargetter 1990), particularly concerning the existence of properties, has been very influential on me. This shows through in chapter 2, on the metaphysics of space, although he actually holds a substantivalist view of space, in opposition to the property view of space that I support in that chapter. I would also like to thank him for asking me to assist him with his work on laws of nature, which has resulted in a joint paper, Leckey and Bigelow (1995), as well as many joint papers at conferences and philosophy department seminars. During the course of this collaboration, I came up with a new view of laws of nature. I have not included this work in this thesis, but I have included, bound at the back of the thesis, a reprint of the joint paper in which my theory of laws of nature is put forward. As explained in chapter 2, I feel that a theory of laws of nature of this general type is needed to support a property view of space. I would also like to thank my co-supervisor Bill Wignall, from the University of Melbourne School of Physics, for his help, especially with his assistance with many aspects of quantum physics. His uncompromisingly realist approach to quantum physics and his particular interpretation of quantum theory (Wignall 1993) have been stimulating in the development of my own realist interpretation of quantum theory. Others I would like to thank include the staff and fellow students at Monash University philosophy department, and the staff and fellow students at La Trobe University philosophy department, at which I began this thesis. I spent one quarter of a year at the University of California at Davis in 1994, and I would like to thank the staff and students in the philosophy department there, in particular Paul Teller, my supervisor during that time, and Michael Jubien, for stimulating discussions. Many of the staff and students at the School of Physics and the department of History and Philosophy of Science at the University of Melbourne, and the Physics Department at Monash University, have also been of great assistance. The librarians and secretarial staff at all these universities have been of great assistance in doing many tasks that most students would ordinarily do for themselves, so I owe them xi Acknowledgments a good deal of gratitude. The same could be said for many of the people who have given me special assistance at many conferences, departments and other places around the world. There are many people I have learned from in discussions of this work, and from questions asked at conferences and seminars.