Classical and Quantum Causality in Quantum Field Theory or The Quantum Universe by Jonathan Simon Eakins, M.Sci. Thesis submitted to the University of Nottingham for the degree of Doctor of Philosophy, April 2004 Abstract Based on a number of experimentally verified physical observations, it is argued that the standard principles of quantum mechanics should be applied to the Universe as a whole. Thus, a paradigm is proposed in which the entire Universe is represented by a pure state wavefunction contained in a factorisable Hilbert space of enormous dimension, and where this statevector is developed by successive applications of operators that correspond to unitary rotations and Hermitian tests. Moreover, because by definition the Universe contains everything, it is argued that these operators must be chosen self-referentially; the overall dynamics of the system is envisaged to be analogous to a gigantic, self-governing, quantum computation. The issue of how the Universe could choose these operators with- out requiring or referring to a fictitious external observer is addressed, and this in turn rephrases and removes the traditional Measurement Problem inherent in the Copenhagen interpretation of quantum mechanics. The processes by which conventional physics might be recovered from this fundamental, mathematical and global description of reality are particularly investigated. Specifically, it is demonstrated that by considering the changing properties, separabilities and factori- sations of both the state and the operators as the Universe proceeds though a sequence of discrete computations, familiar notions such as classical distinguishability, particle physics, space, time, special relativity and endo-physical experiments can all begin to emerge from the proposed picture. A pregeometric vision of cosmology is therefore discussed, with all of physics ultimately arising from the relationships occurring between the elements of the underlying mathematical structure. The possible origins of observable physics, including physical objects positioned at definite locations in an arena of apparently continuous space and time, are consequently investigated for a Universe that incorporates quantum theory as a fundamental feature. Overall, a framework for quantum cosmology is introduced and explored which at- tempts to account for the existence of time, space, matter and, eventually, everything else in the Universe, from a physically consistent perspective. Keywords: Quantum cosmology, Quantum computation, Pregeometry, Emergence, Fac- torisation and Entanglement, Qubit field theory, Quantum Causal sets, Dis- crete time, Information Exchange, Subregisters, Endo-physics, Self-Referential Quantum automata. Contents 1 Introduction 1 2 History and Background 5 2.1 Time in Mathematics, Philosophy and Biology . 5 2.2 Time in Physics . 7 3 The Quantum Universe 15 3.1 Quantum Mechanics . 15 3.2 Quantum Cosmology . 20 3.3 The Stages Paradigm . 31 4 Classicity from Quantum 38 4.1 Factorisation and Entanglement . 43 4.2 Basis Sets and Operators . 51 4.3 Decoherence . 56 4.3.1 The Theory of Decoherence . 56 4.3.2 Problems with Decoherence . 59 4.3.3 Schrodinger’s Cat’s Stages . 62 5 A Quantum Origin of Space 65 5.1 Background . 65 5.1.1 Bottom-Up Approaches . 66 5.1.2 Top-Down Approaches . 68 5.2 Classical Causal Sets . 69 5.3 Splits and Partitions . 77 5.4 Probability Amplitudes and Quantum Causal Sets . 88 5.5 Factorisation and Entanglement of Operators . 98 5.6 Einstein Locality and Quantum Causal Sets . 112 5.7 Physical Examples . 121 5.7.1 The Quantum Big Bang . 121 5.7.2 EPR Paradoxes . 126 5.7.3 Superluminal Correlation . 130 5.7.4 Persistence . 131 5.7.5 Position and Dimension . 133 6 Quantum Registers as Quantum Computers 139 6.1 Computing with CNOT . 139 6.1.1 Classical Computation . 140 i 6.1.2 Quantum Computation . 142 6.1.3 Classical Calculations and the Bell Inequality . 147 6.1.4 Quantum Calculations and the Bell Inequality. 153 6.2 Information Flow in the Quantum Universe . 164 6.2.1 Types of Transformation . 165 6.2.2 Information Change and Exchange . 174 6.2.3 The Ideal Physics Experiment . 182 7 Quantum Field Theory from Quantum Computation 198 7.1 Preliminary Structure . 199 7.2 Dirac Field Theory . 206 7.2.1 The Hamiltonian . 221 7.2.2 The Momentum Operator . 222 7.2.3 Charge . 224 7.3 Field Theory and CNOT . 225 7.4 Discussion . 228 8 The Developing Quantum Universe 235 8.1 Types of Development . 238 8.2 List-Sort Dynamics . 242 8.3 Examples of List-Sort Dynamics . 248 8.3.1 Example I . 250 8.3.2 Example II . 255 8.3.3 Example III . 258 8.3.4 Example IV . 262 8.4 Generated-Sort Dynamics . 265 8.4.1 The Basis Method . 268 8.4.2 The One-to-Many Method . 279 8.4.3 A Type IV Extension . 288 8.4.4 Summary . 292 8.5 Examine-Decision Mechanisms . 295 8.5.1 Preliminary Considerations . 296 8.5.2 Selective Global Evolution . 297 8.5.3 Selective Local Evolution and Endophysics . 304 8.5.4 A Two Qubit ‘Physicist-Sample’ Universe . 309 8.5.5 A Two Qubit ‘Double Experiment’ Universe . 322 8.6 Concluding Remarks . 329 8.6.1 Self-referential Quantum Computation . 329 ii 8.6.2 The Real Universe . 331 8.6.3 Memory and Information . 332 8.6.4 Reduction without Observers . 335 8.6.5 Time without Time . 337 9 Summary, Conclusions and Future Directions 344 A Classical and Quantum Computation 355 A.1 Classical Computation . 356 A.2 Quantum Computation . 362 B The Dirac Field 372 B.1 Lagrangian Dynamics . 372 B.2 Quantisation . 374 B.3 The Hamiltonian Operator . 377 B.4 The Momentum Operator . 380 B.5 The Charge Operator . 382 References 384 List of Figures 5.1 The Hasse diagram of Hasse diagrams featuring those Causal Sets contain- ing up to three members. 71 456 78 5.2 The family structures present in the transition amplitude from Ψn = Θ123• 23 678 to Ψn+1 = Φ145• . ................................ 93 5.3 One possible network of families produced as a toy-universe develops over five jumps. 94 5.4 Pictorial representation of the relationship between operators and factor states. The process described in Figure a) is forbidden, whereas those of Figures b) and c) are allowed. 112 5.5 Causal set structure for the state of an expanding universe of four qubits. 125 5.6 The cubic lattice formed from eight points, with the corners labelled ac- cording to a set of orthogonal axes through the origin (0, 0, 0). 134 6.1 Classical probability ‘tree’ for a developing bit. 150 8.1 Probability, P (Ψn = [c/d]), of an entangled universe after n steps. 257 8.2 Probability of an entangled universe after n steps for Q = 1/2. 261 iii Acknowledgements Due to his continual guidance, ideas, discussions and suggestions during the entire course of my research, I first thank my supervisor, George Jaroszkiewicz. I would also especially like to thank my family for their neverending support and encouragement, not only over the past few years of my Ph.D., but also throughout the whole of my education and life. On a further note, I am additionally grateful to Dr. Christopher Hide for help with LATEX code and document formatting, and to Drs. Nikolaos Demiris, Cornelius J. Grif- fin, Hilmar Hauer and Andrew Whitehead for illuminating discussions on many different aspects of mathematics. Appreciation also to Professor A. Sudbery and Professor V. Belavkin for constructive criticism and comments on the content of the thesis. Lastly, special thanks to all those friends, colleagues, acquaintances and others who gave me the inspiration and often unexpected moments of insight that made this what it is; if you don’t know who you are, at least I do. I acknowledge the EPSRC (UK) for a research studentship. iv To-morrow, and to-morrow, and to-morrow, Creeps in this petty pace from day to day, To the last syllable of recorded time; - W. Shakespeare Macbeth v 1 Introduction Time permeates just about every sphere of life. Indeed, human civilisations are dominated by regularising and synchronising the events in the surrounding World, and it is easy to argue that without agreeing on a common standard of time, society would be unable to function in the way that it does. In fact, time has played a crucial role in all of Mankind’s development. From agri- cultural dependencies on the cyclicity of the Sun, to the machine timings that structured the Industrial Revolution; from the variable ‘t’ present in most of the equations used in science and engineering, to the parameter vital in rationalising the study of history; from the timetables essential for efficient national and global travel, to the Time Machines of literature and imagination; from the rhythms governing the lives of animals and plants, to the cadences of music and speech; from the measurements necessary for road safety laws, to the quantity specified with pinpoint accuracy when coordinating extraterrestrial exploration; time plays a role in everything. Moreover, human life often appears obsessed with the passage of time. Modern society frequently revolves around questions such as “When is...?”, “What time did...?”, “How long until...?”, and so on, and the ever present threat of mortality seems to heighten the sense that time is a precious commodity to be ‘saved’ or ‘made the most of’ wherever possible. Time is something that employers buy, and the cautious bide. Time is something that ‘waits for no man’, but can appear to ‘fly’, drag’ or ‘stand still’. Time is even something whose effects medical research attempts to ‘hold back’. Even primitive Man recognised the importance of the nature of time for his continued existence. Basic subsistence and the quest for food relied heavily upon an understand- ing of the temporal durations of the seasons, and many archaeologists now believe that ancient monuments such as Stongehenge and the Egyptian Pyramids were used partly as astronomical calendars.