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Analysing Emergent Time Within an Isolated Universe Through the Application of Interactions in the Conditional Probability Approach

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Analysing Emergent Time Within an Isolated Universe Through the Application of Interactions in the Conditional Probability Approach View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by South East Academic Libraries System (SEALS) Department of Physics & Electronics Analysing emergent time within an isolated Universe through the application of interactions in the conditional probability approach A thesis submitted in fulfilment of the requirement for the degree of Doctor of Philosophy of Rhodes University by KLH Bryan March 2020 ii Abstract Time remains a frequently discussed issue in physics and philosophy. One interpretation of growing popularity is the ‘timeless’ view which states that our experience of time is only an illusion. The isolated Universe model, provided by the Wheeler-DeWitt equation, supports this interpretation by describing time using clocks in the conditional probability interpretation (CPI). However, the CPI customarily dismisses interaction effects as negligible creating a potential blind spot which overlooks the potential influence of interaction effects. Accounting for interactions opens up a new avenue of analysis and a potential challenge to the interpretation of time. In aid of our assessment of the impact interaction effects have on the CPI, we present rudimentary definitions of time and its associated concepts. Defined in a minimalist manner, time is argued to require a postulate of causality as a means of accounting for temporal ordering in physical theories. Several of these theories are discussed here in terms of their respective approaches to time and, despite their differences, there are indications that the accounts of time are unified in a more fundamental theory. An analytic analysis of the CPI, incorporating two different clock choices, and a qualitative analysis both confirm that interactions have a necessary role within the CPI. The consequence of removing interactions is a maximised uncertainty in any measurement of the clock and a restriction to a two-state system, as indicated by the results of the toy models and qualitative argument respectively. The philosophical implication is that we are not restricted to the timeless view since including interactions as agents of causal interventions between systems provides an account of time as a real phenomenon. This result highlights the reliance on a postulate of causality which forms a pressing problem in explaining our experience of time. iii Declaration I hereby declare that this thesis has not been submitted, either in the same or different form, to this or any other university for a degree and that it represents my own work. I know the meaning of plagiarism and declare that all of the work in this thesis, save for that which is properly acknowledged, is my own. —————————————– KLH Bryan March 2020 iv Acknowledgements Firstly I would like to thank my supervisor, Prof. A.J. Medved, without whom this thesis would not have been possible. Thank you for your guidance, continual assistance, and wide-ranging conversations topics. I would also like to thank Francis Williamson and Pedro Tabensky for their guidance on the philosophical topics and many interesting conversations over coffee. The following funding contributions which supported this research are also owed a debt of gratitude: The Henderson Scholarship, provided by Rhodes University • The NRF/DAAD Bursary (Grant Numbers: 111616) • The NRF bursary through Competitive Programme Grant 93595 • Incorporated in this is the tireless efforts of Mr John Gillam and the Rhodes University Postgraduate Funding Office to whom I will always be grateful. I would also like to thank the many members of the Rhodes Physics Department who have supported me throughout this journey. To my Mom and Dad, thank you for all you have done and continue to do for me. This work represents a culmination of your efforts and love in assisting me in more ways than I can list. To my sisters, Megan and Laura, thank you for being behind me all the way. I cannot wait to force you to pretend to read this. To my grandmother, Ganny, thank you for your continued support in this endeavour and believing I could accomplish the task at hand. Finally, to Michael, thank you, you know why. Regardless of what you may say, this would not have happened without you. Contents Contents v List of Figures viii 1 Introduction 1 1.1 Determining a starting position . 1 1.2 Definitions . 3 1.2.1 Time and change . 3 1.2.2 Static versus frozen systems . 3 1.2.3 Being versus becoming . 5 1.2.4 The flow of time . 6 1.2.5 The arrow of time . 6 1.2.6 Causality . 7 1.3 Our Investigation . 8 2 Time in Physics 10 2.1 A brief history of the “problem of time” . 10 2.1.1 Quantum gravity . 10 2.1.2 Emerging theories . 11 2.2 Newtonian time . 13 2.2.1 Absolute time and relative motion . 14 2.2.2 Absolute time and causality . 14 2.3 Quantum time . 16 Contents vi 2.3.1 The troublesome time operator . 16 2.3.2 An emergent notion of time . 17 2.4 Relativistic time . 17 2.4.1 Special and general relativity . 18 2.4.2 Quantum field theory . 23 2.5 Hope for a resolution through fundamental time . 24 2.5.1 Emerging theories . 24 2.5.2 Interpreting the metric in relativity . 25 2.5.3 Incorporating the quantum field theory metric . 27 2.5.4 Conclusions . 28 3 The Isolated Universe 30 3.1 The Wheeler-DeWitt Equation . 30 3.2 Time in the Page-Wootters Method . 33 3.2.1 An outline of the method . 34 3.3 Criticisms and their responses . 37 3.3.1 The problem of ambiguous clocks and the non-interacting clock solution 37 3.3.2 Kuchar’s criticism: measuring successive times . 40 3.3.3 The refined CPI solution . 41 3.4 Considering interactions . 43 4 Interacting Clocks 44 4.1 The toy Universe . 45 4.2 The lightly damped harmonic oscillator . 46 4.2.1 The clock state . 46 4.2.2 Recovering time and the ‘run time’ limit . 49 4.2.3 The uncertainty and the decoherence rate . 51 4.3 The atomic clock . 53 4.3.1 Keeping time with atomic clocks . 54 4.3.2 The first pulse: Rabi oscillation . 56 4.3.3 Decoherence during Ramsey time . 57 4.3.4 The second pulse time . 58 4.3.5 Analysing the uncertainty . 60 4.4 Implications for interactions . 62 5 Isolation Versus Interaction 64 Contents vii 5.1 The limitations of isolated clocks . 64 5.1.1 The ‘timelessness’ conclusion . 65 5.1.2 The two-system Universe . 68 5.1.3 Interactions as a matter of principle . 69 5.2 The consequences of including interactions . 70 5.2.1 Multi-system description . 71 5.2.2 Ambiguity as superficial issue . 75 6 Philosophical Implications 77 6.1 The experience of time as an illusion . 78 6.1.1 The A-series and B-series of time . 78 6.1.2 The Block Universe in physics . 80 6.1.3 Quantum timelessness . 82 6.2 The experience of time as a physical reality . 84 6.2.1 Interactions and interpretations of time in the CPI . 85 6.3 The role of causality . 87 7 Discussion and Speculations 89 8 Conclusion 95 A Contrasting the conditional probability expressions 98 B The damped harmonic oscillator 101 B.1 The clock state . 101 B.2 The expectation value . 102 B.3 Uncertainty calculation . 103 C The atomic clock 105 C.1 The atomic clock . 105 C.2 The evolution operator . 107 D Research Outputs 108 Bibliography 110 . List of Figures 1.1 An illustration of a static system. This corresponds here to being........... 4 1.2 An illustration of a system undergoing change, corresponding here to both being and becoming. ....................................... 4 1.3 A illustration of a frozen system depicting the lack of conclusive definitions surrounding the concept. 5 2.1 An illustration adapted from the depiction in [27]. This represents a ’phase space’ showing the five versions of string theory along with the underlying M-theory . 12 2.2 A schematic representation of the hierarchy of theories in physics. The question mark represents the as yet unknown ‘theory of everything’ while c, ~, and GN represent speed of light, Plank’s constant, and Newton’s gravitational constant respectively. 12 2.3 Two alternative orders for paths through time.The dotted line represents the suc- cessive order of moments a system passes through as it evolves. Diagram A shows a monotonic order corresponding to the timeline ordering while diagram B shows an example of an alternative ordering that would contradict experiment. 15 2.4 A null cone, with one spatial dimension. Within the cone, paths are timelike while paths outside the cone are spacelike. A path following the cone, representing the path of light, is called lightlike. ................................... 20 2.5 A schematic diagram of a lightcone in the flat spacetime of special relativity . 21 2.6 A schematic diagram of lightcones in the curved spacetime of general relativity. 22 List of Figures ix 4.1 Schematic graph showing the probability of finding S in the excited state as a function of !, adapted from [61]. The maximum probability corresponds to the resonance case where ! = !0. The FWHM, from which we get an expression for the uncertainty, is also shown. 55 5.1 A schematic representation of the Hilbert space of the maximally entangled sys- tems C and R which are perfectly isolated. 69 5.2 A schematic representation of the Hilbert space of the maximally entangled sys- tems C and R when interaction effects are permitted. 71 5.3 A schematic representation of two interacting galaxies. In the first scenario, the expansion of the Universe overcome the gravitational force while in the second scenario the gravitational force overcomes the expansion.
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