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Aspects of Time in Quantum Theory

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Aspects of Time in Quantum Theory Imperial College London Department of Theoretical Physics Aspects of Time in Quantum Theory James M Yearsley October 27, 2011 arXiv:1110.5790v1 [quant-ph] 26 Oct 2011 Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Theoretical Physics of Imperial College London and the Diploma of Imperial College London 1 Declaration I herewith certify that all material in this dissertation which is not my own work has been properly acknowledged. James M Yearsley 2 Abstract We consider a number of aspects of the problem of defining time observables in quantum theory. Time observables are interesting quantities in quantum theory because they of- ten cannot be associated with self-adjoint operators. Their definition therefore touches on foundational issues in quantum theory. Various operational approached to defining time observables have been proposed in the past. Two of the most common are those based on pulsed measurements in the form of strings of projection operators and continuous measurements in the form of complex potentials. One of the major achievements of this thesis is to prove that these two operational approaches are equivalent. However operational approaches are somewhat unsatisfying by themselves. To provide a definition of time observables which is not linked to a particular measurement scheme we employ the decoherent, or consistent, histories approach to quantum theory. We focus on the arrival time, one particular example of a time observable, and we use the relationship between pulsed and continuous measurements to relate the decoherent histories approach to one based on complex potentials. This lets us compute the arrival time probability distribution in decoherent histories and we show that it agrees with semiclassical expectations in the right limit. We do this both for a free particle and for a particle coupled to an environment. Finally, we consider how the results discussed in this thesis relate to those derived by coupling a particle to a model clock. We show that for a general class of clock models the probabilities thus measured can be simply related to the ideal ones computed via decoherent histories. 3 For my parents, to whom I owe everything and For Laura, whose love and support made this possible. but more than all(as all your more than eyes tell me)there is a time for timelessness E.E. Cummings 4 Preface The work presented in this thesis was carried out in the Theoretical Physics Group, Imperial College, between October 2007 and September 2011. The supervisor was Prof. Jonathan J. Halliwell. The results presented in Chapters 3 and 7 were published in Journal of Physics A [99] and Physical Review A [100] respectively. The work presented in Chapter 4 was done in collaboration with Jonathan Halliwell and published in Journal of Physics A [51]. Chapters 5 and 6 are based on work done in collaboration with Jonathan Halliwell and published in Physical Review A [49] and Physics Letters A [50]. The work presented in Chapter 8 was done in collaboration with Delius Downs, Jonathan Halliwell and Anna Hashagen and was published in Physical Review A [101]. I am grateful to Carl Bender, Adolfo del Campo, Fay Dowker, Bei-Lok Hu, Seth Lloyd, Gonzalo Muga, Lawrence Schulman and Antony Valentini for their comments and sugges- tions on the material contained in this thesis. Some of the work in this thesis was carried out during a short stay at the Apuan Alps Centre for Physics at the Towler Institute, Italy during the conference \21st Century direc- tions in de Broglie-Bohm theory and beyond." I would like to thank Mike Towler and the other conference organisers for their hospitality. In addition I would like to thank Kate Clements, Ben Hoare, Steven Johnston, Sam Kitchen, Johannes Knoller, Noppadol `Omega' Mekareeya, Tom Pugh and David Weir for many useful, informative and occasionally even relevant discussions. None of the work set out in this thesis would have been possible without the limitless support and encouragement of my supervisor, Jonathan Halliwell. I am grateful for all his help over the four years of my PhD here at Imperial and I consider myself incredibly fortunate to have had him as a teacher and collaborator. 5 Contents Abstract 3 Preface 5 1 Introduction 10 1.1 A Brief History of Time in Quantum Theory . 10 1.2 The Quantum Zeno Effect . 16 1.3 The Backflow Effect . 19 1.4 The Arrival Time Problem in Quantum Mechanics . 20 1.4.1 General Theory . 20 1.4.2 Standard Forms of the Arrival Time Distribution . 23 1.5 The Decoherent Histories Approach to Quantum Theory . 26 1.6 The Decoherent Histories Approach to the Arrival Time Problem: Introduc- tion and History . 29 1.7 Summary and Overview of the Rest of This Thesis . 31 2 The Path Decomposition Expansion (PDX) 34 2.1 Introduction . 34 2.2 The Path Decomposition Expansion (PDX) . 34 2.3 Using the PDX: Scattering States for a Complex Step Potential . 38 2.4 A Semiclassical Approximation . 40 3 The Propagator for the Step and Delta Function Potentials, using the PDX 43 3.1 Introduction . 43 3.2 The Brownian Motion Definition of the Propagator . 45 3.3 The Step Potential . 46 3.4 The Delta Function Potential . 48 6 4 On the Relationship between Complex Potentials and Strings of Projection Operators 52 4.1 Introduction . 52 4.2 Review and Extension of Earlier Work . 55 4.3 Detailed Formulation of the Problem . 57 4.4 Exact Analytic Results . 64 4.5 Asymptotic Limit . 68 4.6 A Time Averaged Result . 69 4.7 Numerical Results . 72 4.8 Timescales . 75 4.9 Summary and Discussion . 77 5 Arrival Times, Complex Potentials and Decoherent Histories 79 5.1 Introduction . 79 5.2 The Classical Arrival Time Problem via a Complex Potential . 82 5.3 Calculation of the Arrival Time Distribution via a Complex Potential . 84 5.4 Decoherent Histories Analysis for a Single Large Time Interval . 89 5.5 Decoherent Histories Analysis for an Arbitrary Set of Time Intervals . 92 5.5.1 Class Operators . 92 5.5.2 An Important Simplification of the Class Operator . 94 5.5.3 Probabilities for Crossing . 95 5.5.4 Decoherence of Histories and the Backflow Problem . 95 5.5.5 The Decoherence Conditions . 96 5.5.6 Checking the Decoherence Condition for Wavepackets . 97 5.6 Summary and Conclusions . 103 6 Arrival Times, Crossing Times and a Semiclassical Approximation 105 6.1 Introduction . 105 6.2 Semiclassical Derivation of the Arrival Time Class Operators . 108 6.3 Arrival Times, Crossing Times and First Crossing Times . 110 6.4 Crossing Class Operators in Decoherent Histories . 113 6.5 Crossing Time Probabilities and Decoherence of Crossing Histories . 115 6.6 Summary and Discussion . 117 7 7 Quantum Arrival Time for Open Systems 119 7.1 Introduction . 119 7.2 Arrival Time for Open Quantum Systems . 122 7.3 Quantum Brownian Motion . 125 7.4 Properties of the Arrival Time Distribution in Quantum Brownian Motion . 129 7.5 The Decoherent Histories Approach to the Arrival Time Problem . 133 7.6 Decoherence of Arrival Times in Quantum Brownian Motion . 136 7.6.1 General Case . 136 7.6.2 The Near-Deterministic Limit . 139 7.7 Summary and Discussion . 144 8 Quantum Arrival and Dwell Times via an Ideal Clock 146 8.1 Introduction . 146 8.1.1 Opening Remarks . 146 8.1.2 Dwell Time . 147 8.1.3 Clock Model . 147 8.1.4 Connections to Earlier Work . 149 8.1.5 This Chapter . 150 8.2 Arrival Time Distribution from an Idealised Clock . 150 8.2.1 Weak Coupling Regime . 150 8.2.2 Strong Coupling Regime . 153 8.3 Dwell Time Distribution from an Idealised Clock . 156 8.4 Conclusion . 159 9 Summary and Some Open Questions 160 9.1 Summary . 160 9.2 Open Questions and Future Work . 161 Appendix: Some Properties of the Current 163 8 List of Figures 1.1 Probabilities for Space-like and Time-like Surfaces . 12 1.2 The Arrival Time Problem . 21 2.1 The Path Decomposition Expansion (PDX) . 36 3.1 A Typical Path From x = 0 to x = 0 On Our Lattice. 46 3.2 The Bijection Between n(k) and Cn. ...................... 48 4.1 The Path Decomposition Expansion for gP .................... 59 4.2 A Plot Comparing the Functions fV (t) and fP (t). 63 4.3 Plot of the Exact Value of fP (t) Compared With an Analytic Approximation. 73 4.4 A Plot of the Function S(t)............................ 74 5.1 Recap of the Arrival Time Problem. 79 5.2 Plot of the Function f(u). 100 5.3 The Arrival Time of a Superposition of Two Wavepackets. 103 6.1 Crossing Times . 106 6.2 Crossing Times and First Crossing Times . 112 7.1 Arrival Times in Terms of Initial States and Propagators. 139 8.1 The Arrival Time Problem Defined Using a Model Clock. 148 8.2 The Dwell Time Problem, Defined Using a Model Clock. 156 8.3 The PDX Used for the Dwell Time Problem. 157 9 1 Introduction The time is out of joint; O curs´edsprite, That ever I was born to set it right! Shakespeare, Hamlet 1.1 A Brief History of Time in Quantum Theory The treatment of time observables is an important loose end in quantum theory [81, 80]. Whilst there is no indication of any disagreement between the predictions of quantum theory and the outcomes of experiment, time observables constitute a class of quantities that are observables in classical mechanics, but for which no satisfactory treatment exists in standard quantum theory. As well as being an important topic in its own right, a consistent treatment of time observables might also shed light on problems in quantum gravity and quantum cosmology.
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