QUANTUM MEASURE THEORY: A NEW INTERPRETATION Yousef Ghazi-Tabatabai Imperial College arXiv:0906.0294v1 [quant-ph] 1 Jun 2009 A Thesis Submitted for the Degree of Doctor of Philosophy of the University of London and the Diploma of Imperial College January 2009 Declaration Unless specifically mentioned otherwise the work presented in this thesis is my own. Yousef Ghazi-Tabatabai January 2009 i ii Abstract Quantum measure theory can be introduced as a histories based reformulation (and generali- sation) of Copenhagen quantum mechanics in the image of classical stochastic theories. These classical models lend themselves to a simple interpretation in which a single history (a single element of the sample space) is deemed to be `real'; we require only that this real history should not be ruled out by the dynamics, the axioms of which ensure that not all histories are precluded. However, applying this interpretation naively to quantum measure theory we can find experimentally realisable systems (notably the Peres-Kochen-Specker system) in which every history is ruled out by the dynamics, challenging us to formulate a deeper realist framework. Our first response is to hold on to our existing interpretative framework and attempt a revision of the dynamics that would reduce quantum measure theory to a classical dynamics. We explore this approach by examining the histories formulation of a stochastic-collapse model on a simple (discrete) null-lattice, concluding that the drawbacks of this approach outweigh the benefits. Our second response is to abandon our classically inspired interpretation in favour of Sorkin's `co-events', a more general ontology that still allows for strict realism. In this case the `potentially real' objects of the theory (the `beables' in Bell's language) are not individual histories but truth valuation maps, or co-events. We develop & evaluate various co-event schemes that have been suggested to date, finally adopting the multiplicative scheme; the current working model of co-event theory and a promising interpretation of quantum measure theory, though still a work in progress. We conclude by exploring the expression of the dynamics & predictions in this new framework. iii iv Acknowledgements Firstly I would like to thank Fay Dowker for her support, encouragement & supervision of this thesis, and the two papers we have written together. Thanks are certainly due to Rafael Sorkin, whose creative ideas I have spent most of this thesis developing. I am indebted to Rafael for his many insightful, profound & encouraging comments, for his time given in proof reading drafts of parts of this thesis, and for the many conversations on the research presented here. Thanks are also due to Petros Wallden, my collaborator for the paper on which the final chapter of this thesis is based, and who has always been a source of practical and perceptive analysis. Thanks also to David Rideout, Surya Sumati and Wajid Mannan for the many useful dis- cussions regarding the research presented in this thesis. Further thanks to Anna Gustavsson, Alexander Haupt and Wajid Mannan for proof reading drafts of parts of this thesis. This research was funded by a PPARC/STFC studentship. v vi Contents Opening Comments 1 I Quantum Measure Theory 3 1 Introducing The Histories Approach 5 1.1 Decoherent Histories . 5 1.1.1 From States to Histories . 5 1.1.2 The Sum Over Histories Approach . 7 1.1.3 The Decoherence Functional . 8 1.1.4 Decoherence and Emergent Classicality . 9 1.1.5 The Axiomatic Approach . 10 1.1.6 The Consistent Histories Interpretation . 11 1.2 Quantum Measure Theory . 12 1.2.1 Stochastic Theories . 13 1.2.2 The Quantum Measure . 14 1.2.3 Null Sets . 15 1.2.4 The Naive Interpretation . 16 1.3 A Simple Example: The Double Slit System . 16 2 The Kochen-Specker Theorem 19 2.1 The Kochen-Specker Theorem . 19 2.1.1 The Peres Formulation . 19 2.1.2 Events and Null Sets . 22 2.1.3 The Failure of the Naive Interpretation . 22 2.2 Expressing the Peres-Kochen-Specker setup in terms of Spacetime Paths . 22 2.2.1 The Stern-Gerlach Apparatus . 23 2.2.2 Using Stern-Gerlach Apparatus to realise the Peres-Kochen-Specker setup . 24 3 Stochastic Collapse 27 3.1 Opening Comments . 27 3.2 Introduction . 27 3.3 The lattice field model . 28 3.3.1 The unitary theory . 29 3.3.2 The collapse model with the Bell ontology . 31 3.3.3 Quantum and Classical . 32 3.3.4 Equivalence to a model with environment . 33 3.4 Discussion . 35 II Co-Event Schemes for Quantum Measure Theory 37 4 Introducing Co-Events 39 4.1 Summary . 39 4.2 Co-Events in Classical Stochastic Theories . 40 4.3 Co-Events for Quantum Mechanics . 41 vii 4.4 The Various Co-Event Schemes . 43 4.4.1 Classical Co-Events . 44 4.4.2 Linear Co-Events . 45 4.4.3 Quadratic Co-events . 45 4.4.4 General n-Polynomial Co-Events . 46 4.4.5 Multiplicative Co-Events . 47 4.4.6 Ideal Based Co-Events . 48 4.5 Evaluating the Co-Event Schemes . 49 4.6 The Interpretation of the Interpretation . 50 5 The Linear Scheme 53 5.1 Basic Properties . 53 5.1.1 The Linear Dual . 53 5.1.2 Primitivity & Weak Emergent Classicality . 56 5.1.3 Unitality . 57 5.2 Simple Examples . 58 5.2.1 The Linear Coin . 58 5.2.2 The Double Slit System . 59 5.2.3 The Triple Slit System . 59 5.3 The Kochen-Specker-Peres System . 61 5.4 The Failure of the Linear Scheme . 65 5.4.1 The Four Slit System . 66 6 Polynomial Schemes 69 6.1 The Quadratic Scheme . 69 6.1.1 Primitivity & Weak Emergent Classicality . 70 6.1.2 Existence . 71 6.1.3 The Four Slit System . 71 6.1.4 The Failure of the Quadratic Scheme . 72 6.1.5 The Sixteen Slit System . 74 6.2 Higher Order Polynomial Schemes . 75 6.2.1 Basic Properties . 75 6.2.2 The Failure of the Polynomial Schemes . 78 6.2.3 The Many Slit System . 81 III The Multiplicative Scheme 85 7 The Multiplicative Scheme 87 7.1 Basic Properties . 87 7.1.1 Filters & Duals . 87 7.1.2 Primitivity & Weak Emergent Classicality . 89 7.1.3 Existence & Compatibility . 89 7.2 Simple Examples . 90 7.2.1 The Double Slit System . 90 7.2.2 The Triple Slit System . 91 7.2.3 The Four Slit System . 92 7.3 The Kochen-Specker-Peres System . 92 7.3.1 Anhomomorphism Exposed . 93 7.4 Strong Emergent Classicality . 96 7.4.1 Consistency & Compatibility . 96 7.4.2 Classical Partitions . 98 7.5 Infinite Theories . 101 7.5.1 Multiplicative Infinity . 102 7.6 Conclusion . 103 viii 8 Dynamics & Predictions 105 8.1 Dynamical Co-Events . 106 8.1.1 Placing a Measure on the Space of Co-Events . 106 8.1.2 Proof of Theorem 6 . 108 8.2 Approximate Preclusion . 111 8.2.1 The Classical Coin . 111 8.2.2 Cournot's Principle . 112 8.2.3 Approximate Co-Events . 114 8.2.4 Can we Achieve Strong Cournot? . 115 8.2.5 Quantum Operational Weak Cournot . 119 8.3 Conclusion . 120 Conclusion 123 IV Appendix 125 A Proof of Lemma 4 127 B The Existence of Quadratic Co-Events 129 B.1 Integer Decoherence Functional . 129 B.2 General Decoherence Functional . 131 C The Many Slit System 133 C.1 The Double Slit Gedankenexperiment . 133 C.2 The Many Slit Gedankenexperiment . 134 ix x Opening Comments New analysis of a series of now famous experiments early in the 20th century led to the falsification and abandonment of previous physical theories and the adoption of General Relativity and Quantum Mechanics, which became the two pillars of 20th century physics. While both theories proved remarkably successful in their predictive ability, Quantum Me- chanics remained open to interpretative questions leading eventually to the adoption of the Copenhagen Interpretation (usually associated with Bohr & Heisenberg), though there were notable dissenters (in particular Einstein & Schrodinger) and the search for a `realist' and `observer independent' interpretation continued. Indeed, today the interpretation of quan- tum mechanics has become a field in its own right, and is given additional impetus by the growing realisation that an observer independent understanding of quantum mechanics.