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Jia Ding.Pdf (1.136Mb) Quantum Indefinite Spacetime: Part II by Ding Jia A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Applied Mathematics Waterloo, Ontario, Canada, 2018 c Ding Jia 2018 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract We adopt an operational mode of thinking to study spacetime fluctuations and their impacts on questions of quantum gravity. Particular emphasis is put on studying causal structure fluctuations. The study is guided by the principle of causal neutrality, which says that fundamental concepts and laws of physics should be stated without assuming a definite spacetime causal structure. Most traditional concepts and theories of physics violate this principle. A major theme of this thesis is to upgrade the traditional concepts and theories in accordance with the principle of causal neutrality. Among other things, the thesis include works on the following. (1) An axiomatic derivation of the complex Hilbert space structure of quantum theory without assum- ing definite causal structure. (2) A so-called causally neutral quantum field theory (CNQFT) framework for algebraic quantum physics allowing indefinite causal struc- ture. (3) A proposal that causal fluctuation regularizes quantum field ultraviolet divergences (UV) and that the UV regularizing correlation functions with causal fluc- tuations characterize the UV structure of physical states. (4) A study on quantita- tive measures of causality. (5) A so-called \correlation networks" framework to study quantum theory with indefinite causal structure, which generalizes previous frame- works. (6) New definitions of entanglement and entanglement measures that conform to the causal neutrality principle. (7) Ideas on finding a causally neutral analogue of Einstein's equation for quantum spacetime. (8) A simple theorem showing that the communication capacities of general correlations (with definite or indefinite causal structure) defined with respect to state transmission can be reduced to those of chan- nels. iii Acknowledgements I would like to thank my dear supervisors Lucien Hardy and Achim Kempf for their supports in academics and in life. Many times I come to them with vague and preliminary ideas. In hindsight their suggestions/questions/comments often point out exactly the direction along which I need to improve the ideas. I would like to thank my collaborator Nitica Sakharwade for working together and sharing illuminating ideas, and members and guests of Achim's physics of information lab, particularly Jason Pye and Maria Papageorgiou, for a lot of valuable relevant discussions, and the honorary member Yangang Prof Chen for a lot of interesting irrelevant discussions. This work benefits tremendously from the \Quantum Causal Structures" project supported by the John Templeton Foundation, providing multiple opportunities for exchanges among researchers with similar interests. I acknowledge useful discussions especially with Fabio Costa, Ognyan Oreshkov, Caslav Brukner, Paolo Perinotti, Giulio Chiribella, Esteban Castro-Ruiz, Marius Krumm, Philippe Guerin, Costantino Budroni, Xiaobin Zhao, and Yuxiang Yang, Tomas Gonda, TC Fraser and David Schmid. I benefit from discussions/correspondences with many other people, in particular Matti Raasakka, Vern Paulsen, Tian Zhang, Robert Oeckl, Alexander Wilce, Carlo Maria Scandolo, Matthew Graydon, Matt Leifer, Lee Smolin, and Flaminia Giaco- mini. The foundations of QFT reading group Nitica Sakharwade and Doreen Fraser help organize has improved my understanding of quantum field theory. I thank all the participants for the valuable discussions. The work benefits much from the 2017 Spacetime and Information Workshop on the Manitoulin Island. I thank the organizers Natacha Altamirano, Fil Simovic, and Alexander Smith, the host Robert Mann, and all the participants of the workshop. I thank the thesis defence committee members Lucien Hardy, Achim Kempf, Rafael Sorkin, and Florian Girelli, as well as Matti Raasakka, Marius Krumm, and Tian Zhang for their useful comments and suggestions on the draft version of this thesis. I thank the University of Waterloo, the Perimeter Institute and all the people working here for making my research possible. Research at Perimeter Institute is sup- ported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation. This work is partly supported by a grant from the John Templeton Foundation. The opinions expressed in this work are those of the author's and do not necessarily reflect the views of the John Templeton Foundation. Last but not least, I would like to thank my parents Hui Jia (>辉) and Li Jin (金 =), and my partner Yicen Lu (b¦岑) for their gracious understanding and generous support. iv Dedication To my father Hui Jia, who shows me to keep a broad vision, and my mother Li Jin, who shows me to think independently. v Table of Contents List of Figures ix Preface1 1 Foundational considerations2 1.1 The generality of spacetime fluctuations................2 1.2 What fluctuations are fundamental to spacetime?...........5 1.3 Is the fluctuation gravitational?.....................6 1.4 On the operational approach.......................7 1.5 Causal fluctuations............................9 1.5.1 Primacy of causal fluctuations..................9 1.5.2 Causal relation of what?..................... 10 1.5.3 From length and time fluctuations to causal structure fluctuations 12 1.5.4 On the generality of causal fluctuations............. 13 1.5.5 The principle of causal neutrality................ 14 1.6 Questions and opportunities....................... 15 2 Basic framework: operations and correlations 16 2.1 Physical theories as theories of operations and correlations...... 16 2.1.1 Operation............................. 16 2.1.2 Correlation............................ 18 2.1.3 Summary............................. 19 2.2 Example: classical mechanics...................... 19 2.3 Example: quantum theory........................ 21 2.3.1 Quantum networks........................ 21 2.3.2 Operator tensors......................... 23 2.4 Example: local quantum physics/algebraic quantum field theory... 24 2.5 Example: process matrices........................ 25 2.6 Example: operational quantum theory without predefined time.... 26 2.7 Principles for complex Hilbert space quantum theories........ 28 2.7.1 Probabilistic theories....................... 29 2.7.2 Subsystem structures....................... 32 2.7.3 Comments on the framework................... 32 vi 2.7.4 Postulates............................. 33 2.7.5 Derivation............................. 36 3 Causally neutral quantum field theory 37 3.1 QFT in background spacetime...................... 38 3.2 Causaly neutral QFT (CNQFT)..................... 40 3.2.1 A motivating example: the quantum switch.......... 41 3.2.2 The free product algebra..................... 44 3.2.3 An example: the ordinary \two-point" correlation....... 45 3.2.4 State for transition amplitudes without positivity?....... 46 3.2.5 The generalized states...................... 47 3.2.6 The GNS construction...................... 49 3.2.7 On the causal structure...................... 51 3.2.8 On time symmetry........................ 52 3.3 Related works............................... 52 3.4 Ultraviolet structure of the correlation functions............ 54 3.4.1 Field detectors.......................... 55 3.4.2 Field correlation functions.................... 60 3.5 Crossroads: dynamical constraint; quantum gravity.......... 64 3.6 Outlooks.................................. 66 4 Fine-grained causality 68 4.1 Quantitative measures of causal strength................ 69 4.1.1 Causality measures........................ 70 4.1.2 One-shot quantum capacities.................. 73 4.1.3 Quantum causality measures................... 76 4.2 Correlation networks: a special framework............... 80 4.2.1 Basic concepts........................... 82 4.2.2 Quantum theory......................... 83 4.2.3 Compositions........................... 86 4.3 Applications of the special framework.................. 91 4.3.1 Extensions............................. 91 4.3.2 Decomposition.......................... 93 4.3.3 Counterfactual systems...................... 93 4.4 Correlation networks: general frameworks............... 94 4.4.1 Motivations............................ 94 4.4.2 Basic elements.......................... 95 4.4.3 Discussion............................. 96 5 Entanglement in quantum spacetime 97 5.1 Entanglement in quantum spacetime.................. 97 5.1.1 Clarifying some misunderstandings............... 97 5.1.2 Causally neutral entanglement.................. 99 vii 5.1.3 Field entanglement in quantum spacetime........... 100 5.1.4 Aside: Possible lessons for causal set QFT and its entanglement 102 5.2 Einstein constraints?........................... 103 5.2.1 The reconstruction paradigm.................. 103 5.2.2 Horizon entanglement without horizon............. 104 5.2.3 Discussion............................. 106 6 Outlook 108 Appendix A Communication capacities for general correlations 109 Bibliography 111 viii List of Figures 2.1 Quantum network............................. 22 2.2 Operator tensor.............................. 24 2.3 A process matrix setup.........................
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