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Towards a Theory of Quantum Gravity Through Geometrization of Quantum Mechanics

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Towards a Theory of Quantum Gravity Through Geometrization of Quantum Mechanics Towards a Theory of Quantum Gravity Through Geometrization of Quantum Mechanics Thesis by ChunJun Cao In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2018 (Defended May 1, 2018) ii © 2018 ChunJun Cao ORCID: 0000-0002-5761-5474 All rights reserved iii ACKNOWLEDGEMENTS Before I start my long list of appreciations, I think it is worth reminding myself that the greatest lesson I have learned in my past few years is that no accomplishment of mine is that of a single person; rather, it is insignificant in comparison to all the people who have made me the person I am today. I would like to begin by thanking my wonderful advisor, Sean Carroll, for his invaluable support and wisdom in physics as well as in life. I am immensely grateful for all the guidance and encouragement Sean has offered over the years. My academic life at Caltech has been fruitful and exhilarating thanks to Sean’s dedication to physics as well as the well-being of students. I am thankful to the members of my thesis committee, Clifford Cheung, David Hsieh, Anton Kapustin, and John Preskill, for offering their valuable insights and expertise. I also thank my collaborators, Ning Bao, Aidan Chatwin-Davies, Sebastian Fischetti, Nick Hunter-Jones, Cindy Keeler, Junyu Liu, Shaun Maguire, Liam McAllister, Spiros, Michalakis, Jason Pollack, Grant Remmen, Ashmeet Singh, Eugene Tang, Michael Walter, Zitao Wang, Zhao Yang, Yuan Yao, Ariel Zhitnitsky, and Claire Zukowski, with whom I have had a wonderful time exploring the realm of physics. I am especially thankful for Ning Bao and Ariel Zhitnitsky, who are wonderful collaborators and great mentors. I also thank my friend Aidan Chatwin-Davies, who is always considerate, vibrant and full of optimism throughout our five years at Caltech. I very much enjoyed my conversations with Zitao Wang, who has been patient in entertaining my wildest questions and speculations. I am indebted to Tony Bartolotta, Bartek Czech, Dan Harlow, Matthew Heydeman, Alex Kubica, Jenia Mozgunov, Ingmar Saberi, Brian Swingle, Xiao-Liang Qi, Gunther Uhlmann, Mark van Raamsdonk, Guifre Vidal, Tian Wang, Mark Wise,Yi-zhuang You, and many others for the interesting physics discussions. I also thank Caltech, for fostering an intellectually stimulating environment and for support its students. It truly has been a humbling experience to learn and walk among some of the giants of our time. My deepest appreciation to all my friends, who made my PhD years warm and full of life: Bob Briton, Alice Chik, Steven Dong, Frank Hong, Voonhui Lai, Jonas Lippuner, James Liu, John Pang, Alvin Phua, Jamie Rankin, Bekah Silva, Yong Sheng Soh, Zhan Su, Alex Tseng, Becky Wang, Dan Wu, and Lucie Xiang. My iv special thanks to John Pang, for his generousity and hospitality for letting me sleep on the couch and using his car when necessary. We had a lot of good times at Bridges, too. I am immensely grateful for my fellow group mates, Ashmeet Singh, Alec Ridgeway, and James St. Germaine-Fuller, for the consistent social and emotional support as well as wonderful physics interactions. I also thank the Dungeons and Dragons and our PC gaming crew: Aidan Chatwin-Davies, Mike Grudic,David Guszejnov, Florian Hoffmann, Seva Ivanov, Anna Komar, Alex Kubica, Clare Lahey, and Jenia Mozgunov. My thanks to the Caltech Ultimate Frisbee pickup group, and Caltech Concert Band and Chamber Music, which helped maintain my mental sanity during the initial years of graduate school. I recall some of the memory-making restaurants like Sansai for giving us cheap but mostly reliable food when I am too lazy to cook. I also very muched enjoyed my visits to the Counter and Abricott. I still think the Top Restaurant at Allen and Colorado makes the best chicken katsu, disagree with me all you want. Thanks to Sam Woo’s for the interesting food making; it has certainly made an indelible impression on me during my last year. Lastly, I would like to thank my parents, who have been supportive in whatever goal I pursue, and for caring and guiding me throughout the past five years. I am grateful to my girlfriend who has been supportive even in my worse days. And thanks to God who sustains me by His grace and mercy. I would like to acknowledge the Natural Sciences and Engineering Research Council of Canada (NSERC) which help support my research. The research is funded in part by the Walter Burke Institute for Theoretical Physics at Caltech, by DOE grant DE-SC0011632, by the Foundational Questions Institute, by the Gordon and Betty Moore Foundation through Grant 776 to the Caltech Moore Center for Theoretical Cosmology and Physics, and by the Natural Sciences and Engineering Research Council of Canada (NSERC). v ABSTRACT In this thesis, we adapt an approach by assuming quantum mechanics as a funda- mental theory of nature and attempt to recover familiar concepts such as space-time geometry and gravity from quantum wavefunctions and their unitary evolutions. More specifically, we explore a number of approaches in “geometrizing” quantum systems using techniques such as tensor networks and manifold learning. We find that consistency conditions in quantum gravity can be used to put constraints on tensor network models that approximate the anti-de Sitter/Conformal Field Theory correspondence. Furthermore, quantum circuits and tensor networks can also be used to describe cosmological models and reproduce important features of space- time configurations such as de Sitter space. We find that a generic framework using quantum circuit to describe cosmology puts an upper bound on the number of e-folds during the inflationary phase of the Universe’s expansion. In addition to tensor network models, we also propose a Bulk Entanglement Gravity framework that analyzes the entanglement data of a quantum state in a Hilbert space without any a priori assumptions on geometry, such as the likes of a boundary conformal field theory. We find that from an amorphous configuration, one can directly re- cover geometry of bulk space-time from a generic class of wavefunctions that is fully characterized in this thesis via quantum entropy cone techniques. We find that under a number of assumptions, it is possible to derive linearized Einstein’s equation from a version of Jacobson’s entanglement equilibrium conditions for an emergent spacetime geometry in the weak field limit near Minkowski space. We show that non-local entanglement perturbations display features of wormhole-like configurations. We also clarify connections between Bulk Entanglement Gravity and highly generic features in quantum error correction codes that can be used to derive gravity. vi PUBLISHED CONTENT AND CONTRIBUTIONS [1] Ning Bao, ChunJun Cao, Sean M. Carroll, Aidan Chatwin-Davies, Nicholas Hunter-Jones, Jason Pollack, and Grant N. Remmen. “Consistency condi- tions for an AdS multiscale entanglement renormalization ansatz correspon- dence”. In: Phys. Rev. D91.12 (2015), p. 125036. doi: 10.1103/PhysRevD. 91.125036. arXiv: 1504.06632 [hep-th]. C. C participated in the conception of the project as well as the formulation of the main arguments, performed mathematical analysis of the entanglement bounds, and helped write the manuscript. [2] Ning Bao, ChunJun Cao, Sean M. Carroll, and Aidan Chatwin-Davies. “De Sitter Space as a Tensor Network: Cosmic No-Hair, Complementarity, and Complexity”. In: Phys. Rev. D96.12 (2017), p. 123536. doi: 10.1103/ PhysRevD.96.123536. arXiv: 1709.03513 [hep-th]. C. C participated in the conception of the project, drafted and performed mathematical analysis of the arguments, and helped write the manuscript. [3] Ning Bao, ChunJun Cao, Sean M. Carroll, and Liam McAllister. “Quantum Circuit Cosmology: The Expansion of the Universe Since the First Qubit”. In: (2017). arXiv: 1702.06959 [hep-th]. C. C participated in the conception of the project as well as the formulation of the main arguments, performed mathematical analysis, and helped write the manuscript. [4] Ning Bao, ChunJun Cao, Michael Walter, and Zitao Wang. “Holographic entropy inequalities and gapped phases of matter”. In: JHEP 09 (2015), p. 203. doi: 10.1007/JHEP09(2015)203. arXiv: 1507.05650 [hep-th]. C. C participated in the conception of the project, performed analytical and numerical analysis that led to the theorems and their proofs, and helped write the manuscript. [5] ChunJun Cao, Sean M. Carroll, and Spyridon Michalakis. “Space from Hilbert Space: Recovering Geometry from Bulk Entanglement”. In: Phys. Rev. D95.2 (2017), p. 024031. doi: 10.1103/PhysRevD.95.024031. arXiv: 1606.08444 [hep-th]. C. C participated in the conception of the project, drafted and performed mathematical analysis of the arguments, and helped write the manuscript. [6] ChunJun Cao and Sean M. Carroll. “Bulk entanglement gravity without a boundary: Towards finding Einstein’s equation in Hilbert space”. In: Phys. Rev. D97.8 (2018), p. 086003. doi: 10.1103/PhysRevD.97.086003. arXiv: 1712.02803 [hep-th]. C. C participated in the conception of the project, drafted and performed mathematical analysis of the arguments, and helped write the manuscript. vii TABLE OF CONTENTS Acknowledgements . iii Abstract . v Published Content and Contributions . vi Table of Contents . vii List of Illustrations . x List of Tables . xvi Chapter I: Introduction . 1 Chapter II: Consistency Conditions for an AdS/MERA Correspondence . 7 2.1 Introduction . 7 2.2 AdS/MERA . 9 2.2.1 Review of the MERA . 9 2.2.2 An AdS/MERA Correspondence? . 12 2.3 MERA and Geometry . 15 2.3.1 Consistency conditions from matching trajectories . 15 2.3.2 Limits on sub-AdS scale physics . 17 2.4 Constraints from Boundary Entanglement Entropy . 18 2.4.1 MERA and CFT Entanglement Entropy . 19 2.4.2 Constraining SMERA ...................... 21 2.4.3 Matching to the CFT .
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