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Explorations in Quantum Gravity and Conformal Bootstrap

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Explorations in Quantum Gravity and Conformal Bootstrap EXPLORATIONS IN QUANTUM GRAVITY AND CONFORMAL BOOTSTRAP A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Amirhossein Tajdini August 2020 c 2020 Amirhossein Tajdini ALL RIGHTS RESERVED EXPLORATIONS IN QUANTUM GRAVITY AND CONFORMAL BOOTSTRAP Amirhossein Tajdini, Ph.D. Cornell University 2020 In this thesis, I apply numerical and analytical bootstrap techniques for a better understanding of quantum gravity. Each chapter is separated by subject. Chapter two is about a proof of the averaged null energy condition (ANEC) in flat spacetime for relativistic quantum field theories in more than two dimensions. The proof is based on microcausality and isolation of the ANEC in the OPE of primary operators in the lightcone limit. I discuss a no-go theorem for massive higher spin particles in chapter three. The no-go theorem is established on analytic properties of the S-matrix in the Eikonal limit. In chapter four, I explore connections between two-dimensional CFTs and three-dimensional pure gravity using the numerical bootstrap. My collaborators and I developed a fast algorithm for bootstrapping two dimensional CFTs using the modular bootstrap. By bootstrapping theories with large central charge, we found a new bound on the scaling dimension of the spectral gap. Chapter five is about the black hole information paradox. Using the replica trick for computing the entropy of Hawking radiation, we found a new saddle-point in the gravitational path integral. This saddle-point corresponds to a wormhole solution connecting different replica sheets. By including this saddle, we showed that the entropy computation is consistent with unitarity. BIOGRAPHICAL SKETCH Amirhossein Tajdini was born in Tehran, Iran on March 12 1991. At the end of high school, he attended the International Physics Olympiad in Merida, Mexico. He then started his undergraduate studies in physics and mathematics at Sharif University of Technology. During his undergraduate studies, he taught physics to high school students who participated in Physics Olympiad. In 2014, he moved to Cornell University for his PhD and joined Professor Hartman’s group from 2015 to the date this draft is being written. He will continue his research as a postdoctoral researcher at the University of Santa Barbara, California. iii Dedicated to Martha and my family. iv ACKNOWLEDGMENTS First and foremost I would like to thank my advisor Tom Hartman for guiding me and supporting me in the past six years. His depth of knowledge, his originality, and his patience were a real inspiration. He was always very generous with his ideas and his time. I am extremely fortunate to have the chance to do research under his supervision during my PhD. Many thanks to Liam McAllister for his guidance through these years and discussions in group meetings and in person about String theory and Cosmology. I am also grateful for other wonderful teachers I had at Cornell including Eanna Flanagan, Peter Lepage, and Csaba Csaki. I am greatly indebted to my excellent collaborators whom I had the previlege to work with and learn from. I would like to especially thank Nima Afkhami-Jeddi and Sandipan Kundu who have been my collaborators on most of the projects. I would also like to thank Juan Maldacena, Edgar Shaghoulian, Ahmed Almheiri, Jorrit Kruthoff, Venkatesa Chandrasekaran, Kanato Goto, Henry Cohn, and Yikun Jiang with whom I collaborated on other projects. I benefited greatly from conversations and dicussions with many people through- out my PhD, many of them now friends of mine. I would like to thank David Simmons-Duffin, Arvin Shahbazi-Moghaddam, David Meltzer, Eric Perlmutter, Mudassir Moosa, Sachin Jain, John Stout, Murat Kologlu, Ofri Telem, Mehmet Demirtas, Ibrahim Shehzad, Ryan Bilotta, Sungwoo Hong, Gowri Kurup, Naomi Gendler, Reza Javadi-Nezhad, Alex Grant, Pouya Asadi, Allic Sivaramakrishnan, Soner Albayrak, and Junkai Dong. There is also life outside physics and graduate school. I am grateful for many friends who made my time in Ithaca so enjoyable. I would like to thank Najva, Saeed, Behrooz, Meraj, Hedyeh, Omid, Hamidreza, Saghar, Ali and Sajjad. v Thanks to my family for believing in me. Without your support and sacrifices throughout my life, I would not be where I am now. Despite being far away from each other, your constant love and support have been my main source of encouragement in the past six years. Finally, I would like to thank my dearest Martha for your care, support and everything else. You changed the graduate school experience for me and you always make me so happy. vi CONTENTS Biographical Sketch.............................. iii Dedication................................... iv Contents.................................... vi List of Tables.................................x List of Figures................................. xi 1 Introduction1 1.1 Conformal Bootstrap..........................4 1.1.1 Bootstrap in Lorentzian Signature...............6 1.1.2 Bootstrapping 3d Gravity...................8 1.2 Information loss in Quantum Gravity.................9 2 Averaged Null Energy Condition 11 2.1 Introduction............................... 12 2.2 Derivation of the ANEC........................ 17 2.3 Average null energy in the lightcone OPE............... 21 2.3.1 Lightcone OPE......................... 21 2.3.2 Contribution to correlators................... 23 2.3.3 Scalar example......................... 26 2.4 Sum rule for average null energy.................... 27 2.4.1 Analyticity in position space.................. 27 2.4.2 Rindler positivity........................ 28 2.4.3 Bound on the real part..................... 31 2.4.4 Factorization.......................... 33 2.4.5 Sum rule............................. 34 2.4.6 Non-conformal QFTs...................... 36 2.5 Hofman-Maldacena bounds....................... 37 2.5.1 Conformal collider redux.................... 38 2.5.2 Relation to scattering with smeared insertions........ 39 2.5.3 Relation to the shockwave kinematics............. 41 2.6 New constraints on higher spin operators............... 42 2.6.1 Es in the lightcone OPE.................... 43 2.6.2 Sum rule and positivity..................... 44 2.6.3 Comparison to other constraints and spin-1-1-4 example.. 45 3 A no-go theorem for Massive Higher Spin Particles 48 3.1 Introduction............................... 49 3.2 Higher Spin Fields in Flat Spacetime................. 56 3.2.1 Eikonal Scattering....................... 58 3.2.2 Higher Spin-graviton Couplings................ 62 3.2.3 Eikonal Kinematics....................... 65 3.2.4 Bounds on Coefficients..................... 67 vii 3.2.5 D = 4 .............................. 70 3.2.6 Comments............................ 75 3.3 Higher Spin Fields in AdSD ...................... 80 3.3.1 Causality and Conformal Regge Theory............ 82 3.3.2 D > 4 .............................. 88 3.3.3 AdS4/CFT3 ........................... 94 3.3.4 Maldacena-Zhiboedov Theorem and Massless Higher Spin Fields.............................. 97 3.3.5 Comments............................ 99 3.4 Restoring Causality........................... 102 3.4.1 Make CFT Causal Again.................... 102 3.4.2 Stringy Operators above the Gap............... 105 3.5 Cosmological Implications....................... 108 4 Fast Conformal Bootstrap and Constraints on 3d Gravity 113 4.1 Introduction............................... 114 4.2 Truncating the Primal Bootstrap................... 117 4.2.1 Setup.............................. 117 4.2.2 Truncation............................ 119 4.2.3 Comments on Monotonicity.................. 120 4.3 Dual Bootstrap............................. 121 4.3.1 Review.............................. 121 4.3.2 Functionals parameterized by zeroes............. 122 4.3.3 Duality............................. 125 4.3.4 From primal solutions to extremal functionals........ 126 4.4 Modular Bootstrap Algorithm..................... 127 4.4.1 Setup.............................. 128 4.4.2 Algorithm............................ 130 4.5 Modular Bootstrap Results....................... 133 4.5.1 3d gravity and summary of existing bounds......... 133 4.5.2 Bound as a function of c .................... 135 4.5.3 c = 12 and the modular j-function.............. 137 4.5.4 Algorithm Benchmarks..................... 138 4.6 Discussion................................ 139 5 Replica Wormholes and the Entropy of Hawking Radiation 143 5.0.1 The island rule for computing gravitational von Neumann entropies............................. 147 5.0.2 Two dimensional eternal black holes and the information paradox............................. 149 5.0.3 Replica wormholes to the rescue................ 152 5.1 The replica trick for the von Neumann entropy............ 156 5.1.1 The replicated action for n ∼ 1 becomes the generalized entropy157 5.1.2 The two dimensional JT gravity theory plus a CFT..... 160 viii 5.2 Single interval at finite temperature.................. 167 5.2.1 Geometry of the black hole................... 169 5.2.2 Quantum extremal surface................... 170 5.2.3 Setting up the replica geometries............... 171 5.2.4 Replica solution as n → 1 ................... 173 5.2.5 Entropy............................. 175 5.2.6 High-temperature limit..................... 176 5.3 Single interval at zero temperature.................. 178 5.3.1 Quantum extremal surface................... 179 5.3.2 Replica wormholes at zero temperature............ 180 5.4 Two
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