Energy and Entropy in Quantum Field Theories by Adam Levine A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Raphael Bousso, Chair Professor Richard Borcherds Professor Yasunori Nomura Summer 2019 Energy and Entropy in Quantum Field Theories Copyright 2019 by Adam Levine 1 Abstract Energy and Entropy in Quantum Field Theories by Adam Levine Doctor of Philosophy in Physics University of California, Berkeley Professor Raphael Bousso, Chair Energy conditions play an important role in constraining the dynamics of quantum field theories as well as gravitational theories. For example, in semi-classical gravity, the achronal averaged null energy condition (AANEC) can be used to prove that it is always slower to traverse through a wormhole than to travel around via its exterior. Such conditions prevent causality violations that would lead to paradoxes. Recent advances have been made in proving previously conjectured energy conditions directly in quantum field theory (QFT) as well as in uncovering new ones. This thesis will be an exploration of various energy inequalities in conformal field the- ories (CFTs) as well as semi-classical quantum gravity. At the core of this work lies the recently proved quantum null energy condition (QNEC). The QNEC bounds the null energy flowing past a point by a certain second shape derivative of entanglement entropy. We will demonstrate that the QNEC represents a deep connection between causality, energy and entanglement in quantum field theories. We explore this connection first in the context of holographic CFTs. Calculations in holographic theories will lead us to conjecture that the so-called diagonal QNEC is saturated in all interacting QFTs. We will then provide further, independent evidence that this conjecture holds for all CFTs with a twist gap by explicitly calculating shape derivatives of entanglement entropy using defect CFT techniques. i To my grandfathers, Bob and Lenny, who I know would have enjoyed learning about quantum entanglement. To Paul and Barbara O'Rourke. ii Contents Contents ii 1 Introduction 1 1.1 Energy conditions in gravity and quantum field theory . 2 1.2 Quantum Focussing and the Quantum Null Energy Condition . 3 1.3 Energy and Entanglement . 5 1.4 Outline . 7 2 Upper and Lower Bounds on the Integrated Null Energy in Gravity 8 2.1 Introduction and Summary . 8 2.2 Review of Induced Gravity on the Brane . 10 2.3 Lower Bound from Brane Causality . 14 2.4 Upper Bound From Achronality . 16 2.5 Applications . 17 2.6 Discussion and Future Directions . 19 3 Geometric Constraints from Subregion Duality 20 3.1 Introduction . 20 3.2 Glossary . 23 3.3 Relationships Between Entropy and Energy Inequalities . 30 3.4 Relationships Between Entropy and Energy Inequalities and Geometric Con- straints . 31 3.5 Discussion . 41 4 Local Modular Hamiltonians from the Quantum Null Energy Condition 44 4.1 Introduction and Summary . 44 4.2 Main Argument . 46 4.3 Holographic Calculation . 49 4.4 Discussion . 51 5 The Quantum Null Energy Condition, Entanglement Wedge Nesting, and Quantum Focusing 54 iii 5.1 Entanglement Wedge Nesting . 56 5.2 Connection to Quantum Focusing . 65 5.3 Discussion . 72 6 Energy Density from Second Shape Variations of the von Neumann En- tropy 76 6.1 Introduction . 76 6.2 Setup and Conventions . 79 6.3 Null Deformations and Perturbative Geometry . 85 6.4 Non-Perturbative Bulk Geometry . 88 6.5 Non-Null Deformations . 94 6.6 Discussion . 97 7 Entropy Variations and Light Ray Operators from Replica Defects 101 7.1 Introduction . 101 7.2 Replica Trick and the Displacement Operator . 106 7.3 Towards saturation of the QNEC . 109 ^ 7.4 Contribution of T++ ............................... 111 7.5 Higher order variations of vacuum entanglement . 114 7.6 Near Vacuum States . 117 7.7 Discussion . 120 Bibliography 123 8 Appendix 134 A Notation and Definitions . 134 B Surface Variations . 138 C z-Expansions . 140 D Details of the EWN Calculations . 142 E The d = 4 Case . 144 F Connections to the ANEC . 147 G Free and Weakly-Interacting Theories . 150 H Modified Ward identity . 155 I Analytic Continuation of a Replica Three Point Function . 157 J Explicit Calculation of c(2) ............................ 162 K Explicit Calculation of γ(1) ............................ 164 L Calculating Fn .................................. 166 M Free Field Theories and Null Quantization . 168 iv Acknowledgments This work would have been impossible without the support and guidance of many people. Chief among them is my advisor, Raphael Bousso, who, despite not yet being my collabo- rator, has been extremely supportive of me in every aspect of this work. I could not have asked for a better advisor. I also benefited immensely from the patient guidance of and collaboration with Stefan Leichenauer. I'm not sure exactly what I would have done without him, but I know it would have been significantly worse. My collaboration with Tom Faulkner, which is detailed in the final chapter of this work, had a large impact on me. Working with Tom was a humbling and inspiring experience. I am extremely grateful for his patience with us graduate students at Berkeley, especially as our project wore on into its second year! I benefited from many influential conversations with Aron Wall, who taught me a ton throughout graduate school. I am also grateful to the Kavli Institute for Theoretical Physics, where I spent a semester as a graduate fellow. While at KITP, I talked frequently with Don Marolf who was extremely supportive, patient and kind to me. To everyone at BCTP - Arvin, Chris, Ven, Illan, Pratik, Vincent, Jason, Zach, Mudassir - thanks for making these past few years so fun. I hope we will continue yelling at each other over physics for many years to come. This work benefited from the support of the Defense Department through the National Defense Science and Engineering Graduate (NDSEG) fellowship. 1 Chapter 1 Introduction The past two decades of research into quantum gravity has seen a rapid convergence of techniques from quantum information, string theory and quantum field theory. Much of this research has been anchored on the fundamental result of Maldacena, who found the first concrete example of the so-called \holographic principle" in string theory [113]. The holographic principle predicts that the information content of gravitating systems - such as a black hole - can be encoded in a theory of one lower dimension [132, 84]. Maldacena found a precise realization of this principle in the so-called AdS/CFT duality, which equates string theory on anti- de Sitter space (AdS) in d + 1 spacetime dimensions with a special non-gravitational theory - a conformal field theory (CFT) - in d spacetime dimensions. Although AdS/CFT provided further evidence for the holographic principle, the idea of a holographic universe arose out of a famous formula, due originally to Jacob Bekenstein, stating that black holes have entropy which scales not with their space-time volume but rather their horizon area [11]. The relationship between entropy and horizon area precisely takes the form Ahorizon SBH = (1.0.1) 4G~ d−2 where G~ = `Planck is the Planck area for a spacetime of dimension d. Although this fundamental formula was originally conjectured using simple thought ex- periments in the 1970s, it was not given a precise, UV realization until 2006, when Ryu & Takayanagi first found that the entropy for some sub-region R in a holographic CFT is in fact given by the area over 4G~ of a specific surface in the dual AdS spacetime [126, 125]. This co-dimension 2 surface, now called the Ryu-Takayanagi (RT) surface, is found by extremizing over all surfaces anchored and homologous to the boundary region R. This result opened the floodgates for research examining the connection between geom- etry and entanglement. To cherry pick a few important examples, Mark van Raamsdonk's work in [121] suggested that space-time connectivity should be related to entanglement. Fur- thermore, the work of Maldacena and Susskind conjectured that [112] entanglement between CHAPTER 1. INTRODUCTION 2 the exterior and interior Hawking modes of an evaporating black hole should holographically generate a geometric, wormhole-like connection. These fascinating ideas together with Ein- stein's equations suggest that if geometry (and therefore spacetime curvature) are related to entanglement there should be a corresponding connection between entanglement and energy density. This latter connection will be the main the subject of this thesis. 1.1 Energy conditions in gravity and quantum field theory To understand the various connections between energy and entanglement, it is instructive to first review various constraints on energy in both semi-classical quantum gravity and quantum field theory. Einstein's equations can always be solved trivially given a metric by computing the Einstein tensor and then declaring that this gives you the stress tensor. Of course, such a procedure for solving Einstein's equations does not tell you whether the solution is physical. For this question, one must consult energy conditions, which constrain the source term in Einstein's equations. In classical gravity, the weakest energy condition which is manifestly true is the null energy condition (NEC). The NEC states that at every point in the spacetime Tkk(x) ≥ 0 (1.1.1) a where k is a null vector in the tangent space at x and where Tµν is the stress tensor of the field theory coupled to gravity. For example, in free, scalar field theory, the null-null component of the stress tensor is Tkk ∼ rkφrkφ (1.1.2) which is manifestly positive. In quantum field theory, this positivity can break due to quantum fluctuations. The canonical example of such a NEC-violating state is the Casimir vacuum.