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The Quantum Focussing Conjecture and Quantum Null Energy Condition

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The Quantum Focussing Conjecture and Quantum Null Energy Condition The Quantum Focussing Conjecture and Quantum Null Energy Condition by Jason Koeller 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 Yasunori Nomura Professor Nicolai Reshetikhin Summer 2017 The Quantum Focussing Conjecture and Quantum Null Energy Condition Copyright 2017 by Jason Koeller 1 Abstract The Quantum Focussing Conjecture and Quantum Null Energy Condition by Jason Koeller Doctor of Philosophy in Physics University of California, Berkeley Professor Raphael Bousso, Chair Evidence has been gathering over the decades that spacetime and gravity are best under- stood as emergent phenomenon, especially in the context of a unified description of quan- tum mechanics and gravity. The Quantum Focussing Conjecture (QFC) and Quantum Null Energy Condition (QNEC) are two recently-proposed relationships between entropy and geometry, and energy and entropy, respectively, which further strengthen this idea. In this thesis, we study the QFC and the QNEC. We prove the QNEC in a variety of contexts, including free field theories on Killing horizons, holographic theories on Killing horizons, and in more general curved spacetimes. We also consider the implications of the QFC and QNEC in asymptotically flat space, where they constrain the information content of gravitational radiation arriving at null infinity, and in AdS/CFT, where they are related to other semiclassical inequalities and properties of boundary-anchored extremal area surfaces. It is shown that the assumption of validity and vacuum-state saturation of the QNEC for regions of flat space defined by smooth cuts of null planes implies a local formula for the modular Hamiltonian of these regions. We also demonstrate that the QFC as originally conjectured can be violated in generic theories in d 5, which led the way to an improved formulation subsequently suggested by Stefan Leichenauer.≥ i Contents Contents i List of Figures iii List of Tables vi 1 Introduction 1 2 Proof of the Quantum Null Energy Condition 4 2.1 Introduction . 4 2.2 Statement of the Quantum Null Energy Condition . 8 2.3 Reduction to a 1+1 CFT and Auxiliary System . 9 2.4 Calculation of the Entropy . 14 2.5 Extension to D = 2, Higher Spin, and Interactions . 25 3 Holographic Proof of the Quantum Null Energy Condition 28 3.1 Introduction . 28 3.2 Statement of the QNEC . 30 3.3 Proof of the QNEC . 33 3.4 Discussion . 44 4 Information Content of Gravitational Radiation and the Vacuum 49 4.1 Introduction . 49 4.2 Asymptotic Entropy Bounds and Bondi News . 53 4.3 Implications of the Equivalence Principle . 58 4.4 Entropy Bounds on Gravitational Wave Bursts and the Vacuum . 64 5 Geometric Constraints from Subregion Duality Beyond the Classical Regime 66 5.1 Introduction . 66 5.2 Glossary . 69 5.3 Relationships Between Entropy and Energy Inequalities . 76 ii 5.4 Relationships Between Entropy and Energy Inequalities and Geometric Con- straints . 77 5.5 Discussion . 90 6 Local Modular Hamiltonians from the Quantum Null Energy Condition 93 6.1 Introduction and Summary . 93 6.2 Main Argument . 95 6.3 Holographic Calculation . 98 6.4 Discussion . 100 7 Violating the Quantum Focusing Conjecture and Quantum Covariant Entropy Bound in d 5 dimensions 103 7.1 Introduction . .≥ . 103 7.2 Violating the QFC in Gauss-Bonnet Gravity . 105 7.3 Violating the Generalized Covariant Entropy Bound . 109 7.4 Discussion . 110 8 The Quantum Null Energy Condition in Curved Space 112 8.1 Introduction and Summary . 112 8.2 Scheme-(in)dependence of the QNEC . 114 8.3 Holographic Proofs of the QNEC . 121 8.4 Discussion . 129 A Appendices 132 A.1 Correlation Functions . 132 A.2 Details of the Asymptotic Expansions . 134 A.3 KRS Bound . 135 A.4 Single Graviton Wavepacket . 138 A.5 Non-Expanding Horizons and Weakly Isolated Horizons . 143 A.6 Scheme Independence of the QNEC . 145 Bibliography 152 iii List of Figures 2.1 The spatial surface Σ splits a Cauchy surface, one side of which is shown in yellow. The generalized entropy Sgen is the area of Σ plus the von Neumann entropy Sout of the yellow region. The quantum expansion Θ at one point of Σ is the rate at which Sgen changes under a small variation dλ of Σ, per cross-sectional area of the variation. The Quantum Focussing Conjecture states that the quantumA expansion cannot increase under a second variation in the same direction. If the classical expansion and shear vanish (as they do for the green null surface in the figure), the Quantum Null Energy Condition is implied as a limiting case. Our proof involves quantization on the null surface; the entropy of the state on the yellow spacelike slice is related to the entropy of the null quantized state on the future (brighter green) part of the null surface. 6 2.2 The state of the CFT on x > λ can be defined by insertions of @Φ on the Euclidean plane. The red lines denote a branch cut where the state is defined. 11 2.3 Sample plots of the imaginary part (the real part is qualitatively identical) of the na¨ıve bracketed digamma expression in (2.74) and the one in (2.78) obtained from analytic continuation with z = m iαij for m = 3 and various values − − of αij. The oscillating curves are (2.74), while the smooth curves are the result of applying the specified analytic continuation prescription to that expression, resulting in (2.78). 23 3.1 Here we show the region (shaded cyan) and the boundary Σ (black border) before and after the null deformation.R The arrow indicates the direction ki, and Tkk is being evaluated at the location of the deformation. The dashed line indicatesh i the support of the deformation. 30 3.2 The surface in the bulk (shaded green) is the union of all of the extremal surfaces anchoredM to the boundary that are generated as we deform the entan- gling surface. The null vector ki (solid arrow) on the boundary determines the deformation, and the spacelike vector sµ (dashed arrow) tangent to is the one µM we construct in our proof. The QNEC arises from the inequality s sµ 0. 38 ≥ iv 4.1 Penrose diagram of an asymptotic flat spacetime. Gravitational radiation (i.e., Bondi news) arrives on a bounded portion of + (red). The asymptotic regions before and after this burst (blue) are RiemannI flat. The equivalence principle requires that observers with access only to the flat regions cannot extract classical information; however, an observer with access to the Bondi news can receive information (see Sec. 4.3). We find in Sec. 4.4 that the asymptotic entropy bounds of Sec. 4.2 are consistent with these conclusions. 50 5.1 The logical relationships between the constraints discussed in this paper. The left column contains semi-classical quantum gravity statements in the bulk. The middle column is composed of constraints on bulk geometry. In the right column is quantum field theory constraints on the boundary CFT. All implications are true to all orders in G~ 1=N. We have used dashed implication signs for those that were proven to all orders∼ before this paper. 68 5.2 The causal relationship between e(A) and D(A) is pictured in an example space- time that violates . The boundary of A's entanglement wedge is shaded. Notably, in Cviolating ⊆ E spacetimes, there is necessarily a portion of D(A) that is timelikeC ⊆ related E to e(A). Extremal surfaces of boundary regions from this portion of D(A) are necessarily timelike related to e(A), which violates EWN. 78 5.3 The boundary of a BCC-violating spacetime is depicted, which gives rise to a violation of . The points p and q are connected by a null geodesic through the bulk. TheC ⊆ boundary E of p's lightcone with respect to the AdS boundary causal structure is depicted with solid black lines. Part of the boundary of q's lightcone is shown with dashed lines. The disconnected region A is defined to have part of its boundary in the timelike future of q while also satisfying p D(A). It follows that e(A) will be timelike related to D(A) through the bulk, violating2 . 79 C ⊆ E 5.4 The surface M and N are shown touching at a point p. In this case, θM < θN . The arrows illustrate the projection of the null orthogonal vectors onto the Cauchy surface. 82 5.5 This picture shows the various vectors defined in the proof. It depicts a cross- section of the extremal surface at constant z. e(A)vac denotes the extremal surface in the vacuum. For flat cuts of a null plane on the boundary, they agree. For wiggly cuts, they will differ by some multiple of ki. 88 6.1 This image depicts a section of the plane u = t x = 0. The region is defined to be one side of a Cauchy surface split by the codimension-two− entanglingR surface @ = (u = 0; v = V (y); y) . The dashed line corresponds to a flat cut of the nullR plane.f . .g . 94 A.1 A short observation (green shaded rectangle) cannot distinguished the reduced graviton state from the vacuum reduced to the same region. The graviton delivers no information to this observer. 139 v A.2 A long observation (green shaded rectangle) can distinguish the reduced graviton state from the reduced vacuum. The graviton carries information to this observer. 141 A.3 A graviton conveys O(1) information as long as it has appreciable support in the region of observation.
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