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Title Entropy Bounds and Entanglement

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Author Fisher, Zachary Kenneth

Publication Date 2017

Peer reviewed|Thesis/dissertation

eScholarship.org Powered by the California Digital Library University of California Entropy Bounds and Entanglement

Zachary Fisher

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

Physics

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor Raphael Bousso, Chair Professor Yasunori Nomura Professor Nicolai Reshetikhin

Spring 2017 Entropy Bounds and Entanglement

Abstract

Entropy Bounds and Entanglement by Zachary Fisher Doctor of Philosophy in Physics University of California, Berkeley Professor Raphael Bousso, Chair

The generalized covariant entropy bound, or Bousso bound, is a holographic bound on the entropy of a region of space in a gravitational theory. It bounds the entropy passing through certain null surfaces. The bound remains nontrivial in the weak-gravity limit, and provides non-trivial constraints on the entropy of ordinary quantum states even in a regime where gravity is negligible. In the first half of this thesis, we present a proof of the Bousso bound in the weak-gravity regime within the framework of quantum field theory. The bound uses techniques from quantum information theory which relate the energy and entropy of quantum states. We present two proofs of the bound in free and interacting field theory. In the second half, we present a generalization of the Bousso bound called the quantum focussing conjecture. Our conjecture is a bound on the rate of entropy generation in a quantum field theory coupled semiclassically to gravity. The conjecture unifies and generalizes several ideas in holography. In particular, the quantum focussing conjecture implies a bound on entropies which is similar to, but subtly different from, the Bousso bound proven in the first half. The quantum focussing conjecture implies a novel non-gravitational energy condition, the quantum null energy condition, which gives a point-wise lower bound on the null-null component of the stress tensor of quantum matter. We give a proof of this bound in the context of free and superrenormalizable bosonic quantum field theory. i

For Melanie, Dennis, Jeremy and Laura. ii

Contents

Contents ii

List of Figures iv

1 Introduction1 1.1 The Holographic Principle...... 1 1.2 The Bousso Bound...... 3 1.3 Holography and Quantum Field Theory...... 5 1.4 Entropy, Energy and Geometry ...... 7

2 The Bousso Bound in Free Quantum Field Theory9 2.1 Regulated Entropy ∆S ...... 10 2.2 Proof that ∆S ∆ K ...... 13 ≤ h i 2.3 Proof that ∆ K ∆A/4GN ~ ...... 13 2.4 Discussion...... h i ≤ 16 2.A Monotonicity of ∆A(c,b) ∆S ...... 21 4GN ~ − 3 The Bousso Bound in Interacting Quantum Field Theory 22 3.1 Entropies for Null Intervals in Interacting Theories ...... 25 3.2 Bousso Bound Proof ...... 30 3.3 Holographic Computation of ∆S for Light-Sheets ...... 33 3.4 Why is ∆S = ∆ K on Null Surfaces?...... 37 3.5 Discussion...... h i 40 3.A Extremal Surfaces and Phase Transitions on a Black Brane Background . . . 43 3.B Toy Model with ∆ K = ∆S = 0...... 49 h i 6 4 The Quantum Focussing Conjecture 53 4.1 Classical Focussing and Bousso Bound ...... 56 4.2 Quantum Expansion and Focussing Conjecture...... 58 4.3 Quantum Bousso Bound...... 62 4.4 Quantum Null Energy Condition ...... 66 4.5 Relationship to Other Works...... 70 iii

4.A Renormalization of the Entropy...... 75

5 Proof of the Quantum Null Energy Condition 82 5.1 Statement of the Quantum Null Energy Condition...... 86 5.2 Reduction to a 1+1 CFT and Auxiliary System...... 87 5.3 Calculation of the Entropy...... 92 5.4 Extension to D = 2, Higher Spin, and Interactions ...... 103 5.A Correlation Functions...... 105

Bibliography 107 iv

List of Figures

2.1 (a) The light-sheet L is a subset of the light-front x− = 0, consisting of points + with b(x⊥) x c(x⊥). (b) The light-sheet can be viewed as the disjoint union of small transverse≤ ≤ neighborhoods of its null generators with infinitesimal areas Ai ...... 11 2.2{ Operator} algebras associated to various regions. (a) Operator algebra associated to the domain of dependence (yellow) of a space-like interval. (b) The domain of dependence of a boosted interval. (c) In the null limit, the domain of dependence degenerates to the interval itself...... 12 2.3 A possible approach to defining the entropy on a light-sheet beyond the weak- gravity limit. One divides the light-sheet into pieces which are small compared to the affine distance over which the area changes by a factor of order unity. The entropy is defined as the sum of the differential entropies on each segment. . . . 19

3.1 The Rényi entropies for an interval A involve the two point function of defect operators D inserted at the endpoints of the interval. An operator in the ith CFT becomes an operator in the (i + 1)th CFT when we go around the defect. . . . . 25 3.2 The functions g(v) in the expression for the modular Hamiltonian of the null slab, for conformal field theories with a bulk dual. Here d = 2, 3, 4, 8, from bottom to top. Near the boundaries (v 0, v 1), we find g 0, g0 ∞ 1, in agreement with the modular Hamiltonian→ of a Rindler→ wedge. We→ also note→ ± that the functions are concave. In particular, we see that g0 1, in agreeement with our general argument of section 3.2...... | | ≤ 36 3.3 Operator algebras associated to various regions. (a) Operator algebra associated to the domain of dependence (yellow) of a space-like interval. (b) The domain of dependence of a boosted interval. (c) In the null limit, the domain of dependence degenerates to the interval itself...... 38 3.A.1The maximum value Emax(p) of E for getting a surface that returns to the boundary (solid line). For comparison, the line E = p 1 is plotted (the dashed line). The extremal surface solutions of interest appear− in the region p > 1, 0 < E < Emax(p). Here, we have taken d = 3...... 45 v

3.A.2Curves of constant ∆x+ (black solid curves) and ∆x− (blue dashed curves), in the logarithmic parameter space defined by (log(p 1), log(Emax(p) E)/Emax(p)). The value p = 1 maps to and p = maps− to− + on the horizontal− axis, −∞ ∞ ∞ while E = 0 maps to 0 and E = Emax(p) maps to + on the vertical axis. The thick blue contour represent the null solutions with∞ ∆x− = 0. Above this contour, the boundary interval is time-like. If ∆x+ & 15 and we follow a contour of constant ∆x+, we find two solutions with exact ∆x− = 0. For all contours of fixed ∆x+, there exists an asymptotic null solution in the limit p ...... 46 3.A.3The vacuum-subtracted extremal surface area versus ∆x− for fixed→ ∆ ∞x+ (∆x+ = 20 and ∆x+ = 10 for d = 3 is shown). This numerical simulation demonstrates that, for sufficiently large ∆x+ (in d = 3, the condition is ∆x+ & 15), there exists a phase transition at finite ∆x− to a different, perturbative class of solutions. At smaller ∆x+, there is no such phase transition...... 49

4.1 (a) A spatial surface σ of area A splits a Cauchy surface Σ into two parts. The generalized entropy is defined by Sgen = Sout+A/4GN ~, where Sout is the von Neu- mann entropy of the quantum state on one side of σ. To define the quantum expansion Θ at σ, we erect an orthogonal null hypersurface N, and we consider the response of Sgen to deformations of σ along N. (b) More precisely, N can be divided into pencils of width around its null generators; the surface σ is deformed an affine parameter lengthA along one of the generators, shown in green. 60 4.2 (a) For an unentangled isolated matter system localized to N, the quantum Bousso bound reduces to the original bound. (b) With the opposite choice of “exterior,” one can also recover the original entropy bound, by adding a distant auxiliary system that purifies the state...... 64 4.3 (a) A portion of the null surface N, which we have chosen to coincide with Σout in the vicinity of the diagram. The horizontal line at the bottom is the surface V (y), and the orange and blue lines represent deformations at the transverse locations y1 and y2. The region above both deformations is the region outside of

V1,2 (y) and is shaded beige and labeled B. The region between V (y) and V1 (y)

is labeled A and shaded lighter orange. The region between V (y) and V2 (y)is labeled C and shaded lighter blue. Strong subadditivity applied to these three regions proves the oﬀ-diagonal QFC. (b) A similar construction for the diagonal part of the QFC. In this case, the sign of the second derivative with respect to the aﬃne parameter is not related to strong subadditivity...... 67 vi

5.1 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 quantum expansionA 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 space-like slice is related to the entropy of the null quantized state on the future (brighter green) part of the null surface...... 83 5.2 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...... 89 5.3 Sample plots of the imaginary part (the real part is qualitatively identical) of the na¨ıve bracketed digamma expression in equation (5.74) and the one in equation (5.78) obtained from analytic continuation with z = m iα for m = 3 and − − ij various values of αij. The oscillating curves are equation (5.74), while the smooth curves are the result of applying the specified analytic continuation prescription to that expression, resulting in equation (5.78)...... 101 vii

Acknowledgments

A journey of this magnitude cannot be undertaken alone. First and foremost, I would like to thank my advisor, Raphael Bousso. His brilliance and leadership made my five years at Berkeley some of the most intellectually challenging and fulfilling of my life. I owe another large debt of gratitude to each of my collaborators: Horacio Casini, Jason Koeller, Stefan Leichenhauer, Juan Maldacena, and Aron Wall. I feel so fortunate to have had the opportunity to work closely with these outstanding scientists. Of course, I cannot forget to acknowledge the central role played by my instructors and mentors through the years. I would especially like to thank Miles Chen, Isaac Chuang, Tom Henning, Petr Hoˇrava, Holger Müller,Yasunori Nomura, Nicolai Reshetikhin, Andrew Shaw and Barton Zwiebach. I could not have reached this point without their leadership and encouragement. I would furthermore like to thank the many people who have encouraged me, laughed with me and taught me over these five years, my dear friends and colleagues. Foremost among them I would like to thank Netta Engelhardt, Chris Mogni, Ben Ponedel, Fabio Sanchez, Sean Jason Weinberg and Ziqi Yan. I would also like to acknowledge the influence and support of the other members of the Bousso group: Christopher Akers, Venkatesh Chandrasekaran, Illan Halpern, Adam Levine, Arvin Moghaddam, Mudassir Moosa, Vladimir Rosenhaus and Claire Zukowski. Finally, I would also like to thank Eugenio Bianchi, William Donnelly, Ben Freivogel, Matthew Headrick, Ted Jacobson, Don Marolf, David Simmons-Duffin and Andrew Stro- minger for comments and suggestions on the papers comprising this thesis. All of these individuals helped guide my development as a scientist. I am deeply grateful to them all. 1

Chapter 1

Introduction

In the last century, much time and effort has been expended on the problem of quantizing gravity. There is now widespread agreement in the community that the problem of quantum gravity will require radically new physical ideas and principles. Indeed, a complete understanding of quantum gravity still eludes us today. However, an excellent candidate for such a framework is string theory. String theory is a beautiful and self-consistent theory from which gravity arises naturally. It confirms many of our expectations about how quantum gravity should work. Unfortunately, even with this powerful tool at our disposal, many of the most important questions about quantum gravity remain open. One of the remaining questions is to understand what features of quantum gravity are visible at low energies, where effective field theories agree with experiment to excellent precision. Therefore, we must study the qualitatively new features of quantum gravity. One of the most surprising new principles that arises in quantum gravity is the holographic principle. Many results in this thesis are motivated by holography and the closely related area of black hole thermodynamics. Therefore, we begin this thesis with a brief tour of the holographic principle.

1.1 The Holographic Principle

This idea that black holes have entropy originated in Jacob Bekenstein’s 1972 publication [9]. In that paper, Jacob Bekenstein made a beautiful and far-reaching observation: because the horizon of a black hole is a point of no return, a black hole is an entropy sink for anything that falls inside it. However, the second law of thermodynamics prohibits entropy from decreasing in a closed system, such as the exterior of a black hole. For example, a scrambled egg can surely never unscramble, but merely by throwing the egg into a black hole, the egg is no longer accessible and the entropy goes to zero! Bekenstein posited that the way to avoid this paradox was to assign an entropy SBH to the black hole horizon, proportional to CHAPTER 1. INTRODUCTION 2

the black hole horizon area A: S A. (1.1) BH ∝ We can now understand heuristically how the paradox might be resolved: when the egg is tossed into the black hole, the entropy of the outside universe goes down, but the mass, and thus the area, of the black hole horizon goes up; the black hole entropy could therefore conceivably compensate for the loss of matter entropy. Additional evidence for Bekenstein’s conjecture came from the Hawking area theorem [87]. This theorem states that, assuming standard classical conditions on energy densities, the area of a black hole horizon can only increase with time: dA 0 . (1.2) dt ≥ For example, the area of a black hole formed from the merger of two black holes is greater than the sum of the areas before the merger. By comparing equations (1.1) and (1.2), we see that the second law of thermodynamics automatically holds in a universe consisting of just black holes. Soon thereafter, in a groundbreaking paper [88], Hawking fixed the proportionality constant in equation (1.1). Hawking’s calculation used the framework of quantum field theory on a black hole background, taking into account possible effects of gravitational backreaction. His result implied that black holes have a finite temperature, which determines the entropy by the first law of thermodynamics, dS = dE/T . Thus the constant in equation (1.1) was fixed1: A SBH = . (1.3) 4GN ~ Black holes radiate away their energy in the same way as any other thermal object. Since TBH GN ~, the effect is a prediction of quantum gravity, which disappears in the classical ∝ limit ~ 0. Collecting→ these results, it is possible to write down a well-motivated definition of the total entropy of a region of space, in a quantum field theory semiclassically coupled to gravity. One simply adds the entropy of the black hole to the entropy of all of the matter outside the black hole. The resulting quantity is called the generalized entropy: A Sgen = + Smatter outside . (1.4) 4GN ~

The conjecture that Sgen is non-decreasing with time in a semiclassical theory is called the generalized second law, or GSL2 [9]. 1In this equation, and throughout this thesis, we will use natural units for the speed of light and Boltz- mann’s constant, 1 = c = kB; the gravitational coupling constant GN and Planck’s constant ~ will remain explicit unless otherwise speciﬁed. 2The name is something of a misnomer, since the ordinary second law of thermodynamics only holds if one includes every physical source of entropy; the only generalization made is assuming that black holes contribute some entropy. CHAPTER 1. INTRODUCTION 3

The generalized second law sets a limit on the entropy content of weakly gravitating matter systems [7] and of certain spacetime regions. Such considerations lead us to the holographic principle. The holographic principle states that the amount of information which can be stored in a region of a space is ﬁnite and bounded by the area of the boundary of the region under consideration. This notion is surprising from the perspective of quantum ﬁeld theory, where the degrees of freedom are local and so the number scales like the volume. A holographic theory has far fewer degrees of freedom, scaling like the area of the boundary of the region. This property of quantum gravity can manifest itself at low energies as a bound on the information content of a physically valid state. Such bounds are called entropy bounds. Heuristically, we expect entropy bounds to hold because of thought experiments wherein an isolated matter system is added to a black hole, or a spherical spacetime region is converted to a black hole of equal area. We then compute the change in the generalized entropy, and demand that it be nonnegative. This procedure can be carried out and turned into a quantitative bound [70].

1.2 The Bousso Bound

A particularly important holographic bound was conjectured by Bousso [25]. The covariant entropy bound relates matter entropy to the area of arbitrary surfaces, not just black hole horizons. The bound is formulated in terms of light-sheets. A light-sheet is a null surface whose null generators are everywhere converging. We will now introduce some important terminology. Denote the infinitesimal area element between null generators by , and define an affine parameter λ for the congruence. The (classical) expansion scalar is definedA as the logarithmic derivative of with respect to λ: A 1 d θ A . (1.5) ≡ dλ A We define a light-sheet as a null surface with θ 0 everywhere (the non-expansion condition). When adjacent light rays converge, θ ,≤ we say that there is a caustic and we terminate the null generator there. For example,→ the −∞ past lightcone of a point in Minkowski space is a light-sheet. A light-sheet can be directed towards the past or the future as long as θ 0. Having established this defintion, we can now state the covariant entropy bound.≤ Con- sider a (codimension-1) region B of space and shoot out null geodesics from its boundary A = ∂B. Some of these congruences will be light sheets. Allow the light-sheet to terminate when the generators reach caustics. The Bousso bound states that the entropy S which crosses through the light-sheet is bounded by the area of the boundary A: Area[A] S . (1.6) ≤ 4GN ~ Flanagan, Marolf and Wald [71] proposed a useful generalization of the Bousso bound. In this conjecture, the generators of the light-sheet are allowed to terminate arbitrarily early, CHAPTER 1. INTRODUCTION 4

i.e. before reaching a caustic, landing on a codimension-2 surface A0. Then the bound says that the entropy crossing through the prematurely-terminated light-sheet is bounded by the difference of the areas ∆A = Area[A] Area[A0]: − Area[A] Area[A0] S − (1.7) ≤ 4GN ~ Formally, this bound is called the generalized covariant entropy bound. Following common parlance, we will take the stronger statement in equation (1.7) to be our working definition of the Bousso bound. Fundamentally, the Bousso bound is a conjecture. It might capture aspects of how spacetime and matter arise from a more fundamental theory [29, 31]. A general proof may not become available until such a theory is found. Nevertheless, it is of interest to prove the bound at least in certain regimes, or subject to assumptions that hold in a large class of examples. In this spirit, the Bousso bound in equation (1.7) has been shown to hold in settings where the entropy S can be approximated hydrodynamically, as the integral of an entropy flux over the light-sheet; and where certain assumptions constrain the entropy and energy fluxes [72, 35]. These assumptions apply to a large class of spacetimes, such as cosmology or the gravitational collapse of a star. Thus they establish validity of the bound in some broad regimes. However, the underlying assumptions in these earlier proofs have no fundamental status. Unlike the stress tensor, entropy is not local, so the hydrodynamic approximation breaks down if the light-sheet is shorter than the modes that dominate the entropy. In this regime, it is not clear how to define the entropy at all. Consider a single photon wavepacket with a Gaussian profile propagating through otherwise empty flat space. In order to obtain the tightest bound, we may take the light-sheet to have initially vanishing expansion. The difference in areas ∆A is easily computed from the stress tensor and Einstein’s equations. For a finite light-sheet that captures all but the exponential tails of the wavepacket, one finds that the packet focuses the geodesics just enough to lose about one Planck area, ∆A/GN ~ O(1) [30]. For smaller light-sheets, ∆A tends to 0 quadratically with the affine length.∼ For larger light-sheets, ∆A can grow without bound. To check if the bound is satisfied for all choices of light-sheet, one would need a formula for the entropy on any finite light- sheet. Globally, the entropy is log n O(1), where n is the number of polarization states. Intuitively this should also be the answer∼ when nearly all of the wavepacket is captured on the light-sheet, but how can this be quantified? (In field theory, the entropy in a finite region would be dominated by vacuum entropy across the initial and final surface, and hence largely unrelated to the photon.) Worse, for short light-sheets, there is no intuitive notion of entropy at all. What is the entropy of, say, a tenth of a wavepacket?3 3Similar limitations apply to the Bekenstein bound [7], which can be recovered as a special case of the generalized covariant bound in the weak-gravity limit [30]: precisely in the regime where the bound becomes tight, one lacks a sharp definition of entropy. CHAPTER 1. INTRODUCTION 5

This issue is resolved in chapters2 and3 in a novel way. We use tools from quantum information theory and quantum ﬁeld theory to prove the Bousso bound in a weak-gravity limit. We will now expand further on how these tools can be used to prove holographic bounds of this type.

1.3 Holography and Quantum Field Theory

The generalized second law and the Bousso bound are physically reasonable expectations of a theory of quantum gravity. However, these entropy bounds can be reexpressed and understood within ordinary (non-gravitational) quantum field theory. This is possible because these bounds remain nontrivial even well below the Planck scale, in the limit GN ~ 0, holding the geometry fixed. → First, we define an entropy function for any quantum state in terms of its density matrix ρ. Often, we will consider the density matrix for the degrees of freedom localized inside some spatial region A and we will denote the state as ρA for clarity. The state ρA is related to the

global state ρ by tracing out the degrees of freedom localized outside A: ρA = trH−A ρ. 4 The entropy we will bound is the von Neumann entropy SA associated to the region A. It is given in terms of ρA via the formula S = tr[ρ log ρ ] . (1.8) A − A A In any quantum theory, the von Neumann entropy satisﬁes a number of important equalities and inequalities. The most important among these is strong subadditivity, which says that given density matrices with support on three disjoint regions A, B, C, S(ρ ) + S(ρ ) S(ρ ) + S(ρ ) . (1.9) ABC B ≥ AB BC In quantum ﬁeld theory, the von Neumann entropy is ultraviolet divergent, so a regulator is employed, usually a lattice spacing in this context. Due to short-range entanglement, von Neumann entropy in QFT obeys an area law: the leading piece in von Neumann entropy in an expansion scales like the area5:

k(d−2) k(d−4) S = + + + finite, where k − Area[∂A] . (1.10) A d−2 d−4 ··· (d 2) ∝ We are usually, but not exclusively, interested in the finite piece of von Neumann entropy. An important quantity closely related to the von Neumann entropy is the relative entropy. Relative entropy is a function S(ρ σ) of two density matrices, both defined in the same Hilbert space. Explicitly, || S(ρ σ) tr[ρ log ρ] tr[ρ log σ] , (1.11) || ≡ − 4The term entanglement entropy is also used for this quantity in the literature, but that name is misleading. There can be contributions to the von Neumann entropy that arise from classical uncertainty, for example arising from a thermal ensemble of states, and which have nothing to do with quantum entanglement. 5In even dimensions, a logarithmic term can appear in this expansion. CHAPTER 1. INTRODUCTION 6

where σ is some ﬁducial state which one usually takes to be the vacuum state. Relative entropy is an asymmetric measure of the distance between the two density matrices in the Hilbert space. Unlike von Neumann entropy, relative entropy is ultraviolet ﬁnite. For our purposes, it is frequently useful to rewrite the relative entropy in the form

S(ρ σ) = ∆ K ∆S (1.12) || h i − where

K log σ (1.13) ≡ − is called the modular Hamiltonian and

∆ K = tr[ρ log σ σ log σ] = K K (1.14) h i − − h iρ − h iσ ∆S = tr[ρ log ρ σ log σ] = S S (1.15) − − ρ − σ are (divergence-subtracted versions of) the expectation value of the modular Hamiltonian in, and the von Neumann entropy of, the state ρ. In order to render these quantities finite, we have subtracted their values in the state σ, which results in the cancellation of divergences6. Remarkably, in any quantum field theory in any number of dimensions d, the modular Hamiltonian of a half-space takes a simple universal form [19]. The modular Hamiltonian is proportional to the generator of spacetime boosts which leaves the boundary invariant. For example, the modular Hamiltonian of the region A = x x0 = 0, x1 > 0 is7 { | } ∞ 1 d−2 1 KA = 2π dx d x⊥ x T00 . (1.16) Z0 Z This expression is remarkable for many reasons: it is universal for any field theory; it involves only local operators, in fact only the stress tensor; and it provides a direct connection between energy and entropy. Relative entropy also obeys a number of important properties. For example, a simple calculation shows that relative entropy is always positive. A more involved calculation is required to show that relative entropy is monotonic under inclusion; that is, given disjoint regions A, B, S(ρ σ ) S(ρ σ ) . (1.17) AB|| AB ≥ A|| A The meaning of this inequality is that more operators are available in the region AB to distinguish two quantum states than are available in just the region A. 6There are circumstances where vacuum subtraction is not sufficient to cancel all of the divergences in the modular Hamiltonian expectation value and the von Neumann entropy [124]. Such examples do not apply to the von Neumann entropy of null surfaces in the interacting proof. More generally, the state-dependent divergences will contribute equally to the modular Hamiltonian and the entropy. We can then circumvent the issue of divergences by modifying the regularization scheme. 7This expression is valid up to a constant (divergent) factor which drops out of ∆ K . h Ai CHAPTER 1. INTRODUCTION 7

These properties of relative entropy, positivity and monotonicity, are remarkably powerful. They interrelate the energy content of a region of spacetime with its entropy, providing constraints. One may ask whether these bounds are related to the holographic bounds of the previous section. Indeed, in a beautiful 2008 paper, Casini [51] showed that a holographic bound called the Bekenstein bound can been formulated and proven in quantum field theory, using the positivity property of relative entropy. It is also possible to formulate a version of the generalized second law as a statement about monotonicity [164]. That proof applies for any causal horizon in a theory of quantum fields minimally coupled to general relativity. As we shall show in this thesis, the Bousso bound can be proven with this technology as well. Chapters2 and3 of this thesis will prove the Bousso bound in weakly gravitating systems, using relative entropy and properties of quantum field theory. These proofs were first presented in [38, 37]. Chapter2 presents the proof in free and superrenormalizable field theory, where the technique of null quantization is employed to simplify the analysis. The proof is highly nontrivial and implies counterintuitive properties of entropies on null surfaces. Chapter3 presents the proof in field theories with nontrivial interactions. The von Neumann entropy exhibits some counterintuitive properties in this context which we will explore and use to prove the Bousso bound.

1.4 Entropy, Energy and Geometry

One of the most intriguing properties of the Bousso bound is that it puts a geometric bound on entropy. This arises from the connection between energy and entropy comes from black hole thermodynamics, and the connection between geometry and energy from Einstein’s equation. Entropy, energy and geometry are intimately related by the holographic principle. We will explore these connections further in the second half of this thesis. Chapter 4 presents a novel conjecture for quantum fields and semiclassical gravity: the quantum focussing conjecture (QFC). This conjecture is a strengthening of the Bousso bound into a form similar to the generalized second law. In short, the generalized second law states that the first derivative of the generalized entropy is positive; the quantum focussing conjecture states that the second derivative of generalized entropy is positive. We conjecture that the QFC holds even when the generalized entropy is evaluated not just for black holes, but for any arbitrary surface in the spacetime that divides a Cauchy surface into an interior and an exterior. The QFC was first presented in [36]. Intriguingly, there is a close relationship between the QFC and the positivity of energy densities in classical physics. In classical physics, one typically assumes the null energy a b condition (NEC). The null energy condition states that Tkk Tabk k 0, where Tab is the stress tensor and ka is a null vector. This condition is satisfied≡ by physically≥ realistic classical matter fields. In Einstein’s equation, it ensures that light-rays are focussed, never repelled, by matter. The NEC underlies the area theorems [87, 32] and singularity theorems [137, 90, 162], and many other results in general relativity [128, 75, 67, 155, 89, 133, 160, 138, 82]. CHAPTER 1. INTRODUCTION 8

However, quantum fields can potentially violate all local energy conditions, including the NEC [66]. The energy density Tkk at any point can be made negative, with magnitude as large as we wish, by an appropriateh i choice of quantum state. An example of a region where the null energy condition is violated is the horizon of an evaporating black hole. In a stable theory, any negative energy must be accompanied by positive energy elsewhere. Thus, positive-definite quantities linear in the stress tensor that are bounded below may exist, but must be nonlocal. For example, a total energy may be obtained by integrating an energy density over all of space; an “averaged null energy” is defined by integrating Tkk along a null geodesic [24, 163, 113, 159, 84, 93]. Some field theories have been shownh toi satisfy quantum energy inequalities, in which an integral of the stress-tensor need not be positive, but is bounded below [74]. The possibility of violations to the null energy condition is a serious drawback to earlier proofs of the Bousso bound [72, 35]. There are realistic quantum states to which these proofs do not apply. Indeed, a desirable feature of the proof of the Bousso bound in chapters2 and 3 is that it does not assume the null energy condition. The quantum focussing conjecture further develops the connection between energy positivity and entropy inequalities. The QFC implies a novel energy condition called the quantum null energy condition (QNEC). It is a generalization of the null energy condition, and reduces to the null energy condition in the limit ~ 0. The QNEC is a bound on the value of the stress tensor at a point in terms of the second→ derivative of a particular von Neumann entropy. In chapter5, we prove the quantum null energy condition in free and superrenormalizable bosonic field theory. This proof was first presented in [39]. 9

Chapter 2

The Bousso Bound in Free Quantum Field Theory

The Bousso bound, as described in section 1.1, states that the entropy ∆S of matter on a light-sheet cannot exceed the difference between its initial and final areas ∆A: ∆A ∆S. (2.1) 4GN ~ ≥ A light-sheet is a null hypersurface whose cross-sectional area is decreasing or staying constant, in the direction away from A. In this chapter, we will present a proof of this bound for the case that matter consists of free fields, in the limit of weak gravitational backreaction. We will provide a sharp definition of the entropy on a finite light-sheet in terms of differences of von Neumann entropies. Our definition does not rely on a hydrodynamic approximation. It reduces to the expected entropy flux in obvious settings. Using this definition, we will prove the Bousso bound. We will not assume the null energy condition.

Outline In section 2.1 we provide a definition of the entropy on a weakly focused light- sheet. We define ∆S as the difference between the entropy of the matter state and the entropy of the vacuum, as seen by the algebra of operators defined on the light-sheet. The proof of the bound then has two steps. In section 2.2, we explain why ∆S ∆ K , where ∆ K is the difference in expectation values for the vacuum modular≤ Hamilto-h i nian. Thish propertyi holds for general quantum theories [51]. In section 2.3, we show that ∆ K ∆A/4GN ~. We first compute an explicit expression for the modular Hamiltonian, inh sectioni ≤ 2.3. For general regions, the modular Hamiltonian is complicated and non-local. However, the special properties of free fields on light-like surfaces enable us to derive explicitly the modular Hamiltonian in terms of the stress tensor. The expression is essentially the same as the result we would obtain for a null interval in a 1+1 dimensional CFT. Finally, in section 2.3, we use the Raychaudhuri equation to compute the area difference ∆A. The area difference comes from two contributions: focussing of light-rays by matter, and potentially, CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 10 a strictly negative initial expansion. Usually one may choose the initial expansion to vanish. If this choice is possible, it will minimize ∆A and provide the tightest bound. However, if the null energy condition is violated, it can become necessary to choose a negative initial expansion, in order to keep the expansion nonpositive along the entire interval in question and evade premature termination of the light-sheet. We find that the two contributions together ensure that ∆A/4GN ~ ∆ K . Combining the two inequalities, we obtain the ≥ h i Bousso bound, ∆A/4GN ~ ∆S. In section 2.4, we discuss≥ possible generalizations of our result to the cases of interacting fields and large backreaction. We comment on the relation of our work to Casini’s proof of Bekenstein’s bound from the positivity of relative entropy [51], to Wall’s proof of the generalized second law [164], and to an earlier proposal for incorporating quantum effects in the Bousso bound [152]. In the Appendix, we prove monotonicity of ∆A/(4GN ~) ∆S under inclusion, a result stronger than that obtained in the main body of the paper. −

2.1 Regulated Entropy ∆S

We will consider matter in asymptotically flat space, perturbatively in GN . Since Minkowski space is a good approximation to any spacetime at sufficiently short distances, our final result should apply in arbitrary spacetimes, if the transverse and longitudinal size of the light-sheet is small compared to curvature invariants. For definiteness, we work in 3+1 spacetime dimensions; the generalization to d + 1 dimensions is trivial. At zeroth order in GN , the metric is that of Minkowski space:

ds2 = dx+dx− + dx2 , (2.2) − ⊥ 2 2 2 where dx⊥ = dy + dz . Without loss of generality, we will consider a partial light-sheet L that is a subset of the null hypersurface H given by x− = 0. Any such light-sheet can be characterized by two piecewise continuous functions b(x⊥) and c(x⊥) with < b c < everywhere: L is the set of points that satisfy x− = 0, b < x+ < c. See figure−∞ 2.1. ≤ ∞ We begin by giving an intrinsic definition of the vacuum state on H in free field theory. + + The generator of a null translation x x + a(x⊥) along H is given by → ∞ 2 + p [a] = dx dx T a(x⊥) , (2.3) + ⊥ h ++i Z Z−∞ a b a where T++ = Tabk k and k = ∂+ is the tangent vector to H. Given any choice of a(x⊥), one can define a vacuum state 0 by the condition p [a] 0 = 0. | ia + | ia In fact, all nowhere-vanishing functions a(x⊥) define the same vacuum, 0 H , because of the following important result [164]: there are neither interactions nor correlations| i 1 between different null generators of H. When restricted to H, the algebra of observables becomes A 1 These statements hold for correlators that have at least one derivative along the plus direction ∂+φ. CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 11

= x+ = x+

c(x ? ) A0 H H

b(x ) A1 ? L A2 L A Ai

x x ? ? (a) (b)

Figure 2.1: (a) The light-sheet L is a subset of the light-front x− = 0, consisting of points + with b(x⊥) x c(x⊥). (b) The light-sheet can be viewed as the disjoint union of small transverse neighborhoods≤ ≤ of its null generators with inﬁnitesimal areas A . { i}

ultralocal in the transverse direction. For any partition Hi of the null generators of H, the algebra can be written as a tensor product { }

(H) = (H ) . (2.4) A A i i Y 0 0 In the limit where the translation is localized to one ray, a(x⊥) = δ(x⊥ x⊥), equation (2.3) reduces to the generator − ∞ + p (x⊥) = dx T , (2.5) + h ++i Z−∞

and p+(x⊥) 0 x⊥ = 0 deﬁnes a vacuum state independently for each generator. By ultralocality, the vacuum| i state on H is a tensor product of these states. (In terms of small transverse

neighborhoods of each generator, Hi, one can write 0 H = i 0 i.) It will be convenient to write the vacuum state on| iH as a density| i operator, Q σ 0 0 . (2.6) H ≡ | iHH h |

Let the actual state of matter on H be ρH ; this state may be mixed or pure. Let σL and ρL be the restriction, respectively, of the vacuum and the actual state to the light-sheet L:

σ tr − σ (2.7) L ≡ H L H ρ tr − ρ (2.8) L ≡ H L H Correlators of φ with no derivatives are non-zero at space-like distances. However, they do not lead to well deﬁned operators along the light front since we cannot control the UV divergences by smearing it along the light front directions. For this reason we do not consider φ as part of the algebra (H). The canonical 2 A stress tensor component T++ (∂+φ) depends only on such derivatives of the ﬁeld in the null direction. For further details, see reference∝ [164]. CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 12

The von Neumann entropy of either of these density matrices diverges in proportion to the sum of the areas of the two boundaries of L (in units of a UV cutoff). However, we may define a regulated entropy as the difference between the von Neumann entropies of the actual state and the vacuum [51, 123, 94]:

∆S S(ρ ) S(σ ) = tr ρ log ρ + tr σ log σ . (2.9) ≡ L − L − L L L L

For ﬁnite energy global states ρH , this expression will be ﬁnite and independent of the regularization scheme. It reduces to the global entropy, ∆S tr ρH log ρH , in the limit where the latter is dominated by modes that are well-localized→ − to L. Examples include large thermodynamic systems such as a bucket of water or a star, but also a single particle wavepacket that is well-localized to the interior of L.

(a) (b) (c)

Figure 2.2: Operator algebras associated to various regions. (a) Operator algebra associated to the domain of dependence (yellow) of a space-like interval. (b) The domain of dependence of a boosted interval. (c) In the null limit, the domain of dependence degenerates to the interval itself.

An important feature is that we are computing these entropies for null segments. It is more common to consider entropies for spatial segments, see figure 2.2. In that case, the algebra of operators includes all the local operators in the domain of dependence of the segment, see figure 2.2(a). We can also consider a boosted the interval as in figure 2.2(b). The domain of dependence changes accordingly. In the limit of a null interval the domain of dependence becomes just a null segment. This is a singular limit of the standard space- like case: the proper length of the null interval vanishes and the domain of dependence degenerates. Despite these issues, we find that the entropy difference between any state and the vacuum, (2.9), is finite and well defined. In the free theory case, the limiting operator algebra has the ultralocal structure described above. CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 13

2.2 Proof that ∆S ∆ K ≤ h i The vacuum state on the light-sheet L deﬁnes a modular Hamiltonian operator KL, via

e−KL σL = , (2.10) tr e−KL

up to a constant shift that drops out below. Expectation values such as tr KLσL and tr KLρL will diverge, but we may deﬁne a regulated (or vacuum-subtracted) modular energy of ρL:

∆ K tr K ρ tr K σ . (2.11) h i ≡ L L − L L For any two quantum states ρ, σ, in an arbitrary setting, one can show that the relative entropy, S(ρ σ) tr ρ log ρ tr ρ log σ , (2.12) || ≡ − is nonnegative [120].2 With the above deﬁnitions, this immediately implies the inequality [51]

∆S ∆ K . (2.13) ≤ h i

To prove the Bousso bound, we will now show that ∆ K ∆A/4GN ~, where ∆A is the area diﬀerence between the two boundaries of the light-sheet.h i ≤

2.3 Proof that ∆ K ∆A/4GN ~ h i ≤ We can think of the null hypersurface H as the disjoint union of small neighborhoods Hi of a large discrete set of null generators; see figure 2.1(b). By ultralocality of the operator algebra, equation (2.4), we have for the vacuum state σH = i σL,i, σL = i σL,i, where the density operators for neighborhood i are defined by tracing over all other neighborhoods [164]. Using σ in equations (2.10) and (2.11), a modular energyQ ∆ K can beQ defined for each i h ii neighborhood, which is additive by ultralocality: ∆ K = i ∆ K i. Strictly, we should take the limit as the cross-sectional area of each neighborhoodh i becomesh i the infinitesimal area 2 P element orthogonal to each light-ray, A d x⊥. However, we find it more convenient to i → think of Ai as finite but small, compared to the scale on which the light-sheet boundaries b and c vary. Since both the modular energy and the area are additive,3 it will be sufficient to show that ∆ K ∆Ai/4GN ~, where ∆Ai is the change in the cross-sectional area Ai produced at h ii ≤ 2Moreover, the relative entropy decreases monotonically under restrictions of ρ, σ to a subalgebra [119]. With the help of this stronger property, our conclusion can be strengthened to the statement that ∆A(c,b) ∆S 4GN ~ decreases monotonically to zero if the boundaries b and c are moved towards each other. This is shown− in the Appendix. 3By contrast, the entropy ∆S is subadditive over the transverse neighborhoods. In equation (2.9), the vacuum state σL factorizes, but the general state ρL can have entanglement across different neighborhoods H . This does not affect our argument since we have already shown directly that ∆S ∆ K . i ≤ h i CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 14

ﬁrst order in GN ~ by gravitational focussing. We will demonstrate this by evaluating ∆ K h ii and bounding ∆Ai. For any given neighborhood Hi, we may take the aﬃne parameter λi to run from 0 to 1 on the light-sheet Li, as x+ runs from bi = b(x⊥) to ci = c(x⊥). For notational simplicity we will drop the index i in the remainder of this section.

Ultralocality and Conformal Symmetry Determine ∆ K h i We compute the modular Hamiltonian KL on the null interval 0 < λ < 1 in two steps. First, we review the modular Hamiltonian for the semi-inﬁnite interval 1 < λ0 < . Then we use ∞ the special conformal symmetry of the algebra of observables to obtain KL by inversion. We can regard the interval 1 < λ0 < as the upper boundaryA of a right Rindler wedge with bifurcation surface λ0 = 1. By tracing∞ the global vacuum σ over the left Rindler wedge, one ﬁnds that the state on the right is given by the thermal density operator

e−KRW σRW = , (2.14) tr e−KRW where the modular Hamiltonian ∞ 2π 2 0 0 K = d x⊥ dλ (λ 1) T 0 0 (2.15) RW − λ λ ~ Z Z1 coincides with the well-known Rindler Hamiltonian. Wall [164] has shown that the horizon algebra on each generator of H is that of the left-moving modes of a 1+1 dimensional conformal ﬁeld theory. General states transform nontrivially, but the vacuum σ is invariant under special conformal transformations. Hence, the modular Hamiltonian on the interval 0 < λ < 1 can be obtained by applying an inversion λ0 λ = 1/λ0 to the Rindler Hamiltonian. Using the law for the transformation of the stress tensor→ under a conformal transformation with vanishing Schwarzian derivative,

dλ 2 T 0 0 = T , (2.16) λ λ λλ dλ0 one ﬁnds for the modular Hamiltonian of the light-sheet L:

1 2π 2 K = d x⊥ dλ λ(1 λ) T . (2.17) L − λλ ~ Z Z0 Let us make some comments. If we were dealing with a two dimensional CFT the formula (2.17) would be familiar. If instead we had a massive free field in two dimensions, then we note that a null interval is conceptually similar to a very small interval. Therefore we are exploring the UV properties of the theory, which are the same as those for a massless free field. When we go to higher dimensions we can understand equation (2.17) as the result of thinking of the free field in terms of a two dimensional massive fields with masses given by a Kaluza-Klein reduction along the transverse dimensions. CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 15

Focussing and Non-expansion Bound ∆A Generally, the expansion of a null congruence is defined as [162] d log δA θ(λ) [ka = (2.18) ≡ ∇a dλ where δA is an infinitesimal cross-sectional area element. Recall that in the present context we consider the transverse neighborhood of one null geodesic, with small cross section Ai, so we may replace δA A . Our task is to compute the change ∆A of this small cross-section, ≈ i i from one end of Li to the other, by integrating equation (2.18). We will drop the index i, as it suffices to consider any one neighborhood. − At zeroth order in GN ~, the light-sheet of interest is a subset of the null plane x = 0 in Minkowski space, and so has vanishing expansion θ and vanishing shear σab everywhere. One may compute the expansion at first order in GN ~ by integrating the Raychaudhuri equation dθ 1 = θ2 σ σab 8πG T , (2.19) dλ −2 − ab − N λλ

The twist ωab vanishes identically for a surface-orthogonal congruence. We will pick λ = 0 as the initial surface and integrate up to λ = 1. The choice of direction is nontrivial, since we must ensure that the defining condition of light-sheets is everywhere satisfied: the cross-sectional area must be nonexpanding away from the initial surface, everywhere on L. As we shall see, this implies that at first order in GN ~, we must allow for a nonzero initial expansion θ0 at λ = 0. The required initial expansion can be accomplished by a small deformation of the initial surface [30], whose effects on ∆ K and ∆S only appear at higher order. (Of course, we could also start at λ = 1 and integrateh i in the opposite direction. For any given state, both ∆A and the initial expansion will depend on the choice of direction. But we will demonstrate that ∆ K ∆A for all states on h i ≤ future-directed light-sheets beginning at λ = 0. By symmetry of KL under λ 1 λ, the same result immediately follows for past-directed light-sheets beginning at λ =→ 1.) − From equation (2.19) we obtain at first order in GN ~:

λ θ(λ) = θ 8πG Tˆˆdλˆ . (2.20) 0 − N λλ Z0 The non-expansion condition is

θ(λ) 0, for all λ [0, 1] . (2.21) ≤ ∈ If the null energy condition holds, T 0, then this condition reduces to θ 0. More λλ ≥ 0 ≤ generally, however, we may have to choose θ0 < 0 to ensure that antifocussing due to negative energy densities does not cause the expansion to become positive, and thus the light-sheet to terminate, before λ = 1 is reached. However, it is always sufficient to take θ0 to be of order 2 ab GN ~, so it was self-consistent to drop the quadratic terms θ , σabσ , in the focussing ∝ CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 16 equation. Note that, in the semiclassical quantization scheme, the σ2 term can be viewed as arising from the stress tensor of the gravitons and can be explicitly included as part of the total stress tensor by separating the gravitational field into long and short distance modes. From the definition of the expansion, equation (2.19), one obtains the difference between initial and final cross-sectional area: ∆A 1 1 = dλθ(λ) = θ + 8πG dλ(1 λ)T , (2.22) A − − 0 N − λλ Z0 Z0 where we have used equation (2.20) and exchanged the order of integration. In order to eliminate θ0 we now use the non-expansion condition: let F (λ) be a function obeying F (0) = 0 1 0 0, F (1) = 1 and F (λ) 0 for 0 λ 1. From equation (2.21), we have 0 0 F θdλ, and thus from equation (2.20≥) and integration≤ ≤ by parts we find ≥ R θ 8πG dλ[1 F (λ)]T . (2.23) 0 ≤ N − λλ Z With the specific choice F (λ) = 2λ λ2, we find from equations (2.22) and (2.23) that the area difference is bounded from below− by the modular Hamiltonian:

1 ∆A A 8πG dλ λ(1 λ) T . (2.24) ≥ × N − λλ Z0 Comparison with equation (2.17) shows that ∆ K ∆A/4GN ~, as claimed. Combined with the earlier result ∆S ∆ Kh ,i this≤ completes the proof of the Bousso ≤ h i bound, ∆S ∆A/4GN ~, for free ﬁelds in the weak gravity limit. ≤ 2.4 Discussion

An interesting aspect of this argument is that we did not need to assume any microscopic relation between energy and entropy. We did have to assume that we had a local quantum field theory at short distances. Therefore the necessary relation between entropy and energy is the one automatically present in quantum field theory, i.e., given by the explicit expression of the modular Hamiltonian in terms of the stress tensor. Our discussion required a careful definition of the entropy that appeared in the bound. In that sense it is very similar to the Casini version [51] of the Bekenstein bound (see also [123, 94]), and also to Wall’s proof of the generalized second law [165, 164]. All these developments underscore the interesting interplay between local Lorentz invariance of the quantum field theory, Einstein’s equations, and information. It has often been speculated that the validity of these entropy bounds would require extra constraints on the matter that is coupled to Einstein’s equations. Here we see that the only constraint is that matter obeys the standard rules of local quantum field theory. (Conversely, it may be possible to view these rules as a consequence of entropy bounds [27].) CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 17

Relation to other work In [121] a possible counterexample to the Bousso bound was proposed. The idea is to feed matter so slowly into an evaporating black hole that the horizon area remains static or slowly decreases during the process. Hence the horizon is a future-directed light-sheet, to which the bound applies. Yet, it would appear that one can pass a very large amount of entropy through the horizon in this way. How is this consistent with our proof? To understand this, consider the simplest case where the stress tensor component T++ is constant on the light-sheet. For the horizon area to stay constant or shrink, oneh musti 4 have T++ 0. By equation (2.17), this implies ∆ K 0, and positivity of the relative entropyh requiresi ≤ ∆S ∆ K . Hence, in this case,h ∆iS ≤ 0. Thus we find that with our definitions, the entropy≤ is negativeh i for an evaporating black≤ hole, even with the addition of some positive, partially compensating flux; and the entropy is at least nonpositive in the static case. Since ∆A 0 by the non-expansion condition, the bound is safe. Strominger and Thompson≥ [152] have also proposed a quantum version of the Bousso bound. Their proposal is analogous to the definition of generalized entropy, in that one adds to the area the von Neumann entropy of quantum fields that are outside the horizon and distinct from the matter crossing the light-sheet. In contrast, we have given a definition which only involves properties of the quantum fields on the light-sheet L, i.e., on the relevant portion of the horizon. A similar distinction must be made when comparing our result to Wall’s proof of the generalized second law [165, 164]. Wall considers the generalized entropy Sgen(A) = Sm(A)+ A/4GN ~ on semi-infinite horizon regions, where A the area of a horizon cross-section, and Sm(A) is the matter entropy on the portion the horizon to the future of A (which is closely related to the matter entropy on spatial slices exterior to A). Given two horizon slices with A2 to the future of A1, monotonicity of the relative entropy under restriction of the semi- infinite null hypersurface starting at A1 to the semi-infinite subset starting at A2 implies the GSL: 0 S (A ) S (A ) . (2.25) ≤ gen 1 − gen 2 The argument applies to causal horizons, such as Rindler and black hole horizons. Unlike our proof of the Bousso bound, Wall’s proof (like that of [152]) does not assume the non-expansion condition. This is as it should be, since the GSL does not require any such condition. Suppose, for example, that the expansion is not monotonic between A1 and A2, because the black hole is evaporating but there is also matter entering the black hole. Then the horizon interval from A1 to A2 is not a light-sheet with respect to either past- or future-directed light-rays. Yet, the GSL must hold. On the other hand, our proof applies to all weakly focussed null hypersurfaces, whereas the GSL applies only to causal horizons. 4We have considered the case where the light-sheet L is a portion of a null plane H in Minkowski space, whereas we are now discussing the case where L is a portion of the horizon H of a black hole. In general, application of our flat space results to general spacetimes would require that the transverse size of L be small compared to the curvature scale. This is not the case for the horizon of a black hole. However, the vacuum states σH and σL can be defined directly on the black hole background; σH is the Hartle-Hawking vacuum. CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 18

Now suppose we consider a case where both the GSL and the Bousso bound should apply, such as a monotonically shrinking or growing portion of a black hole horizon. In this case, it should be noted that our proof and Wall’s proof [165, 164] refer to different entropies. In general the difference in the matter entropy outside A1 and A2 is distinct from the entropy that we have defined directly on the interval stretching from A1 to A2:

S S (A ) S (A ) = ∆S. (2.26) D ≡ m 1 − m 2 6 Because S ∆S is not of definite sign (and because of the different assumptions about non-expansion),D − our result does not imply Wall’s, and his does not imply ours even in the special case where a horizon segment coincides with a light-sheet. Instead, this case gives rise to two nontrivial constraints on two different entropies: one from the GSL and one from the Bousso bound. Our result allows us to connect a number of older works concerning Bekenstein’s bound [7]. It was shown long ago [30] that this bound follows from the Bousso bound in the weak gravity regime. At the time, a sharp definition of entropy for either bound was lacking [26, 28]. A differential definition of entropy was later applied to the right Rindler wedge, and positivity of the relative entropy was shown to reduce to the Bekenstein bound on this differential entropy, in settings where the linear size and the energy of an object are approximately well-defined [51]. Our present work offers two additional routes to the Bekenstein bound, in the sense of providing precise statements that reduce to Bekenstein’s bound in the special settings where the entropy, energy, and radius of a system are intuitively well-defined. Combining our result with [30] proves a Bekenstein bound, while supplementing a definition of entropy for both the Bousso bound and Bekenstein’s bound as the differential entropy on a light- sheet. The bound is in terms of the product of longitudinal momentum and affine width, but this reduces to the standard form 2πER/~, for spherical systems that are well-localized to the light-sheet. Alternatively, we may regard our section 2.2 alone as a direct proof of Bekenstein’s bound. Again the bound is on the differential entropy, but now in terms of the modular energy ∆ K on a finite light-sheet. For a system of rest energy E that is well localized to the centerh i of a light-sheet of width 2R in the rest frame, one has ∆ K 2πER, so [7] is recovered. h i ≈

Extensions An interesting problem is the extension of our proof to interacting theories. For interacting theories the quantization of fields on the light front is notoriously tricky. One could still try to define the entropy as the difference in von Neumann entropies for spatial intervals, in the limit where the spatial interval becomes null. In order to explore the properties of the entropy defined in this way one can consider strongly coupled field theories that have a holographic gravity dual. We have followed the recipe of [21] to obtain the modular Hamiltonian in terms of entropy perturbations. However, we find that ∆S = ∆ K holds exactly, and not just to first order in an expansion for states close to the vacuum. Thath i is, the relative entropy for every state is zero. This means that in the light-like limit, the CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 19

Aﬁnal

Si Ai matter Ainitial Figure 2.3: A possible approach to defining the entropy on a light-sheet beyond the weak- gravity limit. One divides the light-sheet into pieces which are small compared to the affine distance over which the area changes by a factor of order unity. The entropy is defined as the sum of the differential entropies on each segment.

operator algebra on the null interval becomes trivial, and all states on the null interval become indistinguishable. We expect that this property should extend to interacting theories without a gravity dual. One can intuitively understand this as follows. Concentrating on a null interval is equivalent to exploring the theories at large energies, since we want to localize the measurements at x− = 0. In an interacting theory this produces parton evolution as in the DGLAP equation [85,1, 59]. This evolution leads to states that all look the same at high energies. We expect the same equation ∆S = ∆ K to hold for non-superrenormalizable theories because, in contrast to the free theoriesh wei have discussed in this paper, these do not have operators localizable on a finite null surface [142, 150]. We will discuss these issues further in chapter3. Another question is how to extend our definition of entropy, and our proof, to the more general situation of a rapidly evolving light-sheet in a general spacetime. One approach is to divide the light-sheet into small segments along the affine direction in such a way that the change in area is small and then do an approximately flat space analysis for each piece. This is shown in figure 2.3. Here the initial expansion could be large and negative, but this just helps in obeying the bound. Thus, for each segment we obtain a constraint ∆Ai/(4GN ~) ∆Si. To make this argument we need to have a notion of local vacuum in the QFT in order≥ to define the modular Hamiltonian and to compute ∆S. We assume that this is possible. Then, for the original region we end up with a bound of the type ∆A ∆A = i i ∆S (2.27) 4G 4G ≥ i N ~ N ~ i P X where ∆Si are the entropies differences, as in equation (2.9), for each of the consecutive null segments. We can take the right hand side of equation (2.27) as the definition of the total entropy flux.5 It would be desirable to have a definition of the right hand side which involves 5We thank D. Marolf for this suggestion. CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 20 the whole null interval. Nevertheless, already equation (2.27) is a nontrivial bound. In the regime where we have a clear entropy flux, such as a star or a bucket of water, it reduces to the expected entropy flux if one takes the intervals to be large enough to capture many of the infalling particles. 21

Appendix

2.A Monotonicity of ∆A(c,b) ∆S 4GN ~ −

In sections 2.2 and 2.3, we showed that 0 ∆A(c, b)/4GN ~ ∆S. In fact, this difference decreases monotonically to zero as the boundaries≤ b and c are− moved together. To establish this stronger result, it suffices to consider variations of c. We may set b = 0. We first note that ∆ K ∆S is monotonically decreasing when the light-sheet is restricted. This follows immediatelyh i − from the monotonicity property of relative entropy S(ρ σ) = ∆ K ∆S under restriction to a subspace (via a partial trace operation), or more|| generallyh underi − any completely positive trace-preserving map [119]. Thus it only remains to be shown that ~δ(c) ∆A(c, 0)/4GN ∆ K (c, 0) will decrease monotonically under restriction. We will now prove≡ this for the− modularh i Hamiltonian of a free scalar field. Equation (2.22) for the area difference and equation (2.17) for the modular Hamiltonian can easily be generalized to an interval of length c. Their difference is

c 2 2 θ0(c) (c λ) δ(c) = d x⊥ + 2π dλ − T (λ) . (2.28) − 4G c kk Z N Z0 As we vary c, we always choose the initial expansion to be the largest value compatible with the light-sheet condition: λ θ0 = 8πGN inf dλ Tkk(λ) . (2.29) 0≤λ≤c Z0 The monotonicity of δ(c) is established by

c 2 dδ 2 c ∂θ0 θ0 λ = d x⊥ + 2π dλ 1 T (λ) . (2.30) dc −4G ∂c − 4G − c2 kk Z N N Z0 The ﬁrst term is non-negative, since increasing c broadens the range of the inﬁmum in equation (2.29). The latter two terms are together non-negative. This follows from the c non-expansion condition by integrating 0 dη ηθ(η) 0. It follows that δ is monotonically decreasing under restriction (and monotonically increasing≤ under extension) of the light- R sheet. This proves our claim. 22

Chapter 3

The Bousso Bound in Interacting Quantum Field Theory

In the previous chapter, we proved the Bousso bound, or covariant entropy bound [25, 71],

A A0 ∆S − , (3.1) ≤ 4GN ~ for light-sheets with initial area A and final area A0. The proof applies to free fields, in the limit where gravitational back-reaction is small, GN ~ 0, that the change in the area is of → first order in GN . Though this regime is limited, the proof had some interesting features. We made no assumption about the relation between the entropy and energy of quantum states beyond what quantum field theory already supplies. This suggests that quantum gravity may determine some properties of local field theory in the weak gravity limit. In this chapter, we will generalize our proof to interacting theories. We will continue to work in the weakly gravitating regime. In the course of this analysis, we will establish a number of interesting properties of the entropy and modular energy on finite planar light- sheets, for general interacting theories. In the free case, we defined the entropy as the difference of two von Neumann entropies [51, 123]. The relevant states are the reduced density operators of an arbitrary quantum state and the vacuum, both obtained by tracing over the exterior of the light- sheet. Following Wall [164], we were able to work directly on the light-sheet. Let us recall the structure of the proof in the free case. A very general result, the positivity of the relative entropy [120], implies that ∆S ∆ K , where ∆ K is the vacuum- subtracted expectation value of the modular Hamiltonian≤ operatorh i 1 [51].h Fori free theories, 1For any state ρ , the modular energy is ∆ K tr (Kρ ) tr (Kρ ). The modular Hamiltonian K is 1 h i ≡ 1 − 0 the logarithm of the vacuum density matrix K = log ρ0. K is defined up to an additive constant, which can be fixed by requiring that the vacuum expectation− value of K is zero, such that ∆ K = K . Similarly, ∆S = tr(ρ log ρ ) + tr(ρ log ρ ) is the difference between the entropy for the state ρh underi h considerationi − 1 1 0 0 1 and the vacuum ρ0. CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 23 the modular energy is found to be given by an integral over the stress tensor,

2π 1 ∆ K = dd−2y dx+ g(x+) T (x+, y) . (3.2) h i ++ ~ Z Z0

Here x+ is an aﬃne parameter along the null generators, which can be scaled so that the null interval has unit length. The function g is given by

g(x+) = x+(1 x+) . (3.3) − (For d = 2, g takes this form also in the interacting case; but as we shall see, in higher dimensions it will not.) Using Einstein’s equation, the area difference ∆A = A A0 can be written by a local integral over the stress tensor, plus a term that depends on− the initial expansion of the light-rays. The latter must be chosen so that the expansion remains nonpositive everywhere on the null interval. This is the “non-expansion condition” that determines whether a null hypersurface is a light-sheet. equations (3.2) and (3.3), combined with Einstein’s equation and the non-expansion condition, imply that ∆ K ∆A/4GN ~. To generalize this proof to interacting theories,h ai number ≤ of difficulties must be addressed. Wall’s results do not apply, so the entropy and modular Hamiltonian cannot be defined directly on the light-sheet. Instead, we must consider spatial regions that approach the light-sheet. The positivity of the relative entropy, ∆ K ∆S 0, holds for every spatial region [51], so it could still be invoked. But it is no longerh i − useful:≥ for spatial regions, ∆ K is highly nonlocal, and we are unable to compute it before taking the null limit. h i Instead, we benefit from a new simplification, which happens to arise precisely in the case to which our previous proof did not apply: for interacting theories in d > 2.2 In this case, the entropy ∆S must be equal to the modular energy ∆ K in the null limit. To show this, we recall that the von Neumann entropy is analytically determinedh i by the Rényi entropies. The n-th Rényi entropy is given by the expectation value of twist operators inserted at the two boundaries of the spatial slab. The approach to the null limit can thus be organized as an operator product expansion. We argue that, in the limit, the only operators that contribute to ∆S have twist d 2; and that for interacting theories in d > 2, there is only one such operator. This implies− that ∆S becomes linear in the density operator, and hence [21]

∆ K ∆S 0 (3.4) h i − → in the null limit. 2Our original proof applies to theories for which the algebra of observables is nontrivial and factorizes between null generators. This includes free theories but also interacting theories in d = 2 [164]. For d = 2, the area is the expectation value of the dilaton-like ﬁeld Φ that appears in the action as 1 d2xΦ(x)R+ . 16πGN If the d = 2 theory arises from a Kaluza-Klein reduction of a higher dimensional theory, then Φ is the volume··· of the compact manifold. R CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 24 The unique twist-2 operator is the stress tensor. This implies a second key result:

1 2π d−2 + + + ∆S = d y dx g(x ) T++(x , y) . (3.5) ~ Z Z0

Together with equation (3.4), this extends the validity of equation (3.2) to the interacting case: the modular energy is given by a g-weighted integral of the stress tensor. These arguments do not fully determine the form of the function g(x). For interacting conformal field theories with a gravity dual [122], we are able to compute g(x) explicitly from the area of extremal bulk surfaces [140, 99].3 For d > 2 we find that g differs from the free field case, equation (3.3). However, our proof [38] of the Bousso bound did not depend on equation (3.3). Rather, it is sufficient that g satisfies a certain set of properties. We will show that these properties hold in the interacting case. In particular, we will show that the key property,

dg 1 , (3.6) dx+ ≤

can be established by considering highly localized excitations and exploiting strong subadditivity. This will be suﬃcient to establish the extension of our free proof to the interacting case.

Outline This chapter is organized as follows. Sections 3.1 and 3.2 contain the new results sufficient to prove the Bousso bound in the interacting case (in the weakly gravitating limit). In section 3.1 we consider the light-like operator product expansion of the defect operators that compute the Rényi entropies. We derive equations (3.4) and (3.5), thus recovering a key step in the free-field proof: the local form of the modular energy, equation (3.2). We further constrain the modular energy in section 3.2, where we establish equation (3.6) for interacting fields. All remaining parts of the proof extend trivially to the interacting case. In sections 3.3 and 3.4, we explore our intermediate results for the entropy and modular energy on null slabs, which are of interest in their own right. In section 3.3, we compute the ∆S explicitly for interacting theories with a bulk gravity dual. This determines g(x+) for these theories. For d > 2, we find that g(x+) differs from the free field result. The approach to the null limit is studied in detail for an explicit example in appendix 3.A. In section 3.4, we examine the vanishing of the relative entropy in the null limit, ∆S = ∆ K . This arises because the operator algebra is infinite-dimensional for any spatial slab, whereash i no operators can be localized on the null slab. Any fixed operator is eliminated in the limit and thus cannot be used to discriminate between states. Appendix 3.B illustrates this behavior in a discrete toy model. In section 3.5, we summarize our results and discuss a number of open questions. 3Note that the bound we prove concerns light-sheets in the interacting theory when it is weakly coupled to gravity, not light-sheets in the dual bulk geometry. CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 25

i+1 A D D i

Figure 3.1: The R´enyi entropies for an interval A involve the two point function of defect operators D inserted at the endpoints of the interval. An operator in the ith CFT becomes an operator in the (i + 1)th CFT when we go around the defect.

3.1 Entropies for Null Intervals in Interacting Theories

In this section, we will explore the properties of the entropy of a quantum field theory on a spatial slab in the limit where the finite dimension of the slab becomes light-like (null). We consider free and interacting conformal field theories with d 2 spatial dimensions. (We will comment on the non-conformal case at the end.) For interacting≥ theories in d > 2, we will find that the entropy is equal to the modular Hamiltonian, and that both can be expressed as a local integral over the stress tensor. It is convenient to consider the Rényi entropies first. The nth Rényi entropy associated with a spatial region A S (A) = (1 n)−1 log tr ρn (3.7) n − A can be computed by taking the expectation value of a defect operator in a theory, which we denote by CFTn, obtained from taking n copies of a single CFT. The operator in question is a codimension 2 defect operator localized on the boundary ∂A of a spatial region A in the full Euclidean theory. In other words, the second orthogonal direction to the operator is Euclidean time. The defect operator is such that when we go around it, the various copies of the original CFT are cyclically permuted. In other words, an operator φk(x) defined on th th the k CFT is mapped to φk+1(x) on the (k + 1) CFT, and φn(x) is mapped to φ1(x); see figure 3.1.4 This operator implements the boundary conditions for the replica trick [45, 42]. To analyze the light-like limit, we start from the operators in Euclidean space. We then analytically continue them to Lorentzian time. Finally, we take the light-like limit. In this limit, we expect to have an operator product expansion. This expansion differs from the standard Euclidean operator product expansion in two respects. First, we are approaching the light-like separation, where the operators have zero metric distance but do not coincide,

4 These defect operators are oriented: there is a D+ which maps φi φi+1 and a D which maps → − φi φi 1. For an interval, we have the insertion of D+ and one end and of D at the other end. We will not→ explicitly− discuss this distinction. − CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 26 instead of approaching the coincident point along a purely space-like displacement. Second, in d > 2 dimensions, the two operators are extended and not local operators defined at a point. Despite these differences, we expect that there is a kind of operator product expansion that is applicable in this case. To our knowledge, the systematics of operator product expansions of extended operators in the light-like limit has not been explored. For the remainder of this section, we will make reasonable physical assumptions for the form of these operator product expansions. Operator product expansions for space-like regions were considered in [92, 47]. First, we recall the form of the light-like operator product expansion for local operators. We will take the limit x2 0 with x+ x0 + x1 held fixed. The expansion of two scalar operators has the form → ≡

O(x)O(0) x −2τO+τk (x+)sk O . (3.8) ∼ | | k,sk Xk In this equation, the operator O has spin s , scaling dimension ∆ and twist τ ∆ s ; k,sk k k k ≡ k − k and τO is the twist of the operator O. The twist governs the approach to the light-like limit. For finite x+, we sum over all of the contributions with a given twist. In free field theories, there are infinitely many higher spin operators with twist d 2. 1 − These operators contain two free fields, each with twist 2 (d 2). In an interacting theory, all operators with spin greater than 2 are expected to have twist− strictly larger than d 2. Furthermore, the twist is expected to increase as the spin increases [130] (see [114] for a more− recent discussion). The only operator with spin 2 and twist d 2 is the stress tensor, unless we have two decoupled theories. Operators with spin 1 include− conserved currents. Scalar 1 operators and operators with spin 1/2 can have twist τ 2 (d 2), with equality only for free fields. ≥ − As noted above, for d > 2 the defect operators in question are extended along some of the spatial dimensions. We now discuss features of the operator product expansion in this case. Consider first the standard Euclidean OPE (as opposed to the light-like one). For such operators, the OPE is expected to exponentiate and become an expansion of the effective action for the resulting defect operator. In general, new light degrees of freedom could emerge when the two defect operators coincide. However, in our case the two twist operators annihilate each other, leaving only terms that can be written in terms of operators of the original theory. In other words, we expect

d−2 1 D(x)D(0) exp d y d−2−∆ Ok(x = 0, y) (3.9) ∼ ( " x k #) X Z Xk | |

where y denotes the transverse dimensions and Ok denotes local operators on the defect at x = 0. Thus the expansion is local in y. We can view this equation as an expansion of the effective action for the combined defect (consisting of both defects close together) by integrating out objects with a mass scale of order 1/ x . | | CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 27 The leading term in equation (3.9) is given by the identity operator and contributes a d−2 factor of Ay/ x in the exponent (with a coefficient that depends on n), where Ay is the | | n −(n−1)Sn transverse area. This is the expected form of tr ρ0 = e , which gives the vacuum Rényi entropies for the interval. In the vacuum case, all other operators have vanishing expectation values. This contribution cancels when we compute the difference ∆S of the von Neumann entropies of a general state and the vacuum, so we will not consider it further. When we take the light-like limit of the Rényi defect operators, we expect to have an expansion which looks both like equation (3.8) and like equation (3.9). In other words, we expect the expression to be local along the y direction as in equation (3.9), but with terms that are nonlocal along the x+ direction as in equation (3.8). In principle, along the x+ direction, we can have terms which are very nonlocal. The operator Ok(0, y) in equation (3.9) is replaced by an operator of the form on the right hand side of equation (3.8):

d−2 −(d−2)+τk + sk D(x)D(0) light-like exp d y x (x ) Ok,sk . (3.10) | ∼ ( " | | #) Z Xk Note that the operators which appear in equation (3.10) are the operators of CFTn [92, 47]. The generic form of these operators is

O = O O O , (3.11) 1 2 ··· n th where Ok is an operator on the k copy of the original CFT. Some of the factors in equation (3.11) could be the identity, and the simplest operators we consider have only one factor which is not the identity. Performing the replica trick, the operators with a single factor that appear in the OPE of the two defect operators contribute to the entropy proportionally to an operator in the original CFT. Speciﬁcally, we ﬁnd

S = O . (3.12) single h Si Such contributions are linear in the density matrix, and therefore do not give rise to a non-zero value of ∆ K ∆S. The reason is that the operator on the right hand side is necessarily equal to Kh , sincei − K is the only operator localized to the region whose expectation value coincides with ∆S to linear order for any deviation from the vacuum state [21] (see also [171]).

The d > 2 interacting case We will now argue that for interacting theories in d > 2, all operators that contribute to equation (3.10) are of this simple type: they all have only one nontrivial factor. In fact, only the stress tensor contributes. Clearly, operators with τ > d 2 will not contribute; this includes all higher spin operators in an interacting theory. Conserved− spin 1 currents have twist τ = d 2, but cannot appear because the defect operators are uncharged. Next, consider possible− contributions from CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 28 1 operators with twist 2 (d 2) < τ d 2. These operators could appear in representations which are not symmetric− and traceless≤ 5−. However, since the twist operator is invariant under transverse rotations, these operators must appear in pairs; their combined twist would be bigger than d 2. Thus we can− focus on the operators with spin zero. An operator of CFTn consisting of 1 a single-copy scalar operator with twist in the range 2 (d 2) < τ d 2 would contribute to the entropy. This contribution will generically be divergent− in the≤ light-like− limit to ∆S, which is state dependent. In any case, single copy operators would give an equal contribution to ∆ K , so these operators do not contribute to ∆ K ∆S.6 On the other hand, if we h i 1 h i − n had two operators in the range 2 (d 2) < τ d 2 on different CFT copies inside CFT , the total twist will be higher than d− 2 and≤ we will− not get a contribution in the light-like limit. − This leaves the stress tensor, which has τ = d 2 and can contribute in the null limit. However, unless d = 2 (in which case τ = 0), only a− single factor can contribute. Therefore, ∆S = ∆ K for interacting theories in d > 2. Noticeh thati throughout this discussion, we have taken the coupling fixed and then taken the null limit. In particular, if we have a weakly coupled theory, we will get corrections to the result from free field theory which at each fixed order in perturbation theory will contain logs. One must resum the logarithms first, before taking the null limit, to recover the result that only the stress tensor survives. Returning to the Rényi entropyies, we conclude that in interacting conformal theories, the only operator that can contribute to the expansion in the light-like limit is the stress tensor. All of its descendants contribute as well, so equation (3.10) becomes a Taylor expansion around x+ = 0. Discarding the contribution from the identity operator, which will drop out of ∆S, we get

1 D (x)D (0) exp (n 1)2π dd−2y dx+ g (x+) T (x− = 0, x+, y) ] . n n |light-like ∼ − − n ++ Z Z0 (3.13) In this expression, we have set the size of the interval ∆x+ = 1 and extracted an overall factor of n 1 from the exponent. This factor accounts for the vanishing of the exponent for trivial Rényi− operators when n = 1. We have also replaced the sum over descendants by an integral over a function, gn, determined by matching with a Taylor expansion of the 5Examples of such operators are fermion fields, or antisymmetric tensors in four dimensions. 6In some cases, these contributions are not present because of symmetry reasons. An example is the 1 Wilson-Fisher fixed point at small = 4 d. In this case, the dimension of φ is 2 (d 2) + O(). However, due to the φ φ symmetry, this operator− does not appear in the OPE of the defect− operators involved in the replica trick.→ − Another example is the Klebanov-Witten theory [112]. These are four dimensional theories with operators of dimension 3/2 < 2. However, these operators carry a U(1) charge and cannot appear in this OPE. A relevant question here is whether there are theories with scalars with twists in this range which are not charged under any symmetry. If these operators are present, then our definition for ∆S will become divergent and will need to be modified. CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 29 operator T . The integral is restricted inside the null interval because operators outside this range would not commute with the operators that are spatially separated from the interval. The difference of von Neumann entropies of a general state and the vacuum is then given by analytic continuation: 1 ∆S = lim log Dn(x)Dn(0) n→1 1 n h i − 1 d−2 + + − + = 2π d y dx g(x ) T++(x = 0, x , y) Z Z0 = ∆ K . (3.14) h i The function g is as yet undetermined and will be further discussed in the next section. We expect the same holds for non-conformal theories with an interacting UV fixed point. For theories with a free UV fixed point, even if we expect that the modular Hamiltonian K has the same general form in terms of the stress tensor, whether ∆ K = ∆S or not would generically depend on further details. For relevant deformations ofh ai free UV fixed point we expect to have ∆ K ∆S as in the free theories, while we expect ∆ K = ∆S for asymptotically free theories.h i ≥7 h i

The case of free fields or d = 2 interacting fields In free field theory, or if d = 2, states with ∆S < ∆ K are known to exist on a null slab [38]. We close this section by examining why the aboveh argumenti for ∆S = ∆ K does not apply in these cases. h i If the operator (3.11) which appears in equation (3.10) contains more than one nontrivial factor, it can give rise to a contribution to the entropy which is not equal to the expectation value of any operator in the original CFT. These contributions are interesting because they make ∆S < ∆ K possible. In a free field theory, such operators arise from insertions of the fundamentalh fieldi φ in one copy and another field φ in another copy. They have twist τ = d 2 and can contribute in the light-like limit. In− an interacting theory, all such operators gain a non-zero anomalous dimension. In particular, in a unitary theory, the field φ gains a positive anomalous dimension and so will not contribute in the null limit8. However, in a d = 2 interacting theory, multiple copies of the stress tensor can appear. Since τ = d 2 = 0, the total twist will remain equal to d 2 no matter how many times the stress tensor− appears in (3.11). Thus, in d = 2, we can have− ∆S < ∆ K even for interacting theories. h i 7In asymptotically free theories, the coupling runs as g2 1/ log µ as a function of the scale µ. The OPE is not given by a simple power behaviour but we need∝ to integrate the anomalous dimensions of a dµ 2 range of scales as exp[ µ γ(µ)]. Since γ(µ) g (µ) 1/ log µ, this integral diverges at short distances. Therefore, operators with− non-zero anomalous dimensions∼ ∝ do not contribute in the null limit, which involves going to very high scales.R So we also expect equation (3.14) to hold. 8In gauge theories, the fundamental fields are not gauge invariant on their own, and should be sup- plemented with Wilson lines as interactions are turned on. These Wilson lines end at the positions of the defect. CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 30 3.2 Bousso Bound Proof

The modular Hamiltonian for slabs with non-unit aﬃne length ∆λ = c can be obtained from equation (3.14) by a simple coordinate transformation9:

c 2 ∆ K = 2π d x⊥ dλ g(λ, c) T (λ) . (3.15) h i λλ Z Z0 Here we have rescaled the affine parameter and emphasized the dependence of the function g on this scaling by replacing g(λ) g(λ, c). Symmetry under time reversal implies→ g(λ, c) = g(c λ, c), and boost symmetry implies that − g(λ, c) = cg¯(λ¯) , (3.16) where λ¯ = λ/c. We will now show that monotonicity of ∆A ∆ K is guaranteed if the function g satisfies a small number of other simple properties of− g(λh),i including concavity. We have dδ dθ c ∂g = c 0 + θ + dλ 1 T (λ) (3.17) dc − dc − 0 − ∂c λλ Z0 The first term is nonnegative independent of g. The second term is nonnegative if the function ∂g/∂c (viewed as a function of λ, at fixed c) satisfies the following properties:

∂g = 0 , (3.18) ∂c 0 ∂g = 1 , (3.19) ∂c 1 d ∂g 0 . (3.20) dλ ∂c ≥ c d ∂g These equations follow from the non-expansion condition, via 0 0 dλ θ dλ ( ∂c ). By equation (3.16), we have ≥ ∂g ∂g R =g ¯(λ¯) λ¯ . (3.21) ∂c − ∂λ¯ We conclude that the three suﬃcient conditions for monotonicity speciﬁed above are equivalent to the following conditions:

g(0) = 0 (3.22) g0(0) = 1 (3.23) g(1 λ) = g(λ) (3.24) − g¯00(λ¯) 0 (3.25) ≤ 9 We will set 2π/~ = 4GN = 1 in the remainder of the section. CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 31 The first two conditions, equations (3.22) and (3.23), are always satisfied because the modular Hamiltonian must reduce to the Rindler Hamiltonian near any two-dimensional spatial boundary. The third condition, equation (3.24), follows from CPT symmetry. The last condition, equation (3.25), is concavity. Subject to these conditions, the Bousso bound will be satisfied for any state, with monotonically increasing room to spare as the size of the light-sheet is increased. In each of the explicit calculations we made of the modular Hamiltonian, this property was satisfied. However, the proof of the Bousso bound does not rely on this property. It is straightforward to prove that ∆A/4GN ~ ∆S is positive, if not necessarily monotonic, if we replace the condition in equation (3.25−) with the following condition: g0(λ) 1 (3.26) | | ≤ In the remainder of this section, we will derive this remaining property, that the derivative of g(λ) is bounded, and complete the proof of the Bousso bound. This final condition in equation (3.26) arises from the theory of modular Hamiltoni- ans. Let us define an operator on the global Hilbert space which we call the full modular Hamiltonian: Kˆ = K K c , (3.27) V V − V where V c is the region complementary to V . For example, if V is a Rindler wedge, then KˆV is proportional to the boost generator. It is known that these Hermitian operators are monotonous under inclusion [23], that is Kˆ Kˆ 0 (3.28) V − W ≥ is a positive definite operator for any subregion W V . This property can be seen as a consequence of monotonicity of relative entropy and strong⊆ subadditivity of the entropy [20]. Let us first recall the definition of relative entropy S(ρ ρ ) = tr(ρ log ρ ). This is positive 0 ρ0 for any two density matrices. Relative entropy can also|| be rewritten as S(ρ ρ ) = ∆ K ∆S (3.29) || 0 h i − where ∆S tr(ρ log ρ) + tr(ρ log ρ ) (3.30) ≡ − 0 0 ∆ K tr(ρK) + tr(ρ K) , with K log ρ + constant. (3.31) h i ≡ 0 ≡ − 0 The positivity of relative entropy implies that ∆S ∆ K . Now, the monotonicity of relative entropy is the following statement. Suppose we≤ haveh i two regions W V and we 0 ⊆ have two density matricies for the big region, ρV and ρV . We can consider the restrictions 0 of these density matrices to the subregion W , call them ρW and ρW . Monotonicity is the 0 0 statement that S(ρW ρW ) S(ρV ρV ). In the present case,|| we obtain≤ two|| inequalities, one from W V and one from V c W c: ⊆ ⊆ ∆ K ∆S ∆ K ∆S , (3.32) h V i − V ≥ h W i − W ∆ K c ∆S c ∆ K c ∆S c . (3.33) h W i − W ≥ h V i − V CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 32 where we have rewritten the relative entropies using equation (3.29). Now we add these inequalities and separate the terms corresponding to the vacuum ρ0 and the ones corresponding to a state ρ1 different from the vacuum. The terms involving entropy are

0 0 0 0 S S c + S c S = 0 (3.34) V − V W − W which vanishes because the vacuum state is pure, and

1 1 1 1 S S c + S c S 0 (3.35) V − V W − W ≥ which is positive due to strong subadditivity10. The terms with modular Hamiltonians can 1 0 be grouped into KˆV KˆW and KˆV KˆW . This last term is zero since the full modular Hamiltonian is ah symmetry− i generatorh − whichi annihilates the vacuum.11 Hence we end up with the inequality 1 1 1 1 1 Kˆ Kˆ S S c + S c S 0. (3.36) h V − W i ≥ V − V W − W ≥ This holds for any global state ρ1 and implies equation (3.28). Going further, equation (3.28) implies the operator inequality KV KW KV c KW c and hence − ≥ − 1 1 1 1 K K K c K c . (3.37) h V i − h W i ≥ h V i − h W i ˆ 0 0 0 0 Moreover, KV 0 = 0 implies KV = KV c , and similarly, KW = KW c . Subtracting both of those equations,| i we now have

∆ K ∆ K ∆ K c ∆ K c . (3.38) h V i − h W i ≥ h V i − h W i In the null limit, this property is inherited by the full modular Hamiltonians of null slabs. Now, let us consider a state whose stress-energy is positive and highly concentrated near some x+ =x ¯+ W . Such states can be produced by taking a ﬁxed state and boosting it. We expect that∈ in this limit the state outside the slab (in the region W c, and hence also in c V ) is indistinguishable from the vacuum, so that ∆ K c 0, ∆ K c 0. For such h V i → h W i → states, equation (3.38) reduces to ∆ KV ∆ KW 0, and since both modular energies are positive, h i − h i ≥ ∆ K h V i 1 . (3.39) ∆ K ≥ h W i Now let V be a slab with λ [0, 1 + ] and let W V be a slab with λ [0, 1]. Using equation (3.15), we ﬁnd that the∈ modular energies of the⊂ highly localized states∈ satisfy

∆ K (1 + ) g(λ/¯ (1 + )) h V i = . (3.40) ∆ K g(λ¯) h W i 10 The strong subaditivity statement we are using is S(A) + S(B) S(A C) + S(B C) where A, B and C are three disjoint systems. This property is sometimes also called≤ weak∪ monotonicity.∪ 11This property follows from the definition K = log(ρ0 ) and the Schmidt decomposition of the vacuum V − V state across c . HV ⊗ HV CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 33 In the limit as 0, equation (3.39) now implies → dg g , (3.41) dλ ≤ λ Now, we repeat the argument with the region V as the Rindler region with λ [0, + ], and W the slab with λ [0, 1]. For this region V the function g = λ. For a state∈ with∞ a concentrated stress tensor∈ we obtain g(λ) 1 (3.42) λ ≤ From this, we conclude that dg 1 1 . (3.43) − ≤ dλ ≤ where the first inequality is obtained from the g(λ) = g(1 λ) property in equation (3.24). To prove the Bousso bound, we consider without loss of− generality the null slab λ (0, 1). We define F (λ) λ + g(λ), which obeys F (0) = 0 and F (1) = 1 by equations (3.22∈ ) and (3.23). We also have≡ F 0 0 everywhere, by equation (3.43). These properties of the modular Hamiltonian suffice to show≥ that the area difference along the light-sheet bounds the modular energy: ∆A ∆ K . As usual, positivity of the relative entropy implies that ∆S ∆ K 4GN (with equality≥ holdingh i for d > 2 interacting theories). This completes the proof of the≤ Boussoh i bound for both free and interacting theories, in the weakly gravitating limit.

3.3 Holographic Computation of ∆S for Light-Sheets

In this section, we consider interacting quantum field theories that have a gravity dual. In this case, the Ryu-Takayanagi formula [140, 99] allows us to compute the entropy ∆S in the null limit. This will confirm our earlier demonstration that ∆S = ∆ K , and will determine g(x+) explicitly for such theories. First, we consider a CFT; later, weh i will comment on the non-conformal case. Appendix 3.A discusses the approach to the null limit in greater detail. We write the boundary metric as ds2 = dx+dx− + d~y 2. Let us first consider a spatial strip, extended along the y directions. One− end of the interval is at x+ = x− = 0 and the other end is at x+ = x− = ∆x+, a fixed constant. The bulk metric can be written as − dx+dx− + dy2 + dz2 ds2 = − . (3.44) z2 The minimal surface solution was found in [11, 139]. It is given by + udF ( 1 , d , 3d−2 ; u2(d−1)) + − ∆x 2 2(d−1) 2(d−1) x = x = 1 d 3d−2 , − 2 F ( 2 , 2(d−1) , 2(d−1) ; 1) ∆x+ u d z = 1 d 3d−2 = zmax u , (3.45) 2 F ( 2 , 2(d−1) , 2(d−1) ; 1) √ dx+dx− + dz2 A = A − , (3.46) vacuum y zd−1 Z CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 34 where F is the usual hypergeometric function12 and u (0, 1) is a parameter describing the first half of the minimal surface, which is symmetric around∈ x+ = ∆x+/2. The maximum zmax of z is achieved for u = 1. Here Ay is the area in the y directions. The formal expression for the area is UV divergent, but, as usual, we get a finite remaining contribution. We now consider a boosted interval. For that purpose we apply a combination of a boost in the x± plane and a dilation that transforms

x+ x+ , x− η2x− , z ηz , with η 0 . (3.47) → → → → This transformation takes the original space-like interval to a null interval stretched along the x+ direction. The proper length of the interval approaches zero. We also see that the surface is approaching the AdS boundary, in the sense that the largest value of z is going to zero as z η 0. Under these circumstances we find that the expression of the renormalized area (after∼ → subtracting the cutoff dependent piece) goes to minus infinity as 1/ηd−2. This is the expression for the vacuum von Neumann entropy for the interval. Let us now consider a non-vacuum state. We expect that the minimal area surface will continue to approach the AdS boundary as we take the null limit. Near the boundary, the metric approaches the AdS metric plus some small fluctuations. We can parametrize the metric as dz2 + dxαdxβ(η + h ) ds2 = αβ αβ , h t (x)zd + o(zd+1) . (3.48) z2 αβ ∼ αβ Now the minimal surface action can be written as 1 A = dd−2y dx+dx− + dz2 + zd t (dx+)2 + (3.49) zd−1 − ++ ··· Z p where we wrote the part of the action that does not go to zero in the large boost limit, η 0. More precisely, notice that the first two terms inside the square root scale like η2, while→ the last scales like ηd. We will assume that d > 2 and return to the d = 2 case later.

The case of d > 2 For d > 2, the last term in the square root is a small perturbation and we can therefore expand the action. Due to the factor of 1/zd−1 1/ηd−1, the resulting ﬁrst order term gives a ﬁnite answer ∼

d−2 + z t++ A = Avac + d y dx (3.50) 2 dx−/dx+ + (dz/dx+)2 Z − z A A = max dd−2y dx+ t pud(x+) (3.51) − vac 2 ++ Z 12The value of the hypergeometric function evaluated at 1 can be written in terms of gamma functions: 1 d 3d 2 d 1 F ( 2 , 2(d 1) , 2(d−1) ; 1) = √πd Γ( 2(d 1) )/Γ( 2(d 1) ). − − − − CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 35 where the first term is the vacuum contribution in equation (3.46). We have also used that the vacuum contribution is larger and determines the equations of motion for the surface to the order we need in order to evaluate the second term. We then see that the g++ component of the metric gives a finite contribution. By performing a similar expansion, we can check that all other components of the metric do not contribute in the null limit either. For this, it is important to use equation (3.47) to see how various terms behave. As an example, d d consider a component hyy in the metric. The component contains a z η which multiplies the whole action that scales as η2−d. Since d > 2, such a term does∼ not contribute. In a similar way, we discard higher orders in in the expansion of the metric around z = 0. In conclusion, the only part of the metric that matters is the first non-zero term in the expansion of h++. This first-order term is also the term that gives the expectation value of the stress tensor, 16πG t = N T , (3.52) ++ d h ++i

where T++ is the CFT stress tensor (we have set the AdS radius to unity). A similar expansion was performed in [21].13 Using the solution in equation (3.45), we can write (3.51) in the form

+ ∆A ∆x ∆S = = 2π dd−2y dx+ ∆x+g(x+/∆x+) T (x+, y, x− = 0) (3.53) 4G h ++ i N Z Z0 with g deﬁned parametrically by

d d 1 d 3d−2 2(d−1) u u F ( 2 , 2(d−1) , 2(d−1) ; u ) g(v) = 1 d 3d−2 , v = 1 d 3d−2 . (3.54) 2F ( 2 , 2(d−1) , 2(d−1) ; 1) 2F ( 2 , 2(d−1) , 2(d−1) ; 1)

The function g(v) is plotted for several dimensions in figure 3.2. Explicitly, we find g(v) = v(1 v) for d = 2, and in the limit d the function converges to sin(πv)/π. For small v, we− obtain the result g(v) = v + (v→2). ∞ We have thus obtained ∆S in termsO of the expectation value of an operator, namely a certain integral of T++ . According to the general argument discussed in section 3.1, the operator in the righth handi side is ∆ K ; we obtain ∆S = ∆ K . Notice that the relation ∆S =h ∆ iK gives values of theh i entropy on the light-sheet that are very different from naive expectations.h i For example, consider a thermal state. The entropy scales with the size of the interval as (∆x+)2, rather than the na¨ıve (volume- extensive) entropy which grows like ∆x+ and which applies in the large temperature regime. Hence in this regime, we find ∆S is actually much greater than the naive entropy. To check in detail how the extensive entropy for spatial regions turns into a term that goes as (∆x+)2 for null surfaces, we have computed the areas of minimal surfaces in a black hole background. 13 The authors of [21] considered a general surface and then expanded the metric to first order around the AdS metric. Here, the argument is simpler because we only need the first order term in the expansion of the metric near the boundary. Furthermore, we only need to consider the g++ component. CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 36

0.5

0.4

0.3 L v H g 0.2

0.1

0.0 0.0 0.2 0.4 0.6 0.8 1.0 v Figure 3.2: The functions g(v) in the expression for the modular Hamiltonian of the null slab, for conformal ﬁeld theories with a bulk dual. Here d = 2, 3, 4, 8, from bottom to top. Near the boundaries (v 0, v 1), we ﬁnd g 0, g0 1, in agreement∞ with the modular Hamiltonian of a Rindler→ wedge.→ We also note→ that→ the ± functions are concave. In particular, we see that g0 1, in agreeement with our general argument of section 3.2. | | ≤

We find that there is actually a phase transition into a different class of extremal surface solutions as ∆x− 0. This is explained in more detail in appendix 3.A. We can now briefly→ discuss the situation in non-conformal field theories. If we add a relevant deformation to the field theory, we are adding a scalar field in the bulk which has a profile going like φ z∆ for small z. This affects the metric at quadratic order via terms of the form φ2 z2∆.∼ Such terms modify only the diagonal components of the metric, and we have seen that∼ as long as 2∆ > d 2, such terms vanish. The latter is precisely the unitarity condition for a non-free scalar operators.− 14

The case of d = 2 In two dimensions, it is still true that the minimal surfaces (geodesics) approach the boundary, but it is no longer true that we can treat the term involving g++ in a perturbative fashion because it scales in the same way as the other terms. This implies that the final answer is non-linear in T++. This non-linearity allows for ∆S < ∆ K . For simplicity, consider the special case of the theory at finite temperatureh i (or in Rindler space). Since it is related by a conformal transformation to the plane, we can do all the computations explicitly by a simple coordinate transformation. The two point function of the twist operators is 1 Φn(x)Φn(0) = + − (3.55) h i ∆x ∆x 2∆n [sinh(π β ) sinh(π β )] 14 The unitarity condition is 2∆ d 2. For equality, we have a free field in the boundary theory. ≥ − CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 37 with x± = τ σ. This leads to the entropy [42] ± c β2 ∆x+ ∆x− S = log sinh(π ) sinh(π ) . (3.56) 6 π22 β β c ∆x+∆x− The vacuum case is given by the β limit, or S = 6 log 2 . In the null limit ∆x− 0 we get → ∞ → c β ∆x+ ∆S − = log sinh(π ) . (3.57) |∆x =0 6 π∆x+ β This can be expanded as

c x2 x4 ∆S = + , x 1 , (3.58) 6 6 − 180 ··· c ∆S = [x + constant + ] , x 1 , (3.59) 6 ··· where x = π∆x+/β. The first line is what we expect from the expansion of terms involving 2 operators of the form T++, (T++) , possibly integrated at different points, replica copies, etc. The last expression comes from resuming all these operators. In this case, this agrees with what we expect from the operator product expansion, since all these operators have twist zero in d = 2. The important point is that operators on different replica copies survive the limit; see section 3.1. Note that the modular Hamiltonian is

+ ∆x u cx2 ∆ K = 2π du u 1 + T++ = . (3.60) h i 0 − ∆x h i 36 Z since T = cπ . This agrees with the ﬁrst term of the small x expansion in equa- h ++i 12β2 tion (3.58), but in general it gives something larger than ∆S. This is particularly clear + for x ∆x 1, and it can also be seen from the quartic correction in equation (3.58). ∼ β Therefore, in d = 2, we get ∆S ∆ K but we do not get ∆S = ∆ K . Since all these results follow≤ fromh conformali symmetry, it is clear thath i the gravity answer will reproduce them. This computation was done in [139]; one can check that the geodesics approach the boundary but equation (3.57) is reproduced.

3.4 Why is ∆S = ∆ K on Null Surfaces? h i The relation ∆ K = ∆S is startling at first sight. It implies that the relative entropy h i 0 between any state and the vacuum, S(ρV ρV ) = ∆ K ∆S, vanishes in the light-like limit. The relative entropy is a statistical|| measure ofh howi − easy is to differentiate between two states by making measurements. In general, the probability of confounding two states − || 0 by making N measurements falls off exponentially no faster than e NS(ρV ρV ) (see e.g. [158]). CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 38 If ∆ K = ∆S, then the vacuum cannot be differentiated from any other state by making measurementsh i of operators localized to a null surface. In other words, all states look the same as we approach this surface. A related puzzle is the following: it is a general property of relative entropy that 0 0 S(ρV ρV ) = 0 implies ρV = ρV , but this in turn would give ∆ K = ∆S = 0. However, the prediction|| ∆S = 0 is not what we have found holographically.h i

(a) (b) (c)

Figure 3.3: Operator algebras associated to various regions. (a) Operator algebra associated to the domain of dependence (yellow) of a space-like interval. (b) The domain of dependence of a boosted interval. (c) In the null limit, the domain of dependence degenerates to the interval itself.

In this section, we explain both of these puzzles by noting that the quantities ∆S and ∆ K are defined as limits for vanishing ∆x−. At finite ∆x−, the states are distinguishable byh operatorsi included in the algebra (∆x+, ∆x−) on the causal development of the spatial interval, but as we take the limit ∆x−A 0, any fixed operator eventually drops out from the → + − 15 algebra. No operator remains in the intersection of all algebras, ∆x− (∆x , ∆x )=1. See figure 3.3. The same reason explains how ∆S and ∆ K can∩ beA non-zero while the relative entropy is zero. This result cannot be correct for statesh i on a fixed algebra, but it is a possibility for these quantities defined as limits on vanishing algebras. We describe how this can be accomplished using a toy model of an infinite chain of qubits in appendix 3.B. In general, the relation ∆ K = ∆S = 0 could not have been possible if the algebras for finite ∆x− were not infinite dimensional.h i 6 This phenomenon requires the full QFT, taking the UV cutoff to zero before taking ∆x− 0. Otherwise, at finite ∆x−, we would run out of operators and find ∆ K = ∆S = 0. → Let us briefly describeh whati is meant by a full quantum field theory. A quantum field is an operator-valued distribution. In order to produce an operator acting on Hilbert space, a quantum field φ(x) has to be averaged by a smooth function of compact support φα =

15 Note that the operator T++ evaluated on the null interval should not be considered as part of the algebra because it sends states into non-normalizable states, even if we smear the operator along the null interval and the transverse directions. Nevertheless, the expectation value of this operator can be computed and can be different in two different states. CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 39 ddx α(x)φ(x). If α(x) is smooth on a d-dimensional spacetime region, we are guaranteed by the Wightman axioms that φ is a well defined operator whose domain contains the vacuum R α state. The set of these operators where the support of α(x) is included in a spacetime region V generates the algebra of operators acting in V .16 We want to see if a fixed localized operator can be defined for the null plane such that + − it remains in the intersection of the algebras ∆x− (∆x , ∆x ) which implement the null limit. The problem of whether the domain of the∩ testA function α can be reduced to a spatial region or a region on a null plane, as opposed to a spacetime region, was studied in the past [150, 142], mainly in attempts to develop a precise mathematical foundation to the usual canonical formalism of equal time commutation relations (see also [164]). If φα is a well-defined operator, we should have

φ 0 2 = 0 φ† φ 0 = dx dy α(x)∗α(y) 0 φ(x)†φ(y) 0 < . (3.61) k α| ik h | α α| i h | | i ∞ Z This condition on the two point function of the ﬁeld constrains its ultraviolet behavior. The Fourier transform of the two point function with no time ordering φ†φ is h i 2s−2s+−2s− 2s+ 2s− 2 py p+ p− θ(p0)θ( p ) d , (3.62) 2 2 −∆+s − (p+p− + py)

where py is a polynomial in the transverse components. To evaluate equation (3.61), we take the Fourier transform:

+ − d−2 −i(p+x +p−x +pyy) α(x) = dp+ dp− dpy e α(p+, p−, py) . (3.63) Z For α(x) with support on the surface x− = 0, we have

α(p+, p−, py) = α(p+, py) , (3.64)

with α(p+, py) independent of p− and falling oﬀ to zero faster than any polynomial in p+ and py due to the smoothness of α(x). We have

2s−2s+−2s− 2s+ 2s− 2 2 d−2 py p+ p− α(p+, py) φ 0 dp dp− dp | | . (3.65) α + y d − k | ik ∼ 2 2 2 ∆+s Zp <0,p0>0 (p+p− + py)

The test function makes the integral convergent for large p+ and py. However, the integral may not converge for large p−. The best chance we have for it to converge is when s− = 0. Power counting gives a convergent integral if d 2 τ = ∆ s < − , (3.66) − 2 16These are von Neumann algebras. There is a technical point in that these algebras are better described as algebras of bounded operators. Bounded operators can be obtained from φα by taking the projectors in its spectral decomposition [86]. CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 40 which is never the case for a unitary theory. 1 For free ﬁelds, we have ∆ s = 2 (d 2), but the ﬁeld obeys the wave equation so that instead of equation (3.62),− we have− an expression localized on the mass shell p2 = 2 p−p+ + py = 0. In this case, the denominator in equation (3.62) is replaced by the delta − 2 function δ(p ). Eliminating p+ gives

2s−2s+−2s− 2 2s+ 2s− 2 2 2 d−2 2 py (py/p−) p− α(py/p−, py) φ 0 dp− dp Θ(p− + p /p−) | | . (3.67) k α| ik ∼ y y p Z − This integral in p− is logarithmically divergent for a free scalar field with s+ = s− = 0, but converges for ∂+φ and its derivatives with s+ > s−. For a free spin 1/2 field, it converges + for the ψ component (and derivatives). For a Maxwell field tensor Fµν, we must again take the component with s− = 0 and s+ = 1, that is, the components Fy,+. So only for free fields do we expect to have operators localized on the null surface and ∆S = ∆ K for general states. The localized operators can be non-local in the y direction so that6 ∆hS idoes not need to decompose into a sum of contributions from each of the null lines. For non-conformal theories with a free UV fixed point, the localizability of the operators depends on the details of the approach to the fixed point. Using the spectral representation of the two point function for a scalar field in terms of that of a free massive scalar field

0 φ(x)†φ(0) 0 = dm2 ρ(m2) G (x, m2) , (3.68) h | | i 0 Z the general result [142] is that the derivative ∂+φ of this scalar ﬁeld can be localized only if

dm2 ρ(m2) < . (3.69) ∞ Z This condition gives a ﬁnite wave function renormalization, which is expected to hold for superrenormalizable theories but not for marginal renormalizable theories [151].

3.5 Discussion

Summary. We explored some properties of the entropy associated to null slabs in general interacting field theories. We found a general expression for the modular Hamiltonian in terms of a local integral of the stress tensor components along the null slab, equation (3.2). We derived this by considering the light-cone OPE for the defect operators that compute the Rényi entropies; general arguments involving the spectrum of operators then constrain the von Neumann entropy and show that it is equal to the modular Hamiltonian. We also proved certain inequalities obeyed by the function g that multiplies the stress tensor in the modular Hamiltonian. These inequalities, equations (3.41) and (3.43), were then shown to be sufficient for the Bousso bound [25]. Our work extends our earlier proof of the Bousso bound to interacting theories. CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 41 We computed the von Neumann entropy in the null limit for theories with a gravity dual. In the null limit, the minimal surface approaches the AdS boundary. The change in the area can be found from the asymptotic form of the metric. This asymptotic form of the metric also determines the stress tensor. Therefore, we get a result that is in line with our general expectations. We view this as an additional consistency test on the holographic entanglement entropy formula [140, 99] in a strongly Lorentzian context. Our analysis fully determines the function g for such theories, equation (3.54), and it shows that g takes a different form than in the free theory. A curious feature of our result is that, for interacting theories, the change in entropy is exactly given by the change in the expectation value of the modular Hamiltonian: ∆S = ∆ K . In a finite-dimensional Hilbert space, this relation would also imply that both ∆S andh ∆i K are zero. Here, however, they are non-zero. This is possible because we are taking a limith thati involves infinite dimensional algebras. We also saw that no elements remain in the algebra after we take the limit. One can still consider limiting values of expectation values of operators on the null line, but such operators, or their smeared versions on the null surface, do not define reasonable operators on the Hilbert space because their variance is infinite. Physically, this result means that in interacting theories, one cannot distinguish between any two states by making measurements purely on the light-sheet. Appendix B presents a simple toy model involving an infinite number of qubits where similar features are present.

Discussion and open problems. The Bousso bound involves the notion of an entropy flux through the light-sheet. Defining a local notion of entropy current is notoriously difficult in quantum field theory. Here we have defined it through ∆S, the difference in the von Neumann entropies of the interval between two different quantum states. This notion does indeed have properties that suffice to ensure the validity of the corresponding Bousso bound. Nevertheless, the quantity ∆S has some counter-intuitive properties. The most surprising aspect of this definition is that we find ∆S = ∆ K , which means that all ordinary states are indistinguishable by local measurements on theh light-sheet.i We have not found more familiar-looking definitions for the entropy flux, to which a Bousso- type bound might apply. Further research will be needed to better understand the relation between ∆S and more conventional (space-like) definitions for the entropy flux. Notice that the energy flux is given by a local quantity, the expectation value of T++. On the other hand, ∆S is non-local since the function g depends on the positions of the endpoints of the interval. Thus, it cannot be viewed as the flux of a local operator. We expect that ∆S will provide an upper bound to the more familiar concepts of entropy. For example, in a theory where we can define an entropy current, as in hydrodynamics, we expect that ∆S should be larger than the flux of the entropy current on the light-sheet. In the holographic computations involving black branes, this is indeed true. The reason is very simple: the entropy flux scales like the length of the interval, ∆x+, on the other hand ∆S scales like (∆x+)2. The relative coefficient involves the temperature, T . This means that if CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 42 ∆x+ is somewhat greater than β = 1/T , then ∆S will be larger than the entropy flux. We also see this clearly in the two dimensional results, equation (3.57). We expect this to be a general feature of thermal or hydrodynamic states. An interesting conclusion is that information in interacting theories becomes very delo- calized on the light front. Information that is fairly localized along the longitudinal direction in free theories spreads once we include interactions. We also expect that the mutual information between a null interval and any other fixed region should vanish. This follows from the result ∆S = ∆ K . We also see this in the holographic examples. In a CFT with ah gravityi dual, the entropy ∆S for spatial slabs in a thermal state displays a phase transition as the null limit is approached (see [115] and appendix 3.A). This is likely to hold in general for states which start out with a non-zero ∆S for a space-like interval in the large N approximation. In the previous chapter, we considered free field theory. For free theories, one can prove not only the Bousso bound but the stronger result of monotonicity [38]: ∆A/4GN ~ ∆S never decreases under inclusion in a larger light-sheet. This follows from the concavity− of the function g(λ), g00(λ) < 0, which holds in the free case. Here, we found that this property continues to hold for interacting theories with a holographic dual (figure 3.2), so monotonicity 00 of ∆A/4GN ~ ∆S follows in these cases. We leave a general proof of g (λ) < 0 to future work. It would− also be nice to compute the function g to first order in perturbation theory for a weakly coupled CFT, such as = 4 super Yang-Mills. N 43

Appendix

3.A Extremal Surfaces and Phase Transitions on a Black Brane Background

In this appendix, we consider a thermal state in an interacting CFT with a bulk dual, an asymptotically anti-de Sitter planar black brane spacetime. This allows us calculate the von Neumann entropy holographically using the HRT prescription [99]. We are able to study the approach to the null limit in detail, reproducing the result of Narayan et al. [115] that the entropy on sufficiently large slabs undergoes a phase transition at large boost. We reproduce the result ∆S = ∆ K for the null slab. The metric of a black braneh ini AdS is f dt2 + dx2 + dz2/f + dy2 ds2 = − , f = 1 zd/zd . (3.70) z2 − 0 The inverse black hole temperature is 4πz β = 0 , (3.71) d and the energy density is given by (d 1) 1 T00 = − d . (3.72) 16πGN z0 It follows that the null-null component of the stress tensor is d d 1 T = T = . (3.73) ++ 4(d 1) 00 64πG zd − N 0 The extremal surface action is 1 I = dz f t˙2 +x ˙ 2 + 1/f . (3.74) zd−1 − Z q Let the momentum conjugate to x be denoted p. We find 1 x˙ 1/f f t˙2 p = , x˙ = pzd−1 − . (3.75) zd−1 f t˙2 +x ˙ 2 + 1/f 1 p2z2(d−1) − p − p p CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 44 Define a new effective Lagrangian L0 L px˙: ≡ − 1 1 0 ˙2 2 2(d−1) 2 ˙2 L = d−1 1/f f t 1 p z = 2(d−1) p 1/f f t . (3.76) z − − rz − − q p q Writing E for the momentum conjugate to t, we obtain

f t˙ E = , (3.77) 1/f f t˙2 z−2(d−1) p2 − − E t˙ = p p , (3.78) f 3/2 E2/f + z−2(d−1) p2 p − x˙ = p . (3.79) √f E2/f + z−2(d−1) p2 − These are the equations of motion ofp the extremal surfaces. We take E, p > 0 and ﬁx d scale invariance by setting z0 = 1 in the function f(z), so f(z) = 1 z . Integrating these trajectories, we obtain the null coordinates ∆x± of the the extremal− surface solutions at the boundary, zr E 1 ∆x± = 2 dz p . (3.80) 2 −2(d−1) 2 0 f ± √f E /f + z p Z − We can also rewrite the initial action (i.e. area) fromp equation (3.74) as zr 1 I = 2 dz . (3.81) 2(d−1) 2 −2(d−1) 2 0 z √f E /f + z p Z −

In these integrals, the upper limit zr is the returnp point of the trajectory, which is the smallest positive root of the denominators of equations (3.78) and (3.79); that is,

E2 + z−2(d−1) p2 = 0 . (3.82) 1 zd r − − r

The turning point is calculated from the smallest solution of this equation, with zr (0, 1). The ﬁrst two terms in equation (3.82) are positive when z (0, 1), and the second∈ is greater than one. We conclude that an extremal surface which returns∈ to the boundary exists for all p > 1. As we increase E from zero with p > 1, there are solutions for zr only up to a maximum value of E, which we denote by Emax(p). This maximum value of E is simultaneously the solution to equation (3.82) and

d E2 1 + = 0 . (3.83) dz 1 zd 2(d−1) r − r zr

When E > Emax, there are no extremal surface solutions that return to the AdS boundary. CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 45

E 4

0 2 4 6 8 10 p

Figure 3.A.1: The maximum value Emax(p) of E for getting a surface that returns to the boundary (solid line). For comparison, the line E = p 1 is plotted (the dashed line). The − extremal surface solutions of interest appear in the region p > 1, 0 < E < Emax(p). Here, we have taken d = 3.

In figure 3.A.1, we plot the parameter space (p, E) for d = 3. In d = 3, the curve E = Emax(p) runs near the line E = p 1; the plot looks similar in other dimensions. The shaded region contains the extremal surface− solutions. Interestingly, the extremal surface for E = Emax(p) corresponds to a time-like region on the boundary, with ∆t > ∆x. This counter-intuitive result is possible because even if the boundary interval is time-like, we are still considering locally spatial surfaces in the bulk. However, these extremal surfaces cannot be regulated by vacuum-subtraction, because the extremal surface solutions with this boundary region in vacuum AdS do not have a well-defined area. So the parameter space we are interested in is further reduced to E < Enull(p) Emax(p), where Enull(p) denotes the energy for which the extremal surface solution has ∆x−≤= 0. The separation between Emax(p) and Enull(p) in parameter space is, however, exponentially small. We show the relevant contour in figure 3.A.2 in logarithmic variables (for d = 3; other dimensions are similar). Numerical analysis of the solutions shows that, for ∆x+ 1 and smaller17, there are no exact solutions with ∆x− = 0, only an asymptotic set of solutions∼ for which ∆x+ is fixed and ∆x− approaches but never exactly reaches zero. The parameters p and E go to infinity in the limit ∆x− 0, and the extremal surface runs closer to the AdS boundary. We call this family of solution→ the “perturbative solutions,” because ∆S can be computed perturbatively in this case (see section 3.3). For sufficiently large ∆x+ (larger than approximately 15 in d = 3), in addition to the asymptotic solution, there exist two other solutions with finite − + − Enull(p) such that ∆x = 0 exactly. Figure 3.A.2 gives the contour plot of ∆x and ∆x for

17 Recall we have set z0 to unity. CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 46

35 0 120 0 25 40 70 -10-10 25

20 25 L max

E 20 L 15 E - max E

HH 15 -10-8 log -

10 10 -6 - -10 -0.01 -10 4

5 0.001 0.00001 5 0.1 - 5 -1

0 -10 0 10 20 30 logHp-1L

Figure 3.A.2: Curves of constant ∆x+ (black solid curves) and ∆x− (blue dashed curves), in the logarithmic parameter space defined by (log(p 1), log(Emax(p) E)/Emax(p)). The value p = 1 maps to and p = maps to + on− the− horizontal axis,− while E = 0 maps −∞ ∞ ∞ to 0 and E = Emax(p) maps to + on the vertical axis. The thick blue contour represent the null solutions with ∆x− = 0.∞ Above this contour, the boundary interval is time-like. If ∆x+ & 15 and we follow a contour of constant ∆x+, we find two solutions with exact ∆x− = 0. For all contours of fixed ∆x+, there exists an asymptotic null solution in the limit p . → ∞ a region of the parameter space (p, E). We plot the solutions in logarithmic parameter space in figure 3.A.2. Following a contour of constant and sufficiently large ∆x+ from left to right − in this diagram, ∆x+ & 15, we intersect the contour ∆x = 0 twice, corresponding to the two precisely null solutions. The part of the contour to the left of the first intersection (with p 1) is the “thermal” family of solutions. These solutions have a thermal character because most∼ of the surface extends near to the horizon of the black hole. Hence the entropy contains a term that grows like ∆x+, a volume-extensive term that goes with the thermal entropy density. By increasing p along a contour of fixed ∆x+, we again approach the asymptotic perturbative solution. Let us see in more detail how the area behaves in these two solutions. As shown below, there is a third null solution, but it has greater area than the other two and, according to CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 47 the HRT prescription, should not be regarded as the entropy.

Perturbative solution According to section 3.3, we expect the limiting value of the entropy to take the form

1 + 2 ∆S = 2π A⊥ (∆x ) T dv g(v) . (3.84) h ++i Z0 The diﬀerence between the perturbative extremal surface area and the vacuum area is

1 + 2 ∆A = 8πG A⊥ (∆x ) T dv g(v) . (3.85) N h ++i Z0 Using equation (3.73) and the explicit form of the function g(v), we obtain

2 d 1 Γ Γ + 2 d−1 2(d−1) A⊥ (∆x ) ∆A = 2 d . (3.86) − z 32π1/2(d 1) Γ 3d 1 Γ d 0 − 2(d−1) 2(d−1) Setting A⊥ = z0 = 1, we obtain perfect accord with our numerical simulation of the extremal surfaces.

Thermal solution This solution captures the thermal entropy. We expect the diﬀerence in extremal surface areas to approach ∆x+/2 asymptotically at large ∆x+. The thermal solutions track the horizon of the black hole at z0 = 1. In parameter space, this occurs when E is of the same order as (p 1). When this is the case, most of the − contribution to the integral comes from the region where z is order z0 = 1, and we can expand the integrand around that point. First we perform the substitutions

2d p = 1 + 2δ , z = 1 u/(d 1) ,E = √δ2 σ2 . (3.87) − − − d 1 r − In this limit, the integrals become

d 1 du ∆x = − , (3.88) 2 2 2d δ+σ (u δ) σ r Z − − (d 1)√δ2 σ2 du ∆t = − −p , (3.89) 2 2 d δ+σ u (u δ) σ Z − − d 1 du p + Aren = − = ∆x = ∆x /2 . (3.90) 2 2 2d δ+σ (u δ) σ r Z − − p CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 48 The renormalized area is obtained by subtracting the divergent piece with a UV cutoﬀ . Note that the zeros in the denominator occur at u = δ σ, both of which we take to be positive. Additionally, there is a zero at u = 0 in the denominator± of the integral for t˙. The integral over u starts at the largest zero, u∗ = δ + σ, and moves to larger values of u (which corresponds to smaller values of z). We can do these integrals and focus on the potentially large terms at small δ, σ. We obtain

d 1 ∆x = − log σ , (3.91) −r 2d d 1 δ √δ2 σ2 ∆t = − log − − , (3.92) − d σ Aren = ∆x . (3.93)

The last equation implies that the entropy flux is what we expected. We take the ansatz σ γδa with a > 1, where γ is some constant. The expansions become, for small δ, ∼ d 1 ∆x = − a log δ , (3.94) −r 2d d 1 ∆t = − (a 1) log δ . (3.95) − d − Setting ∆x = ∆t for the null solution, we find 1 a = . (3.96) 1 d − 2(d−1) q This means that thermal solutions with exact ∆x− = 0 exist for large ∆x+. However, we are interested not in the renormalized area but in the area difference with respect to the vacuum solution. Using the area for the vacuum solution [139], the area difference for large ∆x+ is

d−1 + d−1 (d−1)/2 d ∆x 2 π Γ( 2(d−1) ) 1 ∆A + d−2 . (3.97) 1 + − ' 2 d 2 Γ( 2(d−1) )! ( ∆x ∆x ) 2 − − Phase transition for large ∆x+ Comparing equations (3.86) and (3.97), wee see that the perturbative solution has less area than the thermal one for sufficiently small ∆x−. This occurs because the perturbative solution has the same negative and finite term as the vacuum solution which grows like − − d−2 (∆x ) 2 . Hence this term does not appear in the area difference in equation (3.86). The thermal solution cannot have this term because it is an exact solution valid for ∆x− = 0; its area cannot depend on ∆x−. Therefore, the area difference, equation (3.97), diverges CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 49 as (∆x−) 0 for the thermal class of solutions. However, for finite values of ∆x− and sufficiently→ large values of ∆x+, the thermal solution must have smaller area, since it increases only linearly with ∆x+ while the perturbative solutions grow quadratically. The phase transition occurs when the area of the two solutions becomes equal, which is approximately given by

3d−1 d+1 d+4 d/2 Γ Γ( d ) + d+2 − d−2 2 π (d 1) 2(d−1) 2(d−1) (∆x ) 2 ( ∆x ) 2 = − . (3.98) − d 2 Γ d Γ( 1 ) − d−1 " 2(d−1) #

Thermal solution "x# $ 20

A 20 " Perturbative solution "x# $ 20 ! " Unique solution "x# $ 10 10 ! "

! " 0 !4 10!5 10 0.001 0.01 0.1 1 !"x!

Figure 3.A.3: The vacuum-subtracted extremal surface area versus ∆x− for fixed ∆x+ (∆x+ = 20 and ∆x+ = 10 for d = 3 is shown). This numerical simulation demonstrates that, for sufficiently large ∆x+ (in d = 3, the condition is ∆x+ & 15), there exists a phase transition at finite ∆x− to a different, perturbative class of solutions. At smaller ∆x+, there is no such phase transition.

We have numerically evaluated ∆A as a function of ∆x− for fixed ∆x+ in d = 3 dimensions. The result is shown in figure 3.A.3. We observe that, as predicted, the thermal solution tends toward infinite area as we take the limit ∆x− 0. One of the perturbative solutions becomes the minimal area solution for ∆x+ = 20 at→ some finite ∆x−. In every case, the minimal area plateaus to a finite, non-zero value as ∆x− 0. → 3.B Toy Model with ∆ K = ∆S = 0 h i 6 In this appendix, we present a toy model with a countable number of degrees of freedom (qubits), which shares the property we found for interacting theories on a null slab: ∆ K = ∆S = 0. This relation is only possible as a limiting statement, because zero relative entropyh i 6 CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 50 between two states ρ1 and ρ0 implies that the states are equal. Moreover, the relation requires an infinite number of degrees of freedom, or else it would be reached before the limit is taken, in contradiction with the previous sentence. To demonstrate the effect in a toy model, we construct a decreasing sequence of algebras An from which any fixed bounded operator will disappear as n ; in other words, An only contains multiples of the identity operator. Consider an infinite→ ∞ sequence of qubits.∩ The algebra generated by the qubits operators for the qubits at position n, n + 1, ... will be denoted by A . The algebras are nested: A A for m > n. The relative entropy of two n m ⊂ n states reduced to An will decrease with n; that is, ∆ K ∆S 0. Consider states which are formed by tensor productsh i of − two-qubit→ states for the kth and (k2)th qubits. This choice for the entanglement is arbitrary, but entanglement between the qubits k and f(k) with f(k) growing much faster than k is necessary to generate more entanglement than the entropy that is lost as we trace over the first n qubits. Entanglement plays an important role in keeping ∆S finite while the relative entropy goes to zero. The classical entropy is monotonous, so without quantum entanglement, the entropies must tend to zero with increasing n. In the quantum case, the entropy is no longer monotonous, but the relative entropy is monotonous and tends to zero instead. Consider generic states of the form

ρ = ρi,i2 . (3.99) i iO=6 k2

In this tensor product, we omit i if i is already included in the product by a previous factor of 2 ρk,i with k = i. The global relative entropy of two states both of the form in equation (3.99) is 1 0 1 0 S(ρ ρ ) = S(ρ 2 ρ 2 ) . (3.100) || i,i || i,i i X We want a finite relative entropy, so a convergent series. We construct a sequence of mixed states ρn by tracing over the first n 1 qubits of ρ. As n tends to infinity, the relative entropy approaches zero: −

∞ 1 0 1 0 0 S(ρn ρn) < S(ρk,k2 ρk,k2 ) 0 , (3.101) ≤ || √ || → kX= n where we have used that the positivity of the relative entropy, and the fact that the relative entropy of two states on a pair of qubits is greater than that of the states reduced to the second qubit of the pair. We want the global ∆S to remain ﬁnite as n goes to inﬁnity:

∆S = ∆S(k, k2) < . (3.102) ∞ Xk CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 51

For the sequence of entropies ∆Sn, we have

∞ n 2 2 ∆Sn = ∆S(k, k ) + ∆Sred(k, k ) . (3.103) √ Xk=n kX= n

Here, ∆Sred denotes the entropy of the reduced states on the second qubit of the pairs. The pairs of qubits with k < √n have been completely traced out, while the pairs with k > n are still completely included in the state. Using equation (3.102), we see that the first sum in equation (3.103) tends to zero as n . For the second sum to have a finite and positive limit, we demand → ∞ c ∆S (k, k2) , (3.104) red ∼ k log k which gives n 2 ∆Sred(k, k ) c(log log n log log √n) = c log 2 . (3.105) √ ∼ − kX= n 2 If ∆Sred(k, k ) decays much faster, we get lim ∆Sn = 0, which is not what we want. To get 2 a non-zero answer, the entropy of the pairs ∆Sred(k, k ) must not be integrable. If it decays at a slower asymptotic rate than equation (3.104), the limiting value of ∆Sn is infinity. 0 1 Now we choose the two qubit states ρk,k2 and ρk,k2 . We impose three conditions: the relative entropies of these pairs should be integrable, equation (3.101); the differential entropies ∆S of these pairs should also be integrable, equation (3.102); and the ∆Sred of the states on the second qubit should have the asymptotic form in equation (3.104), or slower than this if we want to obtain lim ∆Sn . We choose the pair of states→ ∞ to be

ρ0 = p ψ ψ + (1 p) φ φ , (3.106) | ih | − | ih | ρ1 = p0 ψ ψ + (1 p0) φ φ , (3.107) | ih | − | ih | with φ , ψ a pair of orthogonal pure states for the two qubits. (We choose mixed states because| i the| i relative entropy diverges for any two non-identical pure states.) Taking δp p0 p to be small, we ﬁnd ≡ − δp2 S(ρ1 ρ0) , (3.108) || ' 2p(1 p) − 1 p ∆S δp log − . (3.109) ' p Here, p and δp depend on the pair of qubits (k, k2), and the dependence on k is such that these entropies are integrable. To evaluate the reduced entropy, we have to specify the pure states in terms of the qubits. The choice is arbitrary, but there are some restrictions. We cannot choose two orthogonal CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELD THEORY 52 maximally entangled states for ψ and φ , because in this case, the reduced density matrices 0 1 1 | i | i ρ and ρ will both equal 2 I and we obtain ∆Sred = 0. Instead, we take the two orthogonal states

ψ = a 00 + √1 a2 11 , (3.110) | i | i − | i φ = b 01 + √1 b2 10 . (3.111) | i | i − | i Then the entropy is

∆S 2(a b)(a + b) arctanh(1 2b2(1 p) 2a2p) δp . (3.112) red ' − − − − We can tune the dependence of p, δp, a, b on k so that the entropy goes as equation (3.104) and both equation (3.108) and equation (3.109) are integrable. We fix a and b and take δp 1/(k log k). The relative entropy is integrable because it contains a higher power of δp. For' the total ∆S to be finite, we can choose p 1/2+1/k, to get an additional power of 1/k from the logarithm term in equation (3.109). Then' the states converge to a random state in the sub-Hilbert space spanned by φ , ψ . With this choice, both the total ∆S and the relative entropy are finite; the relative{| i entropy| i} goes to zero with n, while the limit of ∆S is

2 2 lim ∆Sn 2(a b)(a + b) arctanh(1 b a ) log(2) . (3.113) n→∞ → − − −

It is also clear that we have ∆ K n ∆Sn 0 in the limit, or equivalently the relative entropy goes to zero. h i − → The limit for ∆Sn can be made much larger (or inﬁnite) by slowing the asymptotic decay of δp. For example, keeping a, b and p as before but setting δp = 1/k causes ∆Sn to diverge with ﬁnite initial ∆S, while the relative entropy remains constant. 53

Chapter 4

The Quantum Focussing Conjecture

In section 1.1 of this thesis, we discussed several aspects of the holographic principle. These ideas are motivated from black hole thermodynamics, which suggests a bound on the amount of information carried by matter in finite regions of space. Some of these bounds do not need to explicitly refer to black holes. The covariant entropy bound (Bousso bound), which was the focus of chapters2 and3, relates matter entropy to the area of arbitrary surfaces, not just black hole horizons [25]. In the generalized form of [71] the bound states that ∆A ∆S , (4.1) ≤ 4GN ~ where ∆S is the matter entropy passing through a nonexpanding null hypersurface bounded by surfaces whose areas differ by ∆A. More details are given in section 4.1; for a full review, see [31]. The Bousso bound reduces to previous heuristic bounds in well-circumscribed settings [25, 71, 30, 38]. But its validity extends to cosmological spacetimes and regions deep inside a black hole. Since such regions cannot be converted to black holes of the same area, the bound hints at a much broader relation between quantum information and geometry that goes beyond black hole thermodynamics. In a similar vein, we will argue here that a generalized entropy should be ascribed not only to black hole and other causal horizons [107], but to a much larger class of surfaces [165, 17, 129, 69, 65]. The definition of generalized entropy, A Sgen = + Sout . (4.2) 4GN ~ allows us to assign a generalized entropy to any surface σ that divides a Cauchy surface into two portions, where Sout can be taken to be the matter entropy on either one of these portions. We will find that this viewpoint leads to a statement more powerful than the Bousso bound: the quantum focussing conjecture. The generalized entropy is a promising and versatile notion because it is finite. Newton’s constant GN and the exterior entropy are separately cutoff-dependent. But over the past CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 54 decades, evidence has mounted1 that the combined divergences cancel, leaving a finite piece that is invariant under RG flow, as originally proposed in [153] and expanded upon in [104, 77]). What we call gravitational entropy and matter entropy depends on the cutoff; but their sum, Sgen, does not. This suggests that Sgen, unlike Sout or A/4GN ~ separately, reflects some information present in the full quantum gravity theory. Presumably, Sgen is a measure of the entropy of the degrees of freedom accessible on one side of that surface, where the area term represents the dominant contribution coming from Planckian degrees of freedom very close to σ, somehow cut off by quantum gravity [148, 78,6, 153, 104, 77]. However, the generalized entropy is only a semiclassical concept, assigning to each surface an entropy proportional to its area, without worrying about whether these degrees of freedom are entangled with each other, or with other systems. Thus it does not capture nonperturbative physics such as the (presumed) unitarity of Hawking evaporation. For example, the GSL does not hold (except in a coarse-grained sense) after the Page time [134], when more than half the entropy has radiated out of the black hole, so that the hidden purity becomes potentially measurable. At this stage, both the fine-grained entropy of the exterior and the area of the horizon decrease, and the correct statement of the second law becomes different. Similarly, the quantum focussing conjecture is a semiclassical statement that may need to be modified in the nonperturbative regime, e.g., for sufficiently old black holes that have information on the horizon. In this article we will confine our analysis to situations where the semiclassical analysis is valid. At a practical level, extending the notion of generalized entropy to arbitrary surfaces yields powerful extensions of classical GR results to the semiclassical level (much as the GSL supersedes the classical area theorem for causal horizons). For example, Penrose’s singularity theorem for trapped surfaces [137] fails for evaporating black holes because it cannot accommodate quantum fluctuations with negative energy. But a more robust theorem guarantees singularities in the presence of quantum trapped surfaces, which are defined in terms of the generalized entropy [165]. Here we will use Sgen to formulate an extension of the classical focussing theorem for surface-orthogonal null congruences. The classical theorem states that light rays never “antifocus” as long as matter has positive energy. Mathematically, the expansion scalar θ cannot increase along a congruence of light rays, where θ is the logarithmic derivative of the area spanned by the light rays: dθ d d /dλ = A 0 , (4.3) dλ dλ ≤ A where is an infinitesimal area element spanned by nearby null geodesics, and λ is an affine parameter.A We review this result in section 4.1. Because quantum fluctuations can have negative energy, the classical focussing theorem fails at the semiclassical level (e.g., near black hole horizons), just as the area theorem and 1For example, the leading divergence of the vacuum von Neumann entropy near the surface σ scales like 2 A/ but 1/GN is renormalized so as to absorb this divergence. Subleading divergences of the entanglement are similarly canceled by appropriate higher-curvature corrections to the gravitational action. We give a review of these results along with extensive references in appendix 4.A. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 55 the singularity theorem fail. In section 4.2, we define a quantum expansion, Θ, as a functional derivative (per unit area) of the generalized entropy along a null congruence orthogonal to the surface σ. We conjecture that Θ cannot increase along any congruence, even in quantum states that would violate the classical focussing theorem: dΘ 0 . (4.4) dλ ≤ We call the bound in equation (4.4) the quantum focussing conjecture (QFC). We derive and explore two important implications of the QFC. In section 4.3, we show that the QFC implies the Bousso bound, but in an improved form. The Bousso bound was initially formulated only for the case where the matter entropy is dominated by isolated systems. In this setting, a finite entropy is easily computed from a density operator for the system, or by integrating an entropy density. In more general settings, the matter entropy cannot be cleanly separated from the divergent vacuum von Neumann entropy across the surface σ. Two inequivalent quantum extensions of the bound have been put forward: in the weakly gravitating regime, the entropy can be regulated by vacuum subtraction [94, 123, 51, 38, 37]; in a more general setting, one must include the vacuum entanglement [152]. We recover a “quantum Bousso bound” by integrating the QFC. On a (quantum) light- sheet, the generalized entropy is initially decreasing, so by the integrated QFC it will continue to decrease. Hence, the initial generalized entropy is greater than the final one. This statement is manifestly cutoff-independent, i.e., it is automatically equipped to deal with the divergences of the von Neumann entropy. It is closely related to an early improvement of the Bousso bound by Strominger and Thompson [152]. Breaking Sgen into Sout + A/4GN ~ and rearranging terms, one recovers the Bousso bound in the familiar form of equation (4.1), ∆S ∆A/4GN ~; see figure 4.2a. In≤ section 4.4 we explore the QFC in settings where the classical expansion vanishes. We find that the QFC implies a novel quantum null energy condition,

2 ~ d Sout Tkk lim , (4.5) h i ≥ A→0 2π dλ2 A where Tkk is the null-null component of the stress tensor. The proof of the quantum null energy condition for free and superrenormalizable ﬁelds in Minkowski space is provided in chapter5. This provides signiﬁcant evidence supporting the QFC, beyond the evidence already supporting the Bousso bound [25, 71, 31, 35, 152, 38, 37]. In section 4.5, we discuss how the QFC relates to other proposals and results, including the GSL, the quantum Bousso bounds of [152] and of chapters2 and3[38, 37], the quantum singularity theorem of [165], the quantum extremal surface barriers of [65], and a novel GSL for quantum holographic screens [32, 34, 33]. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 56

4.1 Classical Focussing and Bousso Bound

In this section we review the classical notion of the expansion of a null congruence, and two statements that involve this expansion: the classical focussing theorem, and the Bousso bound. Both will later be subsumed by the quantum focussing conjecture.

Classical expansion Consider a congruence of light rays emanating orthogonally from a codimension-2 space-like hypersurface. The expansion scalar θ is defined as the trace of the null extrinsic curvature [162] θ ka . (4.6) ≡ ∇a Here ka = (d/dλ)a is the (null) tangent vector to the congruence, normalized with respect to an affine parameter λ. Equivalently, θ is the logarithmic derivative of the area element spanned by infinitesimally neighboring geodesics: A 1 d θ = lim A . (4.7) A→0 dλ A From its definition, it is clear that θ is a local quantity. A caustic (conjugate point, focal point) is a point where θ , which happens when the cross-sectional area vanishes, i.e., when infinitesimally neighboring→ −∞ geodesics intersect. The evolution of the expansion θ along the congruence is determined by the Raychaudhuri equation: dθ 1 = θ2 σ σab R kakb , (4.8) dλ −D 2 − ab − ab − where Rab is the Ricci tensor and D is the spacetime dimension. The shear σab is defined as the tracefree symmetric part of the null extrinsic curvature [162].

Classical focussing theorem

a b In a spacetime which satisﬁes the null curvature condition, namely Rabk k 0 for all null vectors ka, each term on the right-hand side of equation (4.8) is manifestly≥ nonpositive. Physically, this means that light rays can focus but never antifocus, and it implies the following theorem:2 In a spacetime satisfying the null curvature condition, the expansion is nonincreasing at all regular (noncaustic) points of a surface-orthogonal null congruence: dθ 0 . (4.9) dλ ≤ 2There exists another version of the focussing theorem, which also follows from equation (4.8) and the null curvature condition: if θ(p) is strictly negative at some point p on a null geodesic, then there will be a caustic on the geodesic, at aﬃne parameter no further than (D 2)/θ(p) from the point p. We will not consider this theorem here, because its quantum generalization| is− not yet known.| CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 57

In Einstein gravity, the null curvature condition is equivalent to the null energy condition, a b a that the stress tensor obeys Tab k k 0 for all null vectors k . This inequality is broadly obeyed in the classical limit,h andi by coherent≥ states of a quantum field. However, the null energy condition is not universally valid. It is violated by physically reasonable states in the quantum field theory [66], for example by the Casimir effect [49], moving mirrors [55, 56], squeezed states of light [40, 127], and Hawking radiation [88, 57]. In any region where it is violated, one can construct a counterexample to the above focussing theorem, by choosing a congruence with sufficiently small θ and σab.

Bousso bound A light-sheet is a null hypersurface with everywhere nonpositive expansion θ 0. The Bousso bound [25, 29] is the conjecture that the entropy on a light-sheet cannot≤ exceed the area of its initial cross-section in Planck units, A/4GN ~. The bound can be strengthened [71] in the case where the light-sheet is truncated at some nonzero ﬁnal area A0:

A A0 S − , (4.10) ≤ 4GN ~ The Bousso bound is useful in regimes where the quantities it relates are well defined. In the semiclassical regime, the areas of surfaces are sharply defined. In many situations, the entropy is also easy to compute, for example when dealing with well-isolated matter systems, or with a portion of an extensive system large enough for the notion of entropy density to be meaningful. In the semiclassical regime, the areas of surfaces are sharply defined. In many situations the entropy is also easy to compute, for example when dealing with well-isolated matter systems, or with a portion of an extensive system large enough for the notion of entropy density to be meaningful. Within this wide arena, there exist strong counterexamples to all alternative proposals (so far) of the general form S . A/4GN ~ [110]. The Bousso bound evades these counterexamples because of the special properties of light-sheets. Thus, the notion of light-sheets (rather than spatial volumes, or light cones lacking a non-expansion condition [70]), appears to be crucial. It would be nice to broaden the regime for which the entropy S is well defined. A clarification is particularly necessary in the case where the Bousso bound is applied to a system consisting of only a few quanta, such as a single photon wave packet with Gaussian profile. Globally, the entropy will be of order unity (assuming, for example, an incoherent superposition of different polarization states). However, it is not obvious how to define the entropy on a finite light-sheet: some tail of the wavepacket will be missing, so one cannot use the global density matrix. Restricting the density operator to the finite light-sheet, the von Neumann entropy receives a divergent contribution from entanglement across the boundaries at the two surfaces of areas A and A0. This contribution dominates but it is intuitively unrelated to the photon. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 58

A sharp definition of S was given recently for light-sheets in the weak gravity limit, GN ~ 0, with perturbative matter. In this regime one can restrict both the vacuum and the state→ of interest to the same region or light-sheet. In this setting, the entropy S can be defined as the difference of the two resulting von Neumann entropies of the light-sheet states [94, 123, 51]. Because the divergences of the von Neumann entropy are associated with its boundary, this quantity is finite and reduces to the expected entropy for isolated systems and fluids.3 With this definition, the bound was proven [38, 37] to hold in the weak gravity limit in chapters2 and3 of this thesis. However, this definition cannot be applied when gravity is strong, since it is not clear what one would mean by the “same” light-sheet for two states with different geometry. It is therefore necessary to find some other definition of entropy such that a Bousso bound can be precisely formulated (and perhaps proven). We will show below that the quantum focussing conjecture furnishes such a definition. Interestingly, we will find that this definition does not reduce to that of [38, 37] in the weak- gravity limit, where the latter is well defined. Moreover, we will find that the non-expansion condition θ 0, which was strictly preserved in [38, 37], will be modified to a “quantum non- expansion condition”.≤ The resulting conjecture is similar to that of [152]; we will comment on the differences in section 4.5.

4.2 Quantum Expansion and Focussing Conjecture

In this section, we deﬁne the notion of quantum expansion as a functional derivative of the generalized entropy, and we formulate the quantum focussing conjecture.

Generalized entropy for Cauchy-splitting surfaces Generalized entropy was originally deﬁned [9] in asymptotically ﬂat space, as the area A of all black hole horizons (in Planck units), plus the entropy of matter systems outside the black holes: A Sgen Sout + + counterterms . (4.11) ≡ 4GN ~

A rigorous deﬁnition of Sout can be given as the von Neumann entropy of the quantum state of the exterior of the horizon:

S = tr ρ log ρ . (4.12) out − out out

The reduced density matrix ρout is obtained from the global quantum state ρ by tracing out the ﬁeld degrees of freedom behind the horizon:

ρout = trin ρ . (4.13) 3In the interacting case, it reduces to an upper bound on the naive entropy, which suﬃces. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 59

(If the global state of the matter fields is pure, then Sout = Sin, where Sin is the von Neumann entropy of the interior region.) The von Neumann entropy Sout is UV-divergent. However, there is now strong evidence that this divergence is precisely canceled by a renormalization of Newton’s constant in the area term. Subleading divergences are canceled by other geometric counterterms. This is discussed in detail in appendix 4.A; here we assume that the generalized entropy is indeed finite and cutoff-independent. The generalized entropy was introduced in order to salvage the second law of thermodynamics when matter entropy is lost into a black hole. Bekenstein conjectured that a generalized second law [9] (GSL) survives: the area increase of the black hole horizon will compensate or overcompensate for the lost matter entropy, so that the generalized entropy will not decrease. The GSL does appear to hold for realistic matter entering a black hole, and it has been proven in certain settings (for recent proofs see [164, 141, 12, 166], and for a review of previous proofs, see [167] and references therein). The GSL supersedes not only the ordinary second law, but also Hawking’s area theorem. When a black hole evaporates [88], the null energy condition is violated, and the area decreases. However, the emitted radiation more than compensates for this decrease [135]. We now follow references [165, 129, 65] and extend the notion of generalized entropy beyond the context of causal horizons. Let σ be a space-like codimension-2 surface σ that splits a Cauchy surface Σ into two portions. The surface σ need not be connected; for example, it may be the union of several black hole horizons, or of two concentric topological spheres. Nor does it need to be compact; for example, it could be a cross-section of a Rindler horizon. We pick one of the two sides of σ arbitrarily and refer to it as Σout; see figure 4.1a. We use equations (4.11-4.13) to define a generalized entropy. This viewpoint has two important consequences. Suppose we are given any theorem or conjecture about the area of surfaces (such as Hawking’s area theorem), which is valid classically, but which can fail when the null curvature condition is violated. By a judicious application of the substitution A 4GN ~ Sgen , (4.14) → we may obtain a semiclassical statement of much broader validity (such as the GSL). In- deed, the notion of generalized entropy of nonhorizon surface has been profitably applied to Penrose’s singularity theorem [165], to the Ryu-Takayanagi proposal [69, 65], and to a novel area law for holographic screens [32, 34, 33]. Below, we will apply equation (4.14) to the classical focussing theorem, equation (4.9), to obtain a quantum focussing conjecture. Second, the quantity Sgen provides a cutoff-independent measure of entropy in a bounded region, because the geometric terms cancel the divergences of the von Neumann entropy. Unlike vacuum subtraction, this feature does not rely on a weak-gravity limit. We will exploit this to formulate a quantum Bousso bound in terms of the generalized entropy. In fact, we find that an appropriate bound arises simply as a special case of our quantum focussing conjecture, which we will now formulate. Prepared for submission to JHEP

A Quantum Focussing Conjecture

CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 60 Raphael Bousso,a,b Zachary Fisher,a,b Stefan Leichenauer,a,b and Aron C. Wallc aCenter for Theoretical Physics and Department of Physics, λ University of California, Berkeley, CA 94720, U.S.A. bLawrence Berkeley National Laboratory, Berkeley, CA 94720, U.S.A. ! N c Institute for Advanced Study,σΣ = Σ Princeton,∩ N NJ 08540, USA Σ σ • !out A Abstract: Given a surface that need not lie on a horizon, we define its generalized A Figure(a) 1: XXX LET’Sentropy HAVES Agen LEFTas(b) the FIGURE area ofSHOWING in Planck THE units, SIDEWAYS plus theVIEW, von Neumann entropy of its WITH N AT 45 DEGREESexterior. AND Given⌃, ⇢ aout null, A, congruence ALL LABELED. orthogonal ALSO, to CAN,therateofchangeof WE S per Figure 4.1: (a) A spatial surface σ of area A splits a Cauchy surface Σ into two parts. The gen LABEL IN THE RIGHTunit area FIGURE? defines (a) a Aquantum spatial surface expansion of area. WeA conjecturesplits a Cauchy that the quantum expansion generalized entropy is defined by SgenA = Sout + A/4GN ~, where Sout is the von Neumann entropy of the quantum statesurface on one⌃ into side two of σ parts..cannot To The define increase. generalized the quantum This entropy generalizes expansion is defined Θ the at by classicalSgen = S focussingout + A/4 propertyG~, of light-rays to cases where S is the von Neumann entropy of the quantum state on one side of .To σ, we erect an orthogonal null hypersurfaceout N,where and we quantum considermatter the response may ofviolateSgen to the null energy condition. The conjecture, if true, define the quantum expansion ⇥ at , we erect an orthogonal null hypersurface N, deformations of σ along N. (b) More precisely, Nhascan significant be divided implications. into pencils of widthIntegration yields a precise version of the Strominger- around its null generators; theand surface we considerσ is deformed the response an affine of parameterto deformations length ofalong along oneA N. (b) More precisely, ThompsongenQuantum Bousso Bound, with the entropy of quantum fields regulated by of the generators, shown inN green.can be divided into pencils of width around its null generators; the surface is A deformed an ane parameterthe geometric length ✏ terms.along one Applied of the togenerators, locally parallel shown in light-rays, green. the conjecture implies a Quantum expansion Quantum Null Energy Condition: a lower bound on the stress tensor in terms of the second derivative of the von Neumann entropy. We sketch a proof of this relation. As a first step, let us use thefour substitution null hypersurfaces (4.14) to define orthogonal a quantum to expansion, generated. We by now orthogonal need light-rays towards an additional structure: thethe null past geodesics or future orthogonal and towards to σ that⌃ defineL or ⌃ theR (regardless classical expansion. of which one is chosen as ⌃out). In addition to the twofold choiceThe chosen of Σout hypersurface, this faces usN withwill be an terminated additional, by fourfold caustics, choice: or more generally wherever there are four null hypersurfacesnull generators orthogonal orthogonal to σ. They to are intersect. generated Thus,by orthogonalN consists light of one component of the rays toward the past or futureboundary and toward of the Σ past,L or Σ orR of(regardless the future, of of which [1]. one is chosen as Σ ). Again, we may pick any direction, e.g., the one shown in figure 4.1a. (All of the out Through each point y of there passes one generator of N.Wetake to be an statements below will hold under any of the eight possible choices; in particular, the QFC will be conjectured to hold ata thisne parameter broad level.) along this generator, such that =0on and increases away from The chosen hypersurface.N Thiswill defines be terminated a coordinate by caustics, system or ( more,y)on generallyN. wherever null generators orthogonal to σ intersect.A positive Thus, definiteN consists function of oneV (y component) 0definesasliceof of the boundaryN, consisting of the point of the past, or of the future,on of eachσ [162 generator]. y for which = V . Any such slice of N splits a Cauchy surface into Through each point y oftwoσ there parts. passes Hence oneV ( generatory)istheargumentofageneralizedentropyfunctional of N; see figure 4.1b. We take λ to be an affine parameter along this generator, such that λ = 0 on σ and λ increases away from σ. This defines a coordinate system (λ, y) on N. A[V (y)] Sgen[V (y)] = + Sout[V (y)] . (3.3) A positive definite function V (y) 0 defines a slice of N, consisting4G of~ the point on each generator y for which λ = V . Any such≥ slice of N splits a Cauchy surface into two parts. Hence V (y) is the argument of a generalized entropy functional The quantum expansion, like the classical expansion, is defined by deforming a

slice in the neighborhoodA[V (y)] of one generator y1. To be precise, consider a second slice of Sgen[V (y)] = + Sout[V (y)] . (4.15) N which di↵ers4G fromN ~ only in a neighborhood of generators near y1, with inﬁnitesimal area : The quantum expansion, likeA the classical expansion, is deﬁned by deforming a slice in V✏(y) V (y)+✏#(y1) . (3.4) the neighborhood of one generator y1. To be precise, consider a⌘ second slice of N which

–6– CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 61 diﬀers from σ only in a neighborhood of generators near y , with inﬁnitesimal area : 1 A V (y) V (y) + ϑ (y) . (4.16) ≡ y1

Here we define ϑy1 = 1 in a neighborhood of area around a point y1, and ϑy1 = 0 everywhere else; see figure 4.1b. One can differentiateA the generalized entropy with respect to this localized deformation: dS S [V (y)] S [V (y)] gen lim gen − gen . (4.17) d ≡ →0 y1

The quantum expansion is the ﬁnite quantity obtained by dividing this derivative by the inﬁnitesimal unit area , just as in the classical case in equation (4.7): A

4GN ~ dSgen Θ[V (y); y1] lim . (4.18) ≡ A→0 d y1 A

The above construction is equivalent to defining Θ as the functional derivative of Sgen with respect to V (y): 4GN ~ δSgen Θ[V (y); y1] , (4.19) V ≡ g(y1) δV (y1) where Vg is the (finite) area element ofp the metric restricted to σ, inserted to ensure that the functional derivative is taken per unit geometrical area, not coordinate area. The p notation Θ[V (y); y1] emphasizes that the quantum expansion requires the specification of a slice V (y) and is a function of the coordinate y1 on that slice. The classical expansion θ depends only on the infinitesimal neighborhood of a null generator. By contrast, the quantum expansion Θ depends nonlocally on the quantum state of matter on the half-Cauchy-surface Σout, because the von Neumann entropy of the matter can behave differently at y1 if one changes the state of matter elsewhere on Σout. Moreover, the quantum expansion at y1, Θ[V (y), y1], depends on the choice of V (y) away from y1. However, Θ does not depend on the choice of the half-Cauchy-surface attached to the spatial slice V (y) of N, since all Cauchy surfaces are unitarily equivalent. This freedom makes it possible to find a suitable Σout for any deformation V (y); note that portions of Σout may coincide with N without violating the achronality condition on Cauchy surfaces. In the classical limit GN ~ 0 with the classical geometry held fixed, the matter entropy → Sout does not contribute, and Θ reduces to the local geometric expansion θ at each generator y1.

Quantum focussing conjecture The deﬁnition of a quantum expansion allows us to formulate a generalization of the classical focussing theorem, equation (4.9), to the semiclassical regime. The quantum focussing CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 62 conjecture (QFC) is the statement that δ Θ[V (y); y1] 0 . (4.20) δV (y2) ≤

In words, the quantum expansion cannot increase at y1, if the slice of N defined by V (y) is infinitesimally deformed along the generator y2 of N, in the same direction. Here y2 can be taken to be either the same or different from y1. The QFC is nonlocal: as noted above, the generalized entropy depends on all of σ, and so do its first and second functional derivatives. Even the sign of Θ[V (y)] at some point may depend on the choice of V away from this point. We defined the QFC so that it applies regardless of which side of N we choose to compute the generalized entropy and its derivatives. In principle one could distinguish the two sides, since N moves away from one side and toward the other. Thus, one could attempt to formulate a weaker conjecture that applies only to one side. However a sensible conjecture should be time-reversal invariant [25]. Under time-reversal, the putative weaker conjecture would require N to move towards the opposite spatial side. But the left-hand side of equation (4.20) is the same as if we had chosen that spatial side as the exterior with the original time direction, since Θ involves an even number of derivatives. This suggests that if there are any counterexamples to the QFC, then there will be counterexamples to a weaker conjecture that restricts attention to one side of N. In the next two sections, we will provide evidence for the validity of the QFC. We will show that the QFC implies a Quantum Bousso Bound, for which there is already considerable evidence [25, 71, 31, 35, 152, 38, 37]. We will also show that the QFC implies a previously unknown property of nongravitational theories, the quantum null energy condition, and we sketch a proof of this property.

4.3 Quantum Bousso Bound

In this section, we will show that the QFC implies a quantum Bousso bound. For this purpose we will consider finite variations away from an initial surface σ along the null hypersurface N to some final surface σ0. We will take the surface σ to correspond to V (y) 0; σ0 is defined by a choice of V (y) 0 described below. ≡ ≥ The QFC implies a quantum Bousso bound

Suppose that the quantum expansion at the generator y1 is nonpositive (negative) on σ. Then by integrating the QFC, we ﬁnd that Θ will be nonpositive (negative) at y1 at all later times: for any slice deﬁned by a function V (y), we have

Θ[0, y ] 0 ,V (y) 0 = Θ[V (y), y ] 0 , (4.21) 1 ≤ ≥ ⇒ 1 ≤ where if the ﬁrst inequality is strict, so is the second. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 63

Let us further specialize to the case where a later slice σ0 deﬁned by V (y) diﬀers from σ only on generators along which the generalized entropy is initially decreasing:

V (y) 0 if Θ[0, y] 0 ≥ ≤ V (y) = 0 if Θ[0, y] > 0 . (4.22)

Equation (4.21) implies that the generalized entropy decreases on these same generators on every intermediate slice αV (y), 0 α 1.4 Since the slice is deformed only along these generators, it follows that the generalized≤ ≤ entropy must be less on σ0 than on σ:

S [V (y)] S [0]. (4.23) gen ≤ gen To see that this implication is related to the Bousso bound, let us write out the result using equation (4.11): 0 0 A[σ ] A[σ] Sout[σ ] + Sout[σ] + , (4.24) 4GN ~ ≤ 4GN ~ where we have left other counterterms implicit. Rearranging terms, we ﬁnd

0 0 A[σ] A[σ ] Sout[σ ] Sout[σ] − . (4.25) − ≤ 4GN ~ Thus we recover the Bousso bound, equation (4.10), if we identify

A[σ] A, (4.26) ≡ A[σ0] A0 , (4.27) ≡ S [σ0] S [σ] S. (4.28) out − out ≡ However, it is important to note that the terms on the left and right hand side of equation (4.25) are separately cut-off dependent. Thus it is significant that the QFC yields the Bousso bound in the form of equations (4.23) and (4.24), which are well defined independently of a cutoff. The result is goes beyond the original Bousso bound, equation (4.10), not only in that it clarifies how the matter entropy should be regulated for systems that are not well isolated, but more broadly in how ∆S should be defined.5 equation (4.28) implies that the entropy cannot be determined from data on the null surface N between σ and σ0 alone. Instead, equation (4.28) instructs us to consider the von Neumann entropy on half-Cauchy-surfaces 4The argument does not depend on how we interpolate between the initial and final slice, as long as the sequence of deformations is monotonic in the affine parameter. For example, we could begin by deforming σ along one generator all the way, then along some other generator, etc., until the surface has been moved a distance V (y) along each generator y. 5However, in the weak gravity limit there exists an alternative, inequivalent regulator and definition of ∆S. The corresponding quantum version of the Bousso bound was formulated and proven in [38, 37]; see section 4.5. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 64

σ' σ'

Σ' out Σ' out N N matter Σout Σout σ σ matter aux

(a) (b)

Figure 4.2: (a) For an unentangled isolated matter system localized to N, the quantum Bousso bound reduces to the original bound. (b) With the opposite choice of “exterior,” one can also recover the original entropy bound, by adding a distant auxiliary system that puriﬁes the state.

bounded by σ and σ0, and compute their difference. Thus, data far from N can affect the entropy. As first noted by Strominger and Thompson [152], the contributions of distant entropy are helpful in extending the validity of the Bousso bound into a regime where quantum effects on the metric are important, such as the evaporation of a black hole. For example, the Bousso bound in its original form would be violated on the horizon of a quantum black hole whose evaporation is sufficiently nearly balanced by an influx of entropic radiation [121]. Our equation (4.23), like the Strominger-Thompson proposal, evades this violation. Note that the condition of initial quantum non-expansion, equation (4.22), is satisfied on the event horizon only if we consider past-directed light-sheets. Then equation (4.24) is satisfied because the GSL is valid: the Hawking radiation increases the exterior von Neumann entropy by more than it decreases the Bekenstein-Hawking entropy of the black hole [135]. The close relation of our result to the Strominger-Thompson proposal is discussed further in section 4.5.

Recovering the Bousso bound on isolated systems The original Bousso bound is well defined in the hydrodynamic regime, where entropy can be approximated as the integral of an entropy density. More generally, it is well defined in the broad arena where an isolated matter system (e.g., a box of radiation) crosses the light-sheet N between σ and σ0. We assume that the matter system is well-separated from and unentangled with other matter systems, and also that the system is well-localized away from σ and σ0. Recall that we must choose one side of N to define the exterior of the Cauchy surface (Σout), whose renormalized von Neumann entropy is used for computing Sgen. The quantum focussing conjecture is valid regardless of which choice is made. For the purpose of recovering the Bousso bound, we choose, at σ, the spatial side opposite to N (or equivalently, at σ0, the same side as N). This is illustrated in figure 4.2a. We make this choice independently of whether σ lies in the future of σ0 or vice-versa. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 65

With this choice, the exterior of σ0 contains the same degrees of freedom as the union of N with the exterior of σ. Moreover, with the above assumptions on the matter system(s) crossing N, the density operators factorize:

ρ (σ0) = ρ (σ) ρ , (4.29) out out ⊗ N where ρN is the state of the isolated matter system. Hence the von Neumann entropies are additive: S [σ0] = S [σ] tr ρ log ρ . (4.30) out out − N N Thus the vacuum von Neumann entropy can be separated from the “active” matter entropy; the former can be regarded as already included in the geometric counterterms. Then equation (4.25) becomes the original Bousso bound

S tr ρN log ρN ∆A/4GN ~ , (4.31) ≡ − ≤ with the entropy defined intrinsically as that of the isolated matter system(s) crossing the light-sheet N. There is more than one way to recover the Bousso bound for isolated systems. Suppose that in the above setting of an isolated system on the light-sheet N, we make the opposite choice of exterior; see figure 4.2b. This has the effect of exchanging σ and σ0 in equations (4.29) and (4.30). Instead of equation (4.31) we obtain the bound S < ∆A/(4GN ~), − where S = tr ρN log ρN is the entropy of the isolated system. This bound is valid but not very interesting:− it is trivially satisfied for ordinary matter systems, since S > 0 and ∆A > 0 in the classical limit. However, let us now add an auxiliary system to the exterior (far from N) that purifies the mixed state ρN of the matter crossing N. The initial entropy S[σ] receives no contribution from the matter and auxiliary systems because they are both present and in a pure state. But the final entropy S[σ0] is just that of the purification, and hence equal to that of the matter system, S = tr ρN log ρN . Thus, we recover equation (4.31): with the inclusion of a distant purification,− the bound on the isolated matter system is again nontrivial and equivalent to the previous example.

The role of quantum non-expansion As originally formulated, the Bousso bound applies only to light-sheets, i.e., to null hypersurfaces whose classical expansion θ is nonpositive everywhere in the direction from σ to σ0, and which contain no caustics. (It follows that A A0 0.) However, the classical non-expansion assumption played no role in our formulation− of≥ the quantum Bousso bound. In fact, classical non-expansion can be violated in settings where our bound applies. Consider a slice σ of the event horizon of an evaporating black hole. The generalized entropy is decreasing towards the past, so the conjecture applies if σ0 is chosen as an earlier horizon slice. But then A[σ0] > A[σ], so the classical expansion is positive. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 66

Instead, our quantum Bousso bound substitutes quantum expansion for classical expansion: we restricted to deformations along the generators for which the initial quantum expansion at σ is nonpositive. In this sense, equation (4.22) can be taken as the definition of a quantum light-sheet. The QFC then plays an interesting dual role. First, it guarantees that the quantum expansion is nonpositive not only initially, but everywhere on N between the two slices σ and σ0. And second, the resulting nonpositivity of its integral (the difference between the final and initial generalized entropies) becomes the statement of the entropy bound. The classical Bousso bound can formulated in an alternate way [25], more closely analogous to the quantum version we have constructed. Instead of demanding that θ 0 everywhere on a light-sheet, one could have demanded classical non-expansion only initially,≤ but assumed the null energy condition, and added a requirement that light-sheets must be terminated at caustics. The role of the null energy condition would then be to ensure that θ 0 everywhere on the null surface, by the classical focussing theorem. However, the bound≤ itself would still be a separate statement; it does not also follow from the null energy condition.

4.4 Quantum Null Energy Condition

In the previous section, we considered the integrated QFC and showed that it implies a quantum Bousso bound. The considerable evidence for the Bousso bound thus supports the QFC. We now return to the QFC as a constraint on a second functional derivative,

δ 4GN ~ δSgen 0 , (4.32) V δV (y2) g(y1) δV (y1) ≤

and we examine whether there exist otherp limits in which one can ﬁnd explicit evidence or formulate a proof of the conjecture. We begin in section 4.4 with a proof of the QFC in the oﬀ-diagonal case, y1 = y2. In section 4.4 we consider the diagonal part, y1 = y2. We show that it gives rise to an6 interesting limit when the classical null extrinsic curvature vanishes: the quantum null energy condition, a nongravitational implication of the QFC.

General proof of the oﬀ-diagonal QFC

For y1 = y2, the QFC follows from strong subadditivity. Since A[V (y)] is the integral of a 6 V local functional of V (y), and the factor g appearing the QFC is evaluated at y1, it follows that the oﬀ-diagonal second derivative only receives a contribution from S : p out 2 δ 4GN ~ δSgen 4GN ~ δ Sout = (4.33) V V δV (y2) g(y1) δV (y1) g(y1) δV (y1)δV (y2) p p CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 67

V (y) V (y) B B

C A C A

y1 y2 y1 (a) (b)

Figure 4.3: (a) A portion of the null surface N, which we have chosen to coincide with Σout in the vicinity of the diagram. The horizontal line at the bottom is the surface V (y), and the orange and blue lines represent deformations at the transverse locations y1 and y2. The

region above both deformations is the region outside of V1,2 (y) and is shaded beige and

labeled B. The region between V (y) and V1 (y) is labeled A and shaded lighter orange. The

region between V (y) and V2 (y)is labeled C and shaded lighter blue. Strong subadditivity applied to these three regions proves the oﬀ-diagonal QFC. (b) A similar construction for the diagonal part of the QFC. In this case, the sign of the second derivative with respect to the aﬃne parameter is not related to strong subadditivity.

for y = y . The functional derivative can be realized as the following limit: 1 6 2 δ2S Vg(y )Vg(y ) ∆2S out = lim 1 2 lim (4.34) δV (y )δV (y ) A1,A2→0 1,2→0 1 2 p A1A2 1 2 where is an area element located at y , is a deformation parameter for the surface along Ai i i the generator at yi, and

∆2S S [V (y)] S [V (y)] S [V (y)] + S [V (y)] (4.35) ≡ out 1,2 − out 1 − out 2 out To be precise, we deﬁne

V (y) V (y) + ϑ (y) + ϑ (y), (4.36) 1,2 ≡ 1 y1 2 y2 where ϑ = 1 in a neighborhood of area around y . For brevity of notation, when yi Ai i i = 0 we omit it from the subscript of V1,2 (y). The relevant surfaces are depicted in figure 4.3a. It is clear from the figure that the numerator in equation (4.34) is negative by strong subadditivity. Specifically, we split the region outside of V (y) into three subregions.6

6 The entropies Sout[V (y)] are deﬁned in terms of half-Cauchy surfaces Σout, and for the purposes of this discussion we are using the freedom to unitarily deform Σout so that it lies along the null surface N. Thus the subregions A, B, and C which we deﬁne are subregions of N. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 68

The subregion outside of V1,2 (y) will be called region B, so that Sout[V1,2 (y)] = S(B).

The subregion outside of V1 (y) but inside of V1,2 (y) will be called C, so that Sout[V1 (y)] =

S(BC). Subregion A is defined in a similar way so that Sout[V2 (y)] = S(AB). Finally, we have Sout[V (y)] = S(ABC). In this notation, the numerator of equation (4.34) is the standard combination of entropies appearing in the strong subadditivity inequality: S(B) S(BC) S(AB) + S(ABC) 0 . (4.37) − − ≤ This is enough to prove the off-diagonal QFC in general. We also would like to emphasize that the combination of entropies appearing here is finite and cutoff-independent. This has to be the case because the generalized entropy we started with was cutoff-independent, but it is instructive to see this directly at the level of the matter entropy. Sometimes in quantum field theory, entropy inequalities are true because of cutoff-dependent terms, and in the continuum limit reduce to the trivial statement < 0. Here that is not the case. The cutoff-dependent terms in the von Neumann entropy−∞ for a given region are proportional to integrals of geometric quantities along the boundaries of that region. One can check that, for the regions A, B, and C that we have defined, such terms cancel in the combination S(B) S(BC) S(AB) + S(ABC). Our construction here is similar to− the “entanglement− density” of [13, 117], although the QFC is stronger in that it also places a constraint on the diagonal terms with y1 = y2. Strong subadditivity is not helpful in this case: if we attempted to use the same strategy, then (with appropriately modified definitions of the subregions, see figure 4.3b), we would find a combination of entropies S(B) 2S(BC) + S(ABC), which does not have any direct relation with strong subadditivity. Furthermore,− for this combination of entropies the cutoff- dependent terms do not cancel. This is no surprise since the diagonal part of the QFC receives contributions from the area term, and the area term is essential for making Sgen cutoff-independent in general.

Diagonal part of the QFC

The case y1 = y2 corresponds to a deformation of the surface σ along a single null generator orthogonal to it. Specializing to this case, it is convenient to work with ordinary derivatives with respect to the aﬃne parameter along generator y1, denoted by primes. By equation (4.11) we have 4GN ~ 0 Θ = θ + Sout , (4.38) A where θ is the classical expansion.7 The QFC becomes

0 0 4GN ~ 00 0 0 Θ = θ + (S S θ) (4.39) ≥ out − out A 1 4GN ~ 00 0 = θ2 ς2 8πG T + (S S θ) −2 − − h kki out − out 7 A Here and in the remainder of this section we will not explicitly write lim 0 in our expressions, but it should always be understood. A→ CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 69

The derivatives of Sout scale linearly with , matching the scaling of the other terms. Any terms that go like higher powers of willA drop out as 0. The above form shows that the QFCA has several interestingA → limits. The most obvious is the classical limit, ~ 0. In this case, one recovers the null energy condition, Tkk 0, which must hold since→ at any point p one can consider a congruence with tangenth vectori ≥ka, such that the shear ς and the expansion θ both vanish at p.8 For arbitrary congruences it follows that θ0 0. The same special≤ choice of congruence, with θ = ς = 0, becomes more interesting if we do not take ~ 0. In this case, the QFC implies the relation →

~ 00 T S , (4.40) h kki ≥ 2π out A which we shall call the quantum null energy condition (QNEC). Intriguingly, the QNEC does not depend on GN . Thus the QNEC is entirely a statement about quantum field theory. It is the effective quantum replacement for the null energy condition, and unlike the null energy condition it is something that might follow from first principles in quantum field theory. Furthermore, the QNEC is not affected by higher curvature corrections to the gravitational action, at least when the stationary null congruence is also a Killing horizon and the matter is minimally coupled. As discussed in appendix 4.A, such higher-order terms arise due to quantum loop corrections, which add additional terms to the gravitational entropy Sgrav besides the area. These higher-curvature corrections result in modifications to equation (4.38). However, it has been shown [166, 12] that the gravitational equations of motion also change in exactly the right way so that, for linearized metric perturbations to the Killing 00 horizon, Sgrav = 2π Tkk /~. Therefore, the form of equation (4.40) remains the same. We can furtherAh arguei that the QNEC as presented in equation (4.40) is correct without modification for a scalar field nonminimally coupled to the Ricci scalar R. In this case, the theory is equivalent to general relativity after a field definition [105]. Therefore the focussing result still holds. As a first nontrivial check of the QNEC, we observe that it is satisfied by an infinite class of states in any 1+1 CFT, namely those which are conformally related to the vacuum state (or to coherent states in a Gaussian theory). This follows from the anomalous transformation 00 properties of Tkk and Sout under a general conformal transformation, if we note that Sout = 0 9 and Tkk 0 for vacuum/coherent states on a causal horizon [168]. Indeed, it was suggested in [168h ] thati ≥ what we here call the QNEC might hold for more general states and in higher dimensions.

8 When ~ 0, quantum corrections to Tkk proportional to ~ vanish. The null energy condition is → 0 h i recovered for the ~ term in the semiclassical expansion of Tkk . 9In a theory with 1+1 conformal symmetry the QNECh ini fact implies a slightly stronger statement, 2π 6 2 namely Tkk S00 (S0 ) 0, where c is the central charge (and = 1 since in two dimensions ~ out c out σ is a point).h Thati − is because− this quantity≥ transforms as a primary under conformalA transformations, and one can always ﬁnd a conformal frame where Sout0 = 0 [168]. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 70

A proof of the QNEC in the case of free ﬁeld theory and causal horizons is presented in chapter5.

4.5 Relationship to Other Works

In this section, we discuss how our conjecture and its implications are related to older conjectures and results.

Generalized second law for causal horizons The generalized second law (GSL) states that the generalized entropy of a causal horizon is nondecreasing. The QFC can be applied more broadly to any surface that splits a Cauchy surface. But in particular, the QFC can be applied to cross sections of a causal horizon, and it is natural to ask how it relates to the GSL in this setting. A key diﬀerence is that the GSL constrains the sign of the ﬁrst derivative of the generalized entropy, while the QFC constrains the sign of the second derivative. Thus it is clear that the conjectures are not equivalent on causal horizons. However, the conjectures are related. Assuming that the GSL holds at one time, integrating the QFC implies that the GSL holds at all earlier times on the same causal horizon. Moreover, for causal horizons at late times, the classical expansion θ vanishes, and one expects the matter entropy Sout to stop evolving. Thus, one expects that Θ 0 in the asymptotic future for a (future) causal horizon. With this assumption, the QFC→ implies the GSL on the entire causal horizon.

Strominger-Thompson quantum Bousso bound Strominger and Thompson [152] proposed adding the “von Neumann entropy across the surface” to the area of any cross section of a light-sheet, to obtain a quantum Bousso bound. In particular, it was noted that the leading divergences in the von Neumann entropy are canceled by a renormalization of Newton’s constant. The quantum bound we derive in section 4.3 automatically inherits these important features from the QFC. Thus we largely reproduce the Strominger-Thompson proposal as a special case of the QFC. Our formulation of the quantum entropy bound diﬀers in that we consider the generalized entropy as a fundamental object. Hence we do not distinguish between “von Neumann entropy” across σ, and the entropy of other matter outside the surface σ. The latter is treated as a separate contribution in [152], in a hydrodynamic approximation [71, 35]. In general, the distinction between gravitational entropy, von Neumann entropy, and “matter entropy” is ambiguous. By referring only to the generalized entropy, we were able to sidestep this ambiguity. Moreover, the generalized entropy regulates not only the leading (area) divergence of the CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 71 von Neumann entropy, but also subleading divergences such as the logarithmic divergence in 3 + 1 spacetime dimensions. The use of a hydrodynamic approximation also entered into the non-expansion condition that deﬁnes valid light-sheets in the presence of matter [152]. Here this condition is universally given by equation (4.22): a light-sheet is generated by orthogonal light rays with initially nonpositive quantum expansion Θ. It is interesting that in the hydrodynamic limit, the quantum null energy condition, equation (4.40), reduces to one of the assumptions that underly the proof of the quantum Bousso bound, (∂+s+ 2T++ in the notation of [152], a weakened version of an assumption introduced in [35]). ≤

BCFM quantum Bousso bound

vac In the weak gravity limit, GN ~ 0 and GN ( Tµν Tµν ) 0, one can restrict both the vacuum and the state of interest to→ the same regionh ori − light-sheet.→ Then a vacuum-subtracted entropy ∆S can be defined as the difference between the von Neumann entropies of the state and the vacuum [94, 123, 51]. Because the divergences of the von Neumann entropy are associated with its boundary, this quantity is finite and reduces to the expected entropy for isolated systems and fluids.10 With this definition, we proved in chapters2 and3 that a quantum Bousso bound holds on any portion of a light-sheet [38, 37]. When gravitational backreaction of the state is not negligible, the spacetime geometry is very different from that of a vacuum state. Then it is unclear what one would mean by restricting both a general state and the vacuum to the “same” region or light-sheet. In this case, one cannot define a finite entropy by vacuum subtraction. Here, we use a different method to regulate the divergence of the von Neumann entropy of quantum fields in a bounded region: we combined the matter entropy with the gravitational entropy to obtain a cutoff-independent, generalized entropy. This definition requires a semiclassical regime, but not that the gravitational backreaction is small. Therefore, the QFC does not require gravity to be weak; and the associated quantum formulation of the Bousso bound, too, can be stated in settings where gravity is strong. If gravitational backreaction is small, both statements can be applied. This is interesting, because they appear to be inequivalent. In chapters2 and3, the entropy was defined intrinsically on the light-sheet portion of interest, with no reference to distant spatial regions. The generalized entropy, by contrast, generically depends on regions far from the light-sheet. In fact, the light-sheets themselves are defined differently. The bound of chapters2 and3 requires that a light-sheet have nonpositive classical expansion everywhere on L. Our bound requires that the quantum expansion be nonpositive initially on L; the QFC then becomes the statement of the entropy bound. 10In the interacting case and on a light-sheet, it reduces to an upper bound on the naive entropy, which suffices. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 72

Quantum singularity theorem Penrose’s singularity theorem [137] is a seminal result in general relativity. It states that, in a globally hyperbolic spacetime which satisfies the null energy condition, that the presence of certain compact surfaces T on a connected, noncompact Cauchy slice Σ indicates that the spacetime is null geodesically incomplete. The surfaces which signal the impending break- down of the spacetime are trapped surfaces: surfaces for which the congruence of outgoing null light rays have everywhere negative expansion. The proof uses the Raychaudhuri equation to argue that these null geodesics must reach a caustic in finite affine parameter. If the spacetime were null geodesically complete, then each null generator includes its endpoints. Because of the assumption of the null energy condition, Penrose’s theorem is not applicable to quantum matter. Interestingly, there exists a generalization of Penrose’s theorem, where the role of the area is replaced with the generalized entropy [165]. A quantum trapped surface is defined in a globally hyperbolic spacetime as follows. Suppose that on some Cauchy surface Σ, a compact codimension-2 surface exists, and its exterior is noncompact. If N is the null surface generated by outward future-directedT light rays, and if the generalized entropy is decreasing with time with respect to future null deformations, then is called a quantum trapped surface. In the classical limit, the generalized entropy is simplyT the area, so this criteria reduces to the classical notion of a trapped surface. The quantum proof [165] is similar to the classical one. One starts with the assumption of a noncompact Cauchy surface containing a quantum trapped surface. Unlike the classical case which required the null energy condition, one now assumes that the generalized second law holds (i.e., the generalized entropy cannot decrease on causal horizons). The GSL implies (by contradiction) that the null generators reach caustics in finite affine parameter time. From here, the proof is proceeds as in the classical case: the noncompact surface Σ cannot evolve into a compact surface, which implies that the endpoints of the null geodesics do not belong to the spacetime. The QFC was not necessary to complete the proof of the quantum singularity theorem, but it does have interesting consequences for quantum trapped surfaces which makes them more analogous to their classical counterparts. For example, the outgoing null rays from a classical trapped surface define in an obvious way a sequence of additional trapped surfaces on Cauchy slices to the future. This result only becomes valid in the quantum case under the assumption of the weak quantum focussing theorem.

Barriers to quantum extremal surfaces The prescription for holographic von Neumann entropy is now fairly well-understood at leading order in 1/N in the Anti-de Sitter/Conformal Field Theory correspondence [122]. An ample body of evidence supports the proposal [99] of Hubeny, Rangamani, and Takayanagi, which extends an earlier proposal [140] by Ryu and Takayanagi. The new proposal is that, to calculate the von Neumann entropy of a region R of a CFT, we need to find a codimension- 2 surface X such that ∂X = ∂R, and X homologous to R, which extremizes the area CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 73 functional. If there are many such surfaces, we are instructed to pick a surface with the minimum area. The von Neumann entropy of R is then the area of X, in (bulk) Planck units: 2 SR = AX /4GN ~. Since GN ~ 1/N , this conjecture gives the leading order entropy in a 1/N expansion, but as in the black∼ hole case, there will generally be subleading corrections. Recently, the next-to-leading order corrections in 1/N to the von Neumann entropy were calculated. The proposal [69] of Faulkner, Lewkowycz and Maldacena (FLM) is to take the leading order prescription, that is to calculate the area of an extremal surface in the bulk, and to add in the von Neumann entropy of the bulk state restricted to one side of the extremal surface. That is, the proposal is that boundary von Neumann entropy is dual to the generalized entropy of the extremal-area surface: SR = Sgen(X). The FLM proposal passes some nontrivial consistency checks, but is only supposed to provide the next-to-leading order correction in a 1/N expansion. A natural extension of this conjecture is presented in [65]: instead of extremizing the area and solving for the generalized entropy, we find the surface χ which extremizes the generalized entropy subject to ∂χ = ∂R and χ homologous to R. The proposal in [65] is to identify the generalized entropy of χ with the von Neumann entropy of the boundary field theory in the region R: SR = Sgen(χ). These extremal entropy surfaces are called quantum extremal surfaces. While this construction agrees at leading order with the FLM proposal, they differ at higher orders in N. A classical argument in [64] shows that assuming the null energy condition, a null surface shot out from a codimension-2 extremal surface acts as a “barrier” to other extremal surfaces, in the sense that no continuous 1 parameter family of extremal surfaces can be extended across the barrier. This result was extended to quantum extremal surfaces in [65], but the proof required use of the QFC, in order to show that the null surface shot out from the quantum extremal surface becomes quantum trapped. So once again, the focussing conjectures bring the quantum generalizations closer in line with the classical result.

Generalized second law for quantum holographic screens A new classical area law in general relativity was recently formulated and proven [32, 34], assuming the null energy condition. A marginally trapped surface is a compact codimension-2 surface whose classical expansion vanishes in one orthogonal null direction ka and is strictly negative in the other direction, la. A future holographic screen is a hypersurface of indefinite signature, foliated by marginally trapped surfaces called “leaves.” Subject to certain generic conditions, it was shown that the foliation of a future holographic screen evolves monotonically in the la direction. (For example, for a black hole formed by collapse, this is the outside or past− direction.) This implies further that the area of a future holographic screen increases monotonically along the foliation. A similar area law holds for past holographic screens, defined in terms of marginally antitrapped surfaces, and abundant in cosmological solutions such as our own universe. Past or future holographic screens are easily constructed by picking a null foliation of the spacetime. On each codimension-1 null slice one finds the unique codimension-2 surface of CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 74 maximal area. This surface may lie on the conformal boundary, but if gravity is strong it will lie inside the spacetime. One null expansion vanishes by construction, so the sequence of such surfaces form a holographic screen [29]. The screen will be future or past if the sign of the other expansion is definite. The proof of the theorem is elaborate in general but simple in the case of spherical symmetry. It is easy to see that screens that violate the theorem are intersected twice by the same null congruence N with tangent vector ka on two distinct leaves. The generic condition implies θ = 0 between the two leaves. But θ = 0 on both leaves, in conflict with the classical focussing6 theorem, equation (4.3). Like Hawking’s area theorem for event horizons, the new area law fails when the null energy condition is violated. And like Hawking’s theorem, the area theorem for holographic screens can be reformulated as a generalized second law [33], by replacing area with generalized entropy via equation (4.14). This is the first covariant statement of a generalized second law that applies to general quasilocal horizons, and the first that applies to expanding cosmologies regardless of the sign of the cosmological constant. The novel GSL applies to quantum future (or past) holographic screens. These are defined as hypersurfaces foliated by quantum marginally trapped (or antitrapped) surfaces. The latter, in turn, are defined by requiring that the quantum expansion vanishes, Θ = 0, in one null direction and is strictly negative, Θ < 0, in the other. Like classical holographic screens, these objects are easily constructed in general spacetimes. The proof of this novel GSL proceeds exactly as in the classical area theorem, with the assumption of the null energy condition replaced by the QFC. Again, a generic condition implies that Θ = 0 on N between two leaves. The definition of the quantum holographic screen requires Θ6 = 0 on both leaves, in contradiction with the QFC [33]. 75

Appendix

4.A Renormalization of the Entropy

It is well known that the von Neumann entropy Sout on one side of a sharp boundary σ is subject to UV divergences. Thus in order to define the generalized entropy Sgen, we must invoke a renormalization procedure. We start by regulating Sout using a UV cutoff associated () with some distance scale , so that the outside entropy Sout now depends on the regulator. This could be done using e.g. a heat kernel regulator [157, 143], Pauli-Villars [136, 58], a brick wall cutoff [154], the mutual information [50], or a variety of other methods. The leading-order divergence is proportional to the area [148, 22, 149], but in dimensions D 4 there are additional subleading divergences, each proportional to some local geometrical≥ integral on the boundary. In dimension D there are subleading divergent corrections with weight up to D. Since perturbative quantum gravity is a nonrenormalizable theory, one can also even higher curvature corrections by considering 2-loop or higher diagrams involving gravitons. Higher curvature corrections can also arise from stringy effects. When calculating the generalized entropy, each of these divergences is absorbed into a counterterm, i.e., a parameter in the gravitational action I which controls the size of a correction to the gravitational entropy Sgrav. We shall see that the total quantity Sgen = Sgrav + Sout is invariant under the RG flow. Thus these counterterms are important part of the definition of the QFC, although they drop out of the QNEC for the reasons described in section 4.4.

The replica trick The replica trick (reviewed in [42]) is a convenient way to calculate the von Neumann entropy tr(ρ log ρ), without the nuisance of taking the logarithm of a matrix. This trick is based on− instead evaluating the Rényi entropy 1 S ln tr(ρn) (4.41) n ≡ 1 n − and analytically continuing to n = 1 to obtain S. In cases where the state ρ comes from a Euclidean path integral, there is a beautiful geometrical interpretation of the Rényi entropy in terms of an n-sheeted cover M (n) of the manifold, having a conical singularity with total CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 76 angle 2πn on the entangling surface σ. Assuming that one can analytically continue the effective action I = ln Z from the positive integers to n = 1, one then writes eff −

Sreplica = (1 n∂n)Ieﬀ . (4.42) − − n=1

This defines the geometrical or replica entropy for the state of the quantum fields outside of σ [78, 153, 45, 95,6, 144, 43, 44, 48]. Inserting the 1-loop effective action Ieff of a quantum field, we obtain a nonlocal answer for Sreplica, as expected. However, the UV divergences in Ieff are local, allowing us to compute the corresponding divergences in Sreplica. Note that the identification of Sreplica with the von Neumann entropy Sout is somewhat formal, due to the fact that the replicated manifold has a delta function of curvature at the conical singularity.11 Matter fields can couple to the curvature at the tip, producing “contact terms” whose interpretation will be discussed shortly. Now nothing stops us from inserting a classical gravitational action I[gab] into equation (4.42) (treating the n-sheeted cover manifold as a fixed background metric). In this case everything cancels except for a contribution coming from the conical singularity. So in this case we obtain an entropy which is local on σ, which we call the gravitational entropy Sgrav. If I is the Einstein-Hilbert action, Gibbons and Hawking obtained by this method the Bekenstein-Hawking entropy A/4GN ~ [83], thus explaining why this term is included in the generalized entropy. If on the other hand I is a higher-curvature action, one obtains additional correction terms in Sgrav. For the stationary case, in which σ lies on a Killing horizon, the analytic continuation is easy due to the presence of the rotational symmetry about the bifurcation surface, and hence Sgrav is given by the Wald entropy [161, 106, 103, 102], obtained by differentiating the Lagrangian with respect to the Riemann tensor and multiplying by the binormal µν twice:

2π 2 2 ∂I SWald = d x g µνξo. (4.43) − ~ σ ∂Rµνξo Z p However, Wald’s Noether charge method for deriving the entropy is subject to ambiguities for nonstationary horizons [106, 103], and is therefore unable to determine the coeﬃcients of those terms which vanish on stationary horizons, e.g. terms involving products of extrinsic curvatures. Thus, for the nonstationary case, there are additional corrections to the Wald entropy which were only recently calculated for higher-curvature gravity actions, which will be discussed in the next section. 11This curvature is sometimes smoothed out slightly in order to evaluate the entropy [131], although this smoothing does not by itself eliminate the UV divergences associated with coupling to the curvature. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 77

Nonstationary entropy

For several years it was unclear how to calculate Sgrav for a nonstationary surface σ in a higher-curvature gravity theory, although some information was available using various methods such as field redefinitions and the GSL [106, 105, 73], holography [145, 100], and the Randall-Sundrum model [129]. A major breakthrough came when Lewkowycz and Maldacena [118] found a clever way to analytically continue the smoothed out n-sheeted replica trick in the context of calculating the holographic von Neumann entropy (cf. section 4.5) in AdS/CFT. Their calculation involves performing the replica trick on the conformal boundary of the manifold, while requiring the interior to be a smooth solution to the equations of motion, exploiting the dynamical nature of gravity. One can then orbifold by the replica group Zn to find a manifold for which the n 1 limit can be smoothly taken. This can be used to derive the Ryu- Takayanagi formula→ [140]12 in the regime where the bulk theory is governed by the Einstein- Hilbert action. Their calculation was quickly extended to determine the gravitational entropy functional Sgrav for higher-curvature theories. For quadratic gravity, see [80, 60, 46]; for Lovelock see [14, 16, 15]. In the more general case of f(Riemann) actions, Sgrav is given by the Dong entropy [60] But some ambiguities remain, related to the “splitting problem” [126, 125] (these references also make some inroads into the case where the action contains derivatives of the Riemann tensor). Also, Faulkner, Lewkowycz and Maldacena [69] showed how to include the 1-loop matter in the bulk; this corresponds to adding a bulk von Neumann entropy term, thus replacing A with Sgen of the extremal surface, as done many times in this chapter. Although these formulas were derived for holographic entanglement surfaces, they are consistent with the hypothesis that an entropy can be ascribed to more general surfaces. For example, the holographic entropy functional also seems to be the correct one to use when defining the GSL for linearized metric perturbations to Killing horizons, in any higher curvature gravity theory [166] (cf. [12, 141] for some special cases).

Example: 3+1 dimensions For example, in D = 4 semiclassical gravity coupled to free ﬁelds, the one loop corrections to the inverse of Newton’s constant 1/GN are quadratically divergent in :

1 −2 ∆1-loop = fG , (4.44) GN where the constant of proportionality fG depends on the number and type of matter species. But there are also logarithmic divergences in the three parameters α, β, γ associated with 12A similar argument can be used to prove the HRT formula [99], if one analytically continues using a complexified manifold. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 78 the quadratic gravity effective action [80]: R I = d4x √g + αR2 + β(R )2 + γ(R )2 + I (4.45) eff 16πG µν µνoξ nonlocal Z N where, at one loop,

∆α1-loop = fα log() + gα; (4.46)

∆β1-loop = fβ log() + gβ; (4.47)

∆γ1-loop = fγ log() + gγ. (4.48)

In general, coefficients of power law divergences such as fG depend on the details of the renormalization scheme. But the coefficients of the log divergences fα, fβ, fγ are universal, depending only on the field theory [50].13 These log divergences appear because the dimension is even. (The finite piece of the 1-loop effective action is universal in odd dimensions, 0 but in even dimensions this is true only up to local counterterms such as gα, gβ, gγ, since still counts as a power law!) The value of the f coefficients for various spins in 4 dimensions are listed in [52, 18, 157], although there are certain issues with higher spin fields (3/2 and 2) which we will discuss in further detail momentarily. Dimensional analysis reveals five possible covariant terms in the gravitational entropy density of σ due to α, β, γ:

i ij i j a ij RRi Rij Ki aKj a Kij K a, (4.49)

where Rµνξo is the 4D Riemann tensor, Rµν the 4D Ricci tensor, R is the 4D Ricci scalar, a Kij is the extrinsic curvature; also the indices i, j represent directions parallel to σ, which are raised and lowered by qij (the metric restricted to σ), and a represents an index normal to σ. The extrinsic curvature requires an a index because σ is a codimension-2 surface, so that there are two remaining dimensions to bend into. However, because there are only 3 possible terms in the action, only 3 linear combinations of these 5 terms can appear. The methods of the previous section show that the correct entropy functional is [145, 129, 80, 60, 46] A 8π β S = + d2x 2g αR + R i 1 K iaK j + γ R ij K aKij (4.50) grav 4G 2 i − 2 i j a ij − ij a N ~ ~ Zσ p Aside from the extrinsic curvature terms (which are ambiguous in the Noether charge formalism), this expression is the same as the Wald entropy in equation (4.43). We may then deﬁne the generalized entropy as [62] S = S + S (4.51) gen h gravi out 13In the special case of a CFT these parameters are determined by the central charges c and a, which 2 determine the log divergences of the two conformally invariant contributions: Weyl squared (Cµναξ) and µνπρ αξστ the Euler density RµναξRπρστ respectively. Although these two terms would be conformal if they were ﬁnite, their logarithmic dependence on is a conformal anomaly; thus the partition function on curved spacetimes is not scale invariant. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 79 which in this case expands out to:

β() i 1 ia j () A 8π 2 2 α()R + Ri 2 Ki Kj a Sgen = lim Sout + + d x g 2 − →0  4G ()  ij a ij  N ~ ~ Zσ + γ() R K K p ij − ij a   (4.52) In the limit where becomes small, the various dependences on should cancel out, so that Sgen is independent of the choice of cutoff scale. Since the divergences in Sout are local and proportional to the other terms in equation (4.52), it is manifest that there exists a choice of RG flow for the parameters G, α, β, γ for which Sgen becomes cutoff independent. Thus the generalized entropy Sgen is well defined even though Sout is cutoff dependent. As one shifts the cutoff, the entropy simply moves between the different terms.

Interpretation of contact terms

A crucial consistency condition is that the RG flow of the parameters in Sgen (given by equation (4.52) at one loop in 3+1 dimensions) are in fact the same as for the corresponding parameter in the gravitational action in equation (4.45).14 Although the renormalization of the replica trick entropy automatically matches the renormalization of the gravitational action [116, 54], it is not so clear that the replica entropy can always be written as the sum of a horizon piece plus a statistical piece which is literally a von Neumann entropy tr(ρ log ρ), as in equation (4.51). Thus if one for example imposes − a regulator such as a brick wall (or a lattice) in which it is manifest that Sgen has literally a statistical interpretation, and then compares to the replica trick with e.g. a smoothed out conical singularity, it is not a priori obvious that the RG flow of the two definitions of entropy will agree. The question is whether all terms in the replica entropy can be given a statistical interpretation. For minimally coupled scalars and spinor fields, several calculations have shown an exact agreement between the geometric and statistical viewpoints [81, 58, 143,2, 76, 108, 170] (modulo the K2 terms, whose coefficients were unknown in the 1990s but can now be determined and shown to agree by the methods described above).15 There is an apparent mismatch [146, 62] between the replica entropy of nonminimally coupled scalars [143] and gauge fields [108], and their statistical von Neumann entropy. Al- though the renormalization of the replica entropy automatically matches the renormalization of the gravitational action [116, 54], it is not so clear that this geometrical entropy can always be written as the sum of a horizon piece plus a statistical piece, as in equation (4.51). But these concerns can be resolved. In the case of nonminimally coupled scalar fields whose Lagrangian includes the term ξφ2R, an extra “contact term” appears due to coupling to the conical singularity when performing the replica trick. This extra term is proportional to ξ φ2 . It contributes to σh i 14For nonuniversal coefficients, the same regulator must of course be used onR both sides. 15But see [111] for an apparent discrepancy for scalars in odd dimensions, using a Pauli-Villars regulator. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 80

the Sgrav (which in this case is equal to the Wald entropy (4.43)), and hence appears as an extra term in equation (4.52). It also contributes nontrivially to the RG flow of 1/GN , due to the multiplication of φ at coincident points, restoring consistency [62]. In the case of Maxwell fields, there is also a contact term, which however cannot be 16 explained by the addition of any term of the appropriate dimension to Sgrav. Instead, the extra term arises from “edge modes” in the von Neumann entropy due to the fact that boundary gauge symmetries are not gauged, and can be calculated by a path integral over the electric flux through the surface σ [63] (cf. [96]).17 A similar contact term involving gravitons can presumably be resolved in the same way, but here there are additional conceptual problems which we briefly explore next.

The challenge of higher spin fields Defining the von Neumann entropy for spin 3/2 and 2 fields is considerably trickier than for spins 1. First≤ of all, as a result of their relationship to (super)gravity, they are only consistently defined when expanding around a gravitational background where the traceless part of the Ricci tensor vanishes (Einstein manifolds) [52].18 If the trace is nonzero, i.e., if there is a cosmological constant, the fields acquire an effective mass which must be included as in [53].19 This suggests that only some of the coefficients defined above are physically meaningful, namely those of the Weyl squared term and terms involving the Ricci scalar. As a result, the usual bulk replica trick method for defining the entanglement is suspect, since this is invalid off-shell [54]. A possibly related problem is, in the case of gravitons, it is also necessary to decide on a covariant definition of the location of the surface σ. Choosing σ based on its coordinate location would make the results dependent on the choice of gauge. In spite of these troublesome issues, some have plowed ahead and calculated the fG coefficient for linearized gravitons using a spin-2 gauge-fixed wave operator. Fursaev and

16 ij By analogy to nonminimal scalar, one could try to add a term like AiAjg , where Ai is the gauge potential and gij the inverse normal metric [62], but this is not a gauge-invariant⊥ combination. Nor does it appear in the⊥ holographic gravitational entropy [97] when derived by the Lewkowycz-Maldacena method [118]. 17An alternative explanation [4] is that the contact term arises from the total derivative terms in the action. It has also been suggested [3,5] that this would resolve a discrepancy between the field theoretical and holographic entropy calculations in 6 dimensions found by [100]. However, this interpretation departs from the usual principle that total derivatives may be dropped without consequence, and was argued against in [98]. 18In addition to these constraints, spin 3/2 fields require the cosmological constant to be negative or zero. A further issue (unique to fermionic gauge fields) is that when gauge-fixing the spin 3/2 field, one must take into account not only the Faddeev-Popov ghosts but also the Nielsen-Kallosh ghost [132, 109], in order to obtain the proper number of degrees of freedom [101]; see [61] for a correction of the spin 3/2 calculation in reference [146]. 19For this reason, among other errors, the R2 coefficient for gravitons or gravitinos found in [52, 18, 157] should not be used. CHAPTER 4. THE QUANTUM FOCUSSING CONJECTURE 81

Miele [79] found that the partition function of the replica manifold does not reduce in the limit β 2π to the partition function of the smoothed out cone, which usually must agree → [131]. Solodukhin [146, 147] also calculated fG for the graviton, but his results appear to conflict with [79]. More ambitiously still, He et al. [91] calculated the von Neumann entropy for the entire tower of higher-spin fields which appear in string theory, using an alternative definition of the replica trick on orbifold spacetimes where 1/n is an integer. (For spin 3/2 and 2 their calculation agrees with that of [79].) Notwithstanding the conceptual problems above, it should at least be possible to calculate the graviton von Neumann entropy using the Lewkowycz-Maldacena method [118], in which the replica trick is performed on the boundary, and the equations of motion hold everywhere in the bulk. But this would only allow one to calculate the graviton entropy when σ is an extremal surface! It would not be sufficient for applications such as the QFC in which σ may be an arbitrary surface. Also, in order to derive the result of Faulkner et al. [69] for gravitons, one would need an independent bulk definition of the graviton von Neumann entropy. Thus, further exploration into the nature of von Neumann entropy for gravitons and gravitinos is warranted. 82

Chapter 5

Proof of the Quantum Null Energy Condition

In the previous chapter, we presented a lower bound on Tkk , the quantum null energy condition, which follows from the quantum focussing conjectureh i in the weak gravity limit:

~ 00 T S [Σ] . (5.1) h kki ≥ 2π out A In this chapter, we will prove this inequality in field theory. Let us begin by making some brief remarks on the bound. The quantity Sout is divergent, so one might worry that the bound could be vacuous. However, derivatives of the entropy can be finite even if the entropy itself is divergent. (A more rigorous formulation in terms of functional derivatives will be given in the main text.) Further note that the right hand side can have any sign. If it is positive, then the QNEC is stronger than the NEC; but since it can be negative, it can accommodate situations where the NEC would fail. By integrating the QNEC along a null geodesic, we can obtain the averaged null energy condition, or ANEC, 0 in situations where the boundary term Sout vanishes at early and late times. Intriguingly, the QNEC—an intrinsically field theoretic statement—was recognized by studying conjectured properties of the generalized entropy, A[Σ] Sgen[Σ] = + Sout[Σ] , (5.2) 4GN ~ a key concept arising in quantum gravity [9,8, 10]. Here Σ is a codimension-2 surface which divides a Cauchy surface in two, A[Σ] is its area and Sout is the von Neumann entropy of the matter fields on one side of Σ. The generalized second law (GSL) is the conjecture [9] that the generalized entropy cannot decrease as Σ is moved up along a causal horizon. Equation (5.1) first appeared as a sufficient condition for the GSL, satisfied by a nontrivial class of states of a 1+1 dimensional CFT [168]. The QNEC emerged as a general constraint on quantum field theories when it was noted that the quantum focussing conjecture (QFC) implies equation (5.1) in an CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 83

⌃ Sout()

} A {z | Figure 5.1: 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 quantum expansionA 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 space-like slice is related to the entropy of the null quantized state on the future (brighter green) part of the null surface. appropriate limit [36]. We will briefly describe the QFC and outline how the QNEC arises from it. A generalized entropy can be ascribed not only to horizon slices, but to any surface that splits a Cauchy surface [169, 65, 17, 129, 69]. Moreover, one can define a quantum expansion Θ[Σ; y1], the rate (per unit area) at which the generalized entropy changes when the infinitesimal area element of ν at a point y1 is deformed in one of its future orthogonal null directions [36] (see figure 5.1). This quantity limits to the classical (geometric) expansion as ~ 0. The QFC states that the quantum expansion Θ[Σ; y1] will not increase under any → second variation of Σ along the same future congruence, be it at y1 or at some other point y2 [36]. The QFC, in turn, was presented in the previous chapter as a quantum version of the covariant entropy bound (Bousso bound) [25, 29, 71], a quantum gravity conjecture which bounds the entropy on a nonexpanding null surface in terms of the difference between its initial and final area. The QFC implies the Bousso bound; but because the generalized entropy appears to be insensitive to the UV cutoff [153, 104, 144], the QFC remains well- defined in more general settings. (The QFC is distinct from the quantum Bousso bound of chapters2 and3, which defines the entropy by vacuum subtraction [51], a procedure applicable if the gravitational effects of matter are sufficiently small [38, 37].) In the case where y = y , we showed in chapter4 (see also [36]) that the QFC follows 1 6 2 CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 84 from strong subadditivity, an entropy inequality which all quantum systems must obey.1 For y1 = y2, the QFC remains a conjecture in general, but in special cases it can be proven. −1 The QFC constrains a combination of “geometric” terms proportional to GN that stem from the classical expansion, as well as “matter entropy” terms that stem from Sout and do not involve Newton’s constant. The classical expansion is governed by Raychaudhuri’s equation, θ0 = θ2/2 σ2 8πG T .2 If the expansion θ and the shear σ vanish at y , − − − N h kki 1 then the rate of change of the expansion is governed by a term proportional to GN . In this case, all GN ’s cancel in the terms of the QFC, and equation (5.1) emerges as an apparently nongravitational statement.

Outline In this chapter, we will prove the QNEC in an array of settings. Our proof applies to free or superrenormalizable, massive or massless bosonic fields, in all cases where the surface Σ lies on a stationary null hypersurface (one with everywhere vanishing expansion). An important example is Minkowski space, with Σ lying on a Rindler horizon. Such a horizon exists at every point p and with every orientation ka, so the QNEC constrains all null components of the stress tensor everywhere in Minkowski space. A similar situation arises in a de Sitter background, where p and ka specify a de Sitter horizon, and in Anti-de Sitter space, where they specify a Poincaréhorizon. Other examples include an eternal Schwarzschild or Kerr black hole, but in this case our proof applies only to points on the horizon, with ka tangent to the horizon generators. These should all be viewed as fixed background spacetimes with no dynamical gravity; our proof establishes that free scalar field theory on these backgrounds satisfies equation (5.1). We give a brief review of the formal statement of the QNEC in section 5.1. We then set up the calculation of all relevant terms in section 5.2. In section 5.2, we review the null surface quantization of the theory, on the particular null surface N that is orthogonal to Σ with tangent vector ka. Null quantization has the remarkable feature that the vacuum state factorizes in the transverse spatial directions. This reduces any purely kinematic problem (such as ours) to the analysis of a large number of copies of the free chiral scalar CFT in 1+1 dimensions. We then restrict attention to the particular chiral CFT on the infinitesimal pencil that passes through the point p where Σ is varied. The state on this pencil is entangled with an auxiliary quantum system which contains both the information crossing the other generators of N, and the information that does not fall across N at all. In the 1+1 chiral CFT, the pencil state is very close to the vacuum, but not so close that the QNEC would be trivially saturated by application of the first law of von Neumann entropy. To constrain the second order variations of Sout (the Fisher information), we must keep track of the deviation of the pencil state from the vacuum to second order. We discuss the appropriate expansion of the overall state in section 5.2. We write the state in terms of 1Some recent articles [13, 117] considered a different type of second derivative of the entropy in 1+1 field theory. These inequalities involve varying the two endpoints of an interval independently, and therefore follow from strong subadditivity alone, without making reference to the stress-tensor. 2Raychaudhuri’s equation immediately implies that, in cases where the classical geometrical terms dominate, the QFC is true if and only if the classical spacetime obeys the null curvature condition. CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 85 operators inserted on the Euclidean plane corresponding to the pencil and expand in a basis of the auxiliary system. Then in section 5.2, we expand the entropy and identify the parts of our expnsion enter into the second derivative. ~ 00 In section 5.3, we compute the sign of Tkk 2πA Sout. In section 5.3 we review the replica trick for computing the von Neumannh entropyi − by the analytic continuation of Rényi entropies. We extract two terms relevant to the QNEC, which are computed in sections 5.3 and 5.3 respectively. The most subtle part of the calculation is the analytic continuation of the second of these terms, in section 5.3. In section 5.3, we combine the terms and conclude that the QNEC holds for all states. In section 5.4, we extend our result to establish the QNEC also for superrenormalizable scalar fields, and for bosonic fields of higher spin. We also discuss the extension to interacting theories. We expect that the proof we have given can be extended to fermionic fields, but we leave this task for the future.

Discussion Our result establishes a new and surprising link between quantum information and a more familiar physical quantity, the stress tensor. The QNEC identifies the “accelera- tion” of information transfer as a lower bound on the energy density. Equivalently, the stress tensor can be viewed as imposing a constraint on the second derivative of the von Neumann entropy. The latter can be difficult to calculate but plays an important role in quantum information theory, condensed matter, and high energy physics. Our proof of the QNEC requires no assumptions beyond the known properties of free quantum fields, but it is quite lengthy and somewhat involved. Yet, the QNEC follows almost trivially from a statement involving gravity, the quantum focussing conjecture. This perplexing situation is somewhat reminiscent of the proof of the quantum Bousso bound [38], particularly in the interacting case [37]. It is intriguing that the study of quantum gravity can lead us to simple conjectures such as equation (5.1) which can be proven entirely within the nongravitational sector, where they are far from obvious—so far, indeed, that they had not been recognized until they emerged as implications of holographic entropy bounds or of properties of the generalized entropy. It is becoming clear that the structure of known quantum field theories carries a deep im- print of causal and information theoretic properties ultimately dictated by quantum gravity. This adds to the evidence that “quantizing gravity” has nothing to do with the inclusion of one last force in a quantization program. It would be interesting to try to formulate models of quantum gravity in which focussing of the entropy occurs naturally. Remarkably, the QNEC does not seem to follow from any of the standard identities that apply purely at the level of quantum information. Our proof did involve additional structure supplied by quantum field theory. The QNEC is related to the relative entropy S(ρ σ) = tr(ρ ln ρ) tr(ρ ln σ), which equals Sgen (up to a constant) when σ is taken to be the|| vacuum state− [164]. The relative entropy− satisfies positivity, which guarantees that Sgen(ρ) is less than in the vacuum state. It also enjoys monotonicity, which implies that Sgen is increasing under restrictions; this constrains the first derivative, which is the GSL [164]. It CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 86 may appear that the QNEC can be proven using properties of the relative entropy. But the QNEC is a statement about the second derivative of the generalized entropy. It is possible that the QNEC hints at more general quantum information inequalities, which are yet to be discovered. It is interesting that a recently proposed new GSL, which applies in strongly gravitating regions such as cosmology, also can be shown to follow from the QFC [33].

5.1 Statement of the Quantum Null Energy Condition

The statement of the QNEC involves the choice of a point p a null vector ka at p, and a smooth codimension-2 surface Σ orthogonal to ka at p such that Σ splits a Cauchy surface into two portions. The null vector ka is a member of a vector field orthogonal to Σ defined in a neighborhood of p, ka(y). Here and below we use y as a coordinate label on Σ, also called the “transverse direction.” We can consider a family of surfaces Σ[λ(y)] obtained by deforming Σ along the null geodesics generated by ka(y) by the affine parameters λ(y). The deformed surfaces will also be Cauchy-splitting [34]. This allows us to define a family of entropies Sout[λ(y)], which are the von Neumann entropies of the quantum fields restricted to the Cauchy surface on one side of Σ[λ(y)]. The choice of Cauchy surface is unimportant, since by unitarity the entropy will be independent of that choice. The choice of side of Σ[λ(y)] also does not matter, because the QNEC is symmetric with respect to ka ka. → − Once we have defined Sout[λ(y)], we can consider its functional derivatives. In general, the second functional derivative will contain diagonal and off-diagonal terms (present because Sout is a non-local functional), and the diagonal terms will be proportional to a δ-function. We define the second functional derivative at coincident points by factoring out that δ- function: δ2S δ2S out = out δ(y y0) + off-diagonal. (5.3) δλ(y)δλ(y0) δλ(y)2 − Then if the expansion and the shear of ka(y) vanish at p, we have the general conjecture

2 ~ δ Sout Tkk(p) , (5.4) h i ≥ 2π h(p) δλ(p)2 λ(y)=0

p a b where h is the determinant of the induced metric on Σ and Tkk Tabk k . We will ﬁnd it convenient below to work with a discretized version of the functional≡ derivative, obtained by dividing Σ into regions of small area and considering variations locally constant in those regions. Then equation (5.4) reducesA to the form advertised in equation (5.1):

~ 00 T S . (5.5) h kki ≥ 2π out A CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 87

5.2 Reduction to a 1+1 CFT and Auxiliary System

Null quantization The proof that follows applies when Σ is a section of a general stationary null surface N in D > 2 (the case D = 2 will be treated separately, in section 5.4). We consider deformations of Σ along N toward the future, so the deformation vector ka is future-directed, and we choose to take the “outside” direction to be the side towards which ka points. As mentioned above, a proof of this case automatically implies a proof for the opposite choice of outside. By unitary time evolution of the space-like Cauchy data, we can consider the state to be defined on the portion of N in the future of Σ together with a portion of future null infinity. We rely on null quantization on N, which requires that N be stationary [164]. Null quantization is simplest if we first discretize N along the transverse direction into regions of small transverse area . These regions, which are fully extended in the null direction, are called pencils. UltimatelyA we will take the continuum limit 0, and the QNEC will be shown to hold in this limit. At intermediate stages, acts asA a → small expansion parameter.3 This is the reason why we are restricting ourselves toA D > 2 spacetime dimensions for now: without a transverse direction to discretize, there would be no small expansion parameter. Also, while logically independent from the discretization used to define the QNEC in equation (5.1), we will take these two discretizations to be the same. That is, we will consider deformations of the surface Σ which are localized to the same regions of size that define the discretized null quantization. A There is a distinguished pencil that contains the point p; this is the pencil on which we will perform our deformations. The total Hilbert space of the system can be decomposed as = pen aux, where pen refers to the fields on the distinguished pencil and aux is everythingH H else.⊗ H “EverythingH else” includes both the remaining pencils on N restrictedH to the future of Σ, as well as the relevant portion of null infinity. We do not have to be specific about the exact structure of the auxiliary system; our proof does not assume anything about it other than what is implied by quantum mechanics. Beginning with a density matrix on , we obtain a one-parameter family of density matrices ρ(λ) by tracing out the part of the Hpencil in the past of affine parameter λ. When λ the pencil is fully extended, and when λ + the entire pencil has been traced out.→ −∞λ = 0 corresponds to no deformation of the original→ ∞ surface. When restricted to N, the theory decomposes into a product of 1+1-dimensional free chiral CFTs, with one CFT associated to each pencil of N. In particular, this means that the vacuum state factorizes with respect to the pencil decomposition of N [164]. Crucially, when is small, the state of the pencil is near the vacuum. This can be seen as follows. For a regionA of small size , the amplitude to have n particles on the pencil scales like n/2 (so the probability is appropriatelyA extensive), and therefore the coefficient of n m inA the pencil Fock basis expansion of the state scales like (n+m)/2. Hence for small | ih | A 3The dimensionless expansion parameter is in units of a characteristic length scale of the state we are interested in, e.g., the wavelength of typical excitations.A The state remains fixed as 0. A → CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 88

we can write the state as A ρ(λ) = ρ(0) (λ) ρ(0) + σ(λ) , (5.6) pen ⊗ aux (0) where ρpen(λ) is the vacuum state density matrix on the part of the pencil with aﬃne (0) parameter greater than λ, ρaux is some state in the auxiliary system (not necessarily the vacuum), and the perturbation σ(λ) is small: the largest terms are obtained by taking the partial trace of 0 1 and 1 0 in the pencil Fock basis, and these terms have coeﬃcients which scale like| ih1/2|. Entanglement| ih | between the pencil and the auxiliary system is also present in σ; we willA explore the form of σ in more detail in the following section.

Expansion of the state As discussed above, the pencil state can be described in terms of a 1+1-dimensional free chiral CFT, with fields that depend only on the coordinate z = x+t. In this notation, translations ∂ along the Rindler horizon in the 1+1 CFT are translations in z, and are generated by ∂ ∂z . In a chiral theory, this is equivalent to translations in the spatial coordinate x. Therefore≡ the shift in affine parameter λ of the previous section can be replaced by a shift in the spatial coordinate for the purposes of the CFT calculation. In addition, quantization on a surface of constant Euclidean time τ = it = 0 in a chiral theory is equivalent to quantization on the Rindler horizon. Thus when we construct the state we can use standard Euclidean methods for two-dimensional CFTs. We have argued that, at order 1/2, the perturbation σ on the full pencil must be of the schematic form 0 1 (plus HermitianA conjugate). So on the full pencil, we have the state | ih | ρ = ρ( ) = 0 0 ρ(0) + 1/2 ( 0 ψ + ψ 0 ) i j + , (5.7) −∞ | ih | ⊗ aux A | ih ij| | jiih | ⊗ | ih | ··· ij X where i j is a basis of operators in the auxiliary system and “ ” denotes terms which vanish| moreih | quickly as 0. We will argue in section 5.2 that those··· terms are not relevant for the QNEC, and so weA → will ignore them from now on. For later convenience, we will take (0) the basis i in the auxiliary system to be the one in which ρaux is diagonal. The states ψij are single-particle| i states in the CFT, and we have ensured that the state is Hermitian.| Thei CFT part of the state can be constructed by acting on the vacuum with a single copy of the field operator. In a Euclidean path integral picture, we can get the most general single- particle state by allowing arbitrary single-field insertions on the Euclidean plane. This is shown in figure 5.2. To obtain the state at a finite value of λ, we need to take the trace of equation (5.7) over the region x < λ. Alternatively, we can hold fixed the inaccessible region, x < 0, but translate the field operators used to construct the state by λ. From this point of view the vacuum is independent of λ and we write it as

(0) −2πKpen ρpen = e , (5.8) CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 89

⌧

@ x

Figure 5.2: 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.

where, up to an additive constant, the modular Hamiltonian Kpen coincides with the Rindler boost generator for the CFT [156, 19]. Specializing to the case of a single chiral scalar ﬁeld (extensions will be discussed in section 5.4), the trace of equation (5.7) becomes

ρ(λ) = e−2πKpen ρ(0) + 1/2 e−2πKpen drdθ f (r, θ)∂Φ(reiθ λ) i j , (5.9) ⊗ aux A ij − ⊗ | ih | ij X Z where ∂Φ(z) is now a holomorphic local operator on a two-dimensional Euclidean plane4 and (r, θ) are polar coordinates on that plane, with z = reiθ. Rotations in θ are generated by 5 Kpen. Thus the operator ∂Φ is defined by ∂Φ(reiθ) = e−iθeθKpen ∂Φ(r)e−θKpen . (5.10) All of the operators in equation (5.9) are manifestly operators on the Hilbert space corresponding to x > 0, τ = 0. We are taking Φ to be a real scalar field, so in particular ∂Φ is a Hermitian operator for real arguments. Then in order for ρ to be Hermitian, we must have f (r, θ) = f (r, 2π θ)∗. (5.11) ij ji − Aside from this reality condition, letting f be completely general gives all possible single particle states. To facilitate our later calculations, we will modify equation (5.9) in order to put the auxiliary system on equal footing with the CFT. To that end, define Kaux through the (0) equation ρaux = exp( 2πK ). We can invent a coordinate θ for the auxiliary system and − aux declare that evolution in θ is generated by Kaux. Then define the operators E (θ) eθKaux i j e−θKaux = eθ(Ki−Kj ) i j . (5.12) ij ≡ | ih | | ih | 4We insert ∂Φ instead of Φ in order to remove any zero-mode subtleties. We have checked that the proof still works formally if one inserts Φ instead of ∂Φ, and in fact continues to work when an arbitrary number of derivatives, ∂lΦ, are used. This latter fact is not surprising since insertions of Φ alone (or ∂Φ if we drop the zero mode) are sufficient to generate all single particle states. See [41, 164] for details on the zero-mode. 5Here θ is restricted to be in the range [0, 2π).

1 CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 90

Since K is diagonal in the i basis, with eigenvalues K , E (θ) is just a rescaled i j . aux | i i ij | ih | More generally, multiplying i j on either side by arbitrary functions of Kaux results in the same operator up to an (i, j)-dependent| ih | numerical factor. So by making the replacement

f (r, θ) e(2π−θ)Ki eθKj f (r, θ) , (5.13) ij → ij which does not alter the reality condition on f, we can write

ρ(λ) = e−2πKtot + 1/2e−2πKtot dr dθ f (r, θ)∂Φ(reiθ λ) E (θ) , (5.14) A ij − ⊗ ij ij X Z

where Ktot Kpen + Kaux. From now on, we will simply write K for Ktot. Below it≡ will be useful to write σ(λ) as

σ(λ) 1/2ρ(0) (λ) . (5.15) ≡ A O Thus comparing with equation (5.14), we ﬁnd

(λ) = dr dθ f (r, θ)∂Φ(reiθ λ) E (θ) . (5.16) O ij − ⊗ ij ij X Z As a side comment, we note that one could prepare the state in equation (5.14) via a Euclidean path integral over the entire plane with an insertion of and boundary field configurations defined at θ = 0+ and θ = (2π)−. O

Expansion of the entropy In the previous sections we saw that null quantization gives us a state of the form

ρ(λ) = ρ(0) (λ) ρ(0) + σ(λ), (5.17) pen ⊗ aux (0) where ρpen(λ) is the vacuum state reduced density matrix on the part of the pencil with (0) aﬃne parameter greater than λ, ρaux is an arbitrary state in the auxiliary system, and the perturbation σ is proportional to the small parameter 1/2. In this section, we will expand the entropy perturbatively in σ and show that the QNECA reduces to a statement about the (0) (0) (0) contributions of σ to the entropy. We will assume that both ρ(λ) and ρ (λ) ρpen(λ) ρaux are properly normalized density matrices, so tr(σ) = 0. ≡ ⊗ The von Neumann entropy of ρ(λ) is Sout(λ). We will expand it as a perturbation series in σ(λ): S (λ) = S(0)(λ) + S(1)(λ) + S(2)(λ) + (5.18) out ··· where S(n)(λ) contains n powers of σ(λ). At zeroth order, since ρ(0) is a product state, we have

S(0)(λ) = tr ρ(0)(λ) log ρ(0)(λ) = tr ρ(0) (λ) log ρ(0) (λ) tr ρ(0) log ρ(0) . (5.19) − − pen pen − aux aux CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 91

The ﬁrst term on the right-hand side is independent of λ because of null translation invariance of the vacuum: all half-pencils have the same vacuum entropy. The second term is manifestly independent of λ. So S(0) is λ-independent and does not play a role in the QNEC. Now we turn to S(1)(λ):

S(1)(λ) = tr σ(λ) log ρ(0)(λ) = tr σ(λ) log ρ(0) (λ) tr σ(λ) log ρ(0) . (5.20) − − pen − aux Once again, the second term is λ-independent, which we can see by evaluating the trace over the pencil subsystem:

tr σ(λ) log ρ(0) = tr [tr σ(λ)] log ρ(0) = tr σ( ) log ρ(0) . (5.21) aux aux pen aux aux ∞ aux (0) To evaluate the first term, we use the fact that ρpen(λ) is thermal with respect to the boost operator on the pencil. Then we have 2π ∞ tr σ(λ) log ρ(0) (λ) = A dλ0 (λ0 λ) T (λ0) , (5.22) − pen − h kk i ~ Zλ where the integral is along the generator which defines the pencil and the expectation value is taken in the excited state. This is the first λ-dependent term we have in the perturbative expansion of S(λ). Taking two derivatives and evaluating at λ = 0 gives the identity

(0) (1) 00 2π S + S = A Tkk . (5.23) ~ h i 00 Subtracting Sout from both sides of this equation shows that

~ 00 ~ 00 ~ 00 S T = S S(0) S(1) = S(2) + , (5.24) 2π out − h kki 2π out − − 2π ··· A A A where “ ” contains terms higher than quadratic order in σ. The QNEC (equation (5.1)) is the statement··· that this quantity is negative in the limit 0. Earlier we showed that σ was proportional to 1/2. Then S(2) is proportional to A, and → we must check that S(2)00 is negative. However, theA higher order terms S(`) for ` > 2A vanish more quickly with and therefore drop out in the limit 0. A We have shown that the QNECA → reduces to the statement that S(2)00 0 for perturbations from the vacuum. In fact, we have shown something a little stronger.≤ In general, the perturbation σ will have terms proportional to n/2 for all n 1. Our arguments show that only the term proportional to 1/2 mattersA for the QNEC,≥ and furthermore that this term is oﬀ-diagonal in the single-particle/vacuumA subspace. So we can simplify matters by considering states which contain only such a term proportional to 1/2 and no higher powers of . In other words, we can take the state to be of the form inA equation (5.7) with the unwrittenA “ ” terms set equal to zero. Now we only need to show that S(2)00 0 for such states. ··· ≤ CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 92

5.3 Calculation of the Entropy

The replica trick The replica trick prescription is to use the following formula for the von Neumann entropy [45]:

n Sout = tr[ρ log ρ] = (1 n∂n) log tr[ρ ] . (5.25) − − n=1

This can be written as S = log Z˜ (5.26) out D n where6 Z˜ tr[ρn] and the operator is defined by n ≡ D f(n) (1 n∂ )f(n) (5.27) D ≡ − n n=1 ˜ where f(n) is some function of n. Since Zn is only defined for integer values of n, we first must analytically continue to real n > 0 in order to apply the operator. The analytic continuation step is in general quite tricky, and will require careD in our calculation. (Our analytic continuation is performed in section 5.3.) On general grounds discussed above, we must study the second-order term in a perturbative expansion of the entropy about the state ρ(0). Suppressing all λ dependence, we have (0) n Zñ = tr (ρ + σ) . (5.28)

(2)00 Expanding Z˜n to quadratic order to isolate S , we have

− n n 2 Z˜ = tr (ρ(0))n + n tr σ(ρ(0))n−1 + tr (ρ(0))kσ(ρ(0))n−k−2σ + . (5.29) n 2 ··· k=0 X Using the notation introduced in equation (5.15) we can write

− n n 1 Z˜ = tr (ρ(0))n + n tr (ρ(0))n + tr (ρ(0))−k (ρ(0))k (ρ(0))n + . (5.30) n O 2 O O ··· k=1 X We denote by (k) the operator conjugated by (ρ(0))k: O O (k) (ρ(0))−k (ρ(0))k (5.31) O ≡ O = e2πkK e−2πkK . (5.32) O 6 n In the replica trick one often works with the partition function Zn, in terms of which Zñ = Zn/(Z1) . Choosing Zn over Zñ is equivalent to choosing a different normalization for ρ, but we find it convenient to keep tr ρ = 1. CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 93

This is equivalent to a Heisenberg evolution of in the angle θ by an amount 2πk. Since is the integral of operators with angles 0 θ

− n n 1 n2 log Z˜ (k) 2 , (5.35) n ⊃ 2 O O n − 2 hOin k=1 X where we have kept only the part quadratic in . The contribution of the second term to the von Neumann entropy will be proportionalO to , which vanishes because of the tracelessness of σ. Therefore we only need to consider thehOi ﬁrst term. Since we are considering angle-ordered expectation values, we have the identity

n−1 2 n−1 (k) = n (k) , (5.36) O O O n * k=0 ! + k=0 X n X and so from the ﬁrst term in equation (5.35) the relevant part of log Z˜n can be written as

n−1 2 ˜ n 1 (k) log Zn n + . (5.37) ⊃ − 2 hOOi 2 * O ! + Xk=0 n Restoring the λ dependence and taking λ derivatives gives ∂2 S(2)00 = log Z˜ (λ) (5.38) ∂λ2 D n λ=0 2 00 n−1 n 00 1 (k) = − n + . (5.39) D 2 hOOi D 2 * O ! + Xk=0 n 7One could worry that the phase factor in equation (5.10) spoils this relation, but notice that the phase has period 2π in θ and so does not appear when shifting by 2πk. CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 94

00 The ... n notation means take two λ derivatives and then set λ = 0. In the following sectionsh wei will compute these two terms separately. (1) (2) We note that the two terms in equation (5.39) are analogous to δSEE and δSEE of reference [68], where a similar perturbative computation of the entropy was performed. Though the details of the two calculations diﬀer (in particular we have an auxiliary system as well as a CFT), it would be interesting to explore further the connection between our present work and that of reference [68].

Evaluation of same-sheet correlator 00 In this section we consider the term n appearing in equation (5.39). The analytic continuation of this term in n is straightforward.hOOi We first apply : D n n tr e−2πnK [ ] T OO − n = − −2πnK (5.40) D 2 hOOi D 2 tr[e ] = π ∆ K (5.41) − hOO h ii where ∆ K K K is the vacuum-subtracted modular Hamiltonian. When an expectation valueh i ≡ −appears h i without a subscript it is understood to refer to the normalized h· · · i expectation value n with n = 1, i.e., the angle-ordered expectation value with respect to ρ(0). Also noteh· that · · i K appears outside of the angle-ordering in the trace form of the expectation value, which is formally equivalent to being inserted at θ = 0. We now consider the λ dependence. Recall that K is defined to be λ-independent, and the λ-dependence of enters through a shift in the coordinate insertion of ∂Φ (see O equation (5.16)). We first split ∆ K into ∆ K pen and ∆ K aux. The expectation value involving ∆ K will be independenth i of λ becauseh i of translationh i invariance of the CFT, and h iaux so can be ignored. Since Kpen is the CFT boost generator on the half-line x > 0, ∆ K pen has a well-known expression in terms of the energy-momentum tensor of the CFT [156h , i19]: ∞ 1 ∞ ∆ K = dx x T (x) = dx x T (x) . (5.42) h ipen A kk −2π Z0 Z0 Therefore the correlation function in equation (5.41) is expressed in terms of the correlation functions ∂Φ(z λ)∂Φ(w λ)T (x) , which are the same as ∂Φ(z)∂Φ(w)T (x + λ) by translation invariance.h − This makes− the λ-derivativesi easy to evaluate.h We find i n 1 − 00 = T (0) . (5.43) D 2 hOOin 2 hOO i Inserting the explicit form of gives O 1 0 0 (m) (m0) 0 −imθ −im0θ0 T (0) = dr dr dθ dθ f (r)f 0 0 (r )e e hOO i (2π)2 ij i j i,j,i0j0 Z X0 m,m iθ 0 iθ0 0 ∂Φ(re )∂Φ(r e )T (0) E (θ)E 0 0 (θ ) , × h ij i j i D E (5.44) CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 95

where we have introduced Fourier representations of fij(r, θ) deﬁned by

∞ 1 f (r, θ) = f (m)(r)e−imθ . (5.45) ij 2π ij m=−∞ X The correlation functions we need are evaluated in the appendix. Plugging equation (5.104) with n = 1 and equation (5.98) into equation (5.44) yields

T (0) hOO i 0 0 2 dr dr dθ dθ 0 sinh πα 0 0 = − f (m)(r)f (m )(r0)e−π(Ki+Kj ) ij eiθ(−p−m−2)eiθ (p−m −2) (2π)3 (rr0)2 ij ji ip + α i,j,p Z ij m,mX0 0 1 dr dr − − − sinh πα = f (m 2)(r)f ( m 2)(r0)e−π(Ki+Kj ) ij , (5.46) π (rr0)2 ij ji im α i,j,m ij X Z − where we used the Kronecker deltas coming from the θ integration and redefined the dummy variable m m 2, and αij Ki Kj is the difference between two eigenvalues of Kaux. Note that we→ reserve− the letters≡ p and− q throughout to denote integers divided by n, but in this case n = 1 and so p ranges over the integers. Substituting equation (5.46) into equation (5.43), we find

0 n 1 dr dr − − − sinh πα − 00 = f (m 2)(r)f ( m 2)(r0)e−π(Ki+Kj ) ij . (5.47) D 2 hOOin 2π (rr0)2 ij ji im α i,j,m ij X Z − Evaluation of multi-sheet correlator We now turn to the second term in equation (5.39),

00 − 2 1 n 1 (k) . (5.48) 2D * O ! + Xk=0 n The analytic continuation of this term to real n will turn out to be much more challenging than that of the ﬁrst term of equation (5.39), because n appears in the upper summation limit. Using equation (5.16), can write the sum over replicas in equation (5.48) as follows:

2 2 n−1 2πn (k) = dr dθ f (r, θ)∂Φ(r, θ; λ) E (θ) . (5.49) O ij ⊗ ij * ! + * i,j 0 ! + Xk=0 n X Z n This equality comes from interpreting (k) as inserted on the (k + 1)th replica sheet (see equation (5.31)). Summing over sheetsO and integratingO θ [0, 2π] on each one is equivalent ∈ CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 96 to just integrating θ [0, 2πn], which covers the entire replicated manifold. The definition of ∂Φ for angles greater∈ than 2π is given by the the Heisenberg evolution rule, the right hand side of equation (5.10). The field is still holomorphic, but it would be misleading to write it as a function of reiθ since it is not periodic in θ with period 2π. Because the fij(r, θ) are not dynamical, they should be identical on each sheet. In the Fourier representation as in equation (5.45), this means keeping the Fourier coefficients fixed and keeping the m parameters integer. Thus we have

00 n−1 2 1 (k) 1 0 0 (m) (m0) 0 −imθ −im0θ0 = dr dr dθ dθ f (r)f 0 0 (r )e e 2D O D2(2π)2 ij i j * k=0 ! +n i,j,i0,j0 Z X m,mX0 0 0 00 0 ∂Φ(r, θ)∂Φ(r , θ ) E (θ)E 0 0 (θ ) . (5.50) × h in h ij i j in The CFT two point function is calculated in appendix 5.A:

1 w q ∂Φ(z)∂Φ(w) 00 = sign(q)q(q2 1) (5.51) h in n(zw)2 − z |Xq|<1 1 0 = sign(q)P (q, r, r0)eiθ(−q−2)eiθ (q−2) (5.52) n(rr0)2 |Xq|<1 where q takes values in the integers divided by n, and

r0 q P (q, r, r0) q(q2 1) . (5.53) ≡ − r When n = 1 there are no nonzero terms in the sum, but when n > 1 the answer is nonzero. For future convenience, we separated the parts which depend on θ from those that do not. The auxiliary system two point function is calculated in appendix 5.A:

0 0 −2πnKi 1 −ip(θ−θ ) sinh nπαij nπαij Eij(θ)Ei0j0 (θ ) = δij0 δji0 e e e , (5.54) h in ˜aux ip + α πnZn p ij X (0) where p is also an integer divided by n and Z˜aux tr e−2πnKaux is a normalization factor. n ≡ Substituting this equation as well as equation (5.52) intoh equationi (5.50) gives

0 0 1 dr dr dθ dθ 0 0 0 f (m)(r)f (m )(r0)eiθ(−q−p−2−m)eiθ (q+p−2−m ) Dn2(2π)3Z˜aux (rr0)2 ij ji n i,j,p Z m,mX0 sinh πnα ij e−πn(Ki+Kj ) sign(q)P (q, r, r0) . (5.55) × ip + αij |Xq|<1 CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 97

The angle integrations give Kronecker deltas multiplied by 2πn. The result is

0 0 i dr dr (m−2) (−m−2) 0 −πn(Ki+Kj ) sign(q)P (q, r, r ) f (r)f (r ) sinh πnαije D ˜aux (rr0)2 ij ji  q + m + iα  2πZn i,j,m ij X Z |Xq|<1 0  0  i dr dr − − − sign(q)P (q, r, r ) = f (m 2)(r)f ( m 2)(r0) sinh πα e−π(Ki+Kj ) . 2π (rr0)2 ij ji ij D  q + m + iα  i,j,m ij X Z |Xq|<1  (5.56)

In going to the last line, we used the fact that the sum in brackets vanishes when n = 1 d and that, for any two functions f(n), g(n) such that f(1) and dn f(n) n=1 are ﬁnite and g(1) = 0, the following relation holds: (f(n)g(n)) = f(1) g(n) . (5.57) D D We now turn to the analytic continuation and application of on the term in brackets in equation (5.56). We will take care of the awkward sign(q) by writingD the q-dependent part of the sum as two sums with positive argument. We will suppress the (r, r0) dependence for the rest of the calculation: sign(q)P (q) P (q) P ( q) = + − . (5.58) q + m + iα q + m + iα q m iα ij 0

− P (q) P ( q) n 1 P ( k ) P ( k ) + − = n + − n . (5.59) q + m + iα q m iα k k 0

Analytic continuation We need to evaluate

n−1 k k P ( n ) P ( n ) k + k − , (5.60) D n z n + z ! Xk=1 − where P (z) is given by equation (5.52). However, for the remainder of this section, we will consider P (z) to be an arbitrary analytic function whose functional form is independent of n. We will specialize to the form given by equation (5.52) in section 5.3. CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 98

We start by writing the sum in equation (5.60) as

n−1 k k n−1 P ( n ) P (z) P ( n ) P (z) P (z) P (z) k − + − k − + k + k (5.61) D n z n + z ! D n z n + z ! Xk=1 − Xk=1 − and then we evaluate the terms separately. Consider the ﬁrst term in the ﬁrst set of parenthesis. Because P (z) is analytic, we can expand it in a power series with positive powers of ∞ r z: P (z) = r=0 arz . This gives

P n−1 ∞ ( k )r zr a n − . (5.62) D r k r=1 n z Xk=1 X − We can simplify the fraction using polynomial division; for r 1, ≥ − ( k )r zr r 1 k s n − = zr−s−1 , (5.63) k n n z s=0 − X which means the ﬁrst term in the ﬁrst set of parenthesis in equation (5.61) is

− ∞ − − n 1 P ( k ) P (z) r 1 n 1 k s n − = a zr−s−1 . (5.64) D k r D n n z r=1 s=0 Xk=1 − X X Xk=1 The advantage of writing it this way is that it isolates the n dependence into something which can be easily analytically continued. First, recall that overall factors of powers of n don’t matter if the expression they multiply vanishes at n = 1, as in equation (5.57). Next, note that the resulting expression is actually a polynomial in n. It can be expressed this way using Faulhaber’s formula:

− n 1 1 s s + 1 ks = ( 1)j B (n 1)s−j+1 , (5.65) s + 1 − j j − j=0 Xk=1 X

where Bs is the j-th Bernoulli number in the convention that B1 = 1/2. This makes application of straightforward: − D − − n 1 k s n 1 = ks = ( 1)sB . (5.66) D n D − − s Xk=1 Xk=1 Thus for the ﬁrst term in equation (5.61) we have

n−1 P ( k ) P (z) ∞ r−1 n − = a zr−s−1( 1)sB . (5.67) D k − r − s n z r=1 s=0 Xk=1 − X X CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 99

The second term follows completely analogously:

n−1 P ( k ) P (z) ∞ r−1 − n − = a zr−s−1B . (5.68) D k r s n + z r=1 s=0 Xk=1 X X Combining these results, the ﬁrst set of large parenthesis in equation (5.61) is

∞ r−1 a zr−s−1B [( 1)s 1] . (5.69) − r s − − r=1 s=0 X X

For even s this is zero. For odd s > 1, Bs = 0, and so only s = 1 can contribute. Substituting B = 1/2 gives 1 − P (z) a a + 1 + 0 . (5.70) − z2 z z2 We now turn to the second set of parenthesis in equation (5.61). These two terms can be evaluated simultaneously. First, we can multiply through by n/n to give an overall factor of n (which is irrelevant) and convert the denominators to k zn and k + zn. We also pull P (z) through because it is independent of n: − D − n 1 1 1 P (z) + . (5.71) D k zn k + zn Xk=1 − This sum can be evaluated in terms of the digamma function ψ(0)(w), which is deﬁned in terms of the Gamma function Γ(w):

∞ Γ0(w) 1 1 ψ(0)(w) = γ + . (5.72) ≡ Γ(w) − k + 1 − k + w Xk=0 By manipulating the sum, one can show

− n 1 1 = ψ(0)(n w) ψ(0)(1 w) . (5.73) k w − − − Xk=1 − Thus the second set of parenthesis in equation (5.61) is equal to

P (z) ψ(0)(n zn) ψ(0)(1 zn) + ψ(0)(n + zn) ψ(0)(1 + zn) . (5.74) D − − − − We cannot naively apply yet. We ﬁrst have to select the correct analytic continuation to real positive n from the manyD possible analytic continuations of integer n data. This CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 100

is known to be a challenging problem in general.8 Nevertheless, in our context the correct analytic continuation prescription is clear. The digamma function has poles in the complex plane at zero and all negative real integers. Recall that we are ultimately interested in plugging in zm m iαij. Thus if we are not careful, for certain values of m, the digamma functions in≡ equation− − (5.74) will blow up when α 0 near n = 1. On the other hand, on physical grounds we expect our ij → result to be perfectly well-behaved when αij 0, which simply corresponds to a degeneracy in the auxiliary system. The way we avoid→ the poles of the digamma function near n = 1 when α 0 is by using the reflection formula ij → ψ(0)(1 w) = ψ(0)(w) + π cot πw , (5.75) − which produces different analytic continuations given the same integer data. These observa- tions lead to the following prescription: for each value of m, use the reflection formula (5.75) to avoid the poles of the digamma function near n = 1 as αij 0. As an example, consider the term ψ(0)(1 z n) = ψ(0)(1+→mn+α n) in equation (5.74). − m ij When αij = 0, this has a pole when nm 1. Thus for a given m 1, we cannot expect to have a smooth n-derivative at n = 1. The≤ resolution is to use equation≤ (5.75) to get

ψ(0)(1 + mn + α n) = ψ(0)( mn α ) π cot π(mn + α ) (5.76) ij − − ij − ij = ψ(0)( mn α ) π cot πα , (5.77) − − ij − ij where the last equality is only true for integer n. The remaining digamma term is now free of poles for mn 1, which is precisely when there was a problem before the application of the reflection≤ formula, and can now be easily applied. This example illustrates how the correct analytic continuationD depends on the value of m. We must apply this reasoning separately to each term in equation (5.74). After applying this procedure to each digamma function as needed to avoid the poles, it will turn out that all of the extra cotangent terms cancel against each other. There is another way to motivate this prescription. Even for small but finite αij, the analytic continuations picked out by our prescription can be seen to be qualitatively better than the one obtained by using equation (5.74) directly, as illustrated in figure 5.3. Notice that while both curves match for integer n, the curve obtained by applying the prescription outlined above is the only one which smoothly interpolates between the integers. The oscillations of the “wrong” curves get larger and larger as αij is reduced or m is increased. Applying our prescription to equation (5.74), there are three expressions depending on 8See reference [68] for a recent discussion of the difficulties of the analytic continuation. Reference [68] also contains another method for computing the entropy perturbatively that does not rely on the replica trick. Such a method avoids the need to analytically continue, and applying it to the present calculation would serve as a check of our analytic continuation prescription. We leave that check to future work. CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 101

Αij = 1 Αij = 0.5 Αij = 0.1

n n n 1 2 3 1 2 3 1 2 3

Figure 5.3: Sample plots of the imaginary part (the real part is qualitatively identical) of the na¨ıve bracketed digamma expression in equation (5.74) and the one in equation (5.78) obtained from analytic continuation with z = m iα for m = 3 and various values − − ij of αij. The oscillating curves are equation (5.74), while the smooth curves are the result of applying the speciﬁed analytic continuation prescription to that expression, resulting in equation (5.78).

the value of m. We are focussing on the quantity in brackets in equation (5.74):

(0) (0) (0) (0) ψ (1 n nzm) ψ ( nzm) + ψ (n nzm) ψ (1 nzm) m > 0 (0) − − (0)− − (0) − −(0) − ψ (n + nzm) ψ (1 + nzm) + ψ (n nzm) ψ (1 nzm) m = 0 (5.78) (0) − (0) (0) − − (0)− ψ (n + nzm) ψ (1 + nzm) + ψ (1 n + nzm) ψ (nzm) m < 0 − − − Now we are ready to apply . The digammas ψ(0)(w) will turn into polygammas ψ(1)(w) d (0) D ≡ dw ψ (w), which obey the recurrence relation 1 ψ(1)(w + 1) = ψ(1)(w) . (5.79) − w2 This recurrence relation simplifies the result for m > 0 and m < 0 while the recurrence relation along with the reflection formula simplifies the result for m = 0. The result for the second set of parenthesis in equation (5.61) with z = zm is

2 P (zm) π 2 + δ(m)P (zm) 2 . (5.80) zm sinh παij We are now ready to give the ﬁnal expression for equation (5.60). Adding equation (5.70) with z = zm and equation (5.80) we ﬁnd

− n 1 P ( k ) P ( k ) a a π2 n − n 1 0 k + k = + 2 + δ(m)P ( iαij) 2 (5.81) D n zm n + zm ! zm zm − sinh παij Xk=1 − for arbitrary analytic P (zm). CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 102

Completing the proof Now we specialize to the form of P (z) needed for our calculation which came from the particular ∂Φ∂Φ 00 two-point function we were computing (equations (5.52) and (5.53)): h i 0 P (z) = z(z2 1)ez log (r /r) . (5.82) − Thus a = 0, and a = 1. Using equation (5.81) gives 0 1 −

n−1 k k 2 0 −iαij P ( n ) P ( n ) i iπ 2 r k + k − = + δ(m) 2 αij(αij + 1) . (5.83) D n zm n + zm ! im αij sinh παij r Xk=1 − − Plugging this into equation (5.56) and plugging that into equation (5.49) gives the term from equation (5.39) that we have been focussing on in this section:

2 00 n−1 0 1 1 drdr − − − (k) =− f (m 2)(r)f ( m 2)(r0) sinh πα e−π(Ki+Kj ) D2 O 2π (rr0)2 ij ji ij * ! + i,j,m Xk=0 n X Z 1 α r0 −iαij + δ(m) ij π2(α2 + 1) . (5.84) × im α sinh2 πα ij r " − ij ij # Notice that the first term in this expression exactly cancels the contribution to S(2)00 coming from the first term in equation (5.39), presented in equation (5.47). We now consider −π(Ki+Kj ) 2 2 the second term, and define the manifestly positive quantity Mij e π (αij + 1) to clean up the notation. Then we have ≡

0 0 −iαij 00 1 drdr r α S(2) = − f (−2)(r)f (−2)(r0) ij M . (5.85) 2π (rr0)2 ij ji r sinh πα ij i,j ij X Z The integrals over r, r0 factorize, giving

∞ ∞ 00 1 − − α S(2) = − dr riαij −1f ( 2)(r) dr r−iαij −1f ( 2)(r) ij M . (5.86) 2π ij ji sinh πα ij i,j 0 0 ij X Z Z Recall the constraint on the test functions derived previously by requiring the density matrix ∗ be Hermitian (equation (5.11)): fij(r, θ) = fji(r, 2π θ) . In Fourier space, this implies (m) (m) ∗ − fji (r) = fij (r) . Inserting this into equation (5.86) we see that the factors in brackets are complex-conjugates of each other. Furthermore, because sinh παij always has the same sign as αij, the overall sign of the entire term is negative and so we ﬁnd

S(2)00 0 . (5.87) ≤ As discussed after equation (5.24), this proves the QNEC. CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 103

5.4 Extension to D = 2, Higher Spin, and Interactions

In D = 2, there are no transverse directions, and so it is not possible to use the fact that the state is very close to the vacuum. Nevertheless, once one has proven the QNEC for a free scalar field in D > 2, one can use dimensional reduction to prove it for free scalar fields in D = 2. Let Φ(z, y) be the chiral scalar on N in D > 2, where y labels the D 2 transverse coordinates. One can isolate a single transverse mode by integrating Φ(z, y)− against a real transverse wavefunction, and this defines an effective two-dimensional field:

Φ (z) dy ψ(y)Φ(z, y) , (5.88) 2D ≡ Z 2 where ψ is normalized such that ψ = 1. Correlation functions of Φ2D and its derivatives exactly match those of a two-dimensional chiral scalar, and so our dimensional reduction is R defined by the subspace of the D-dimensional theory obtained by acting on the vacuum with Φ2D. In any such state, one can integrate the D-dimensional QNEC along the transverse direction to find 1 δ2S dy T (y) dy out . (5.89) h kk i ≥ 2π δλ(y)2 Z Z Here we have suppressed the value of the affine parameter as a function of the transverse direction. The effective two-dimensional change in the entropy is defined by considering a total variation in all of the generators which is uniform in the transverse direction. For such a variation we have δ2S δ2S S00 = dy dy0 out dy out , (5.90) 2D δλ(y)δλ(y0) ≤ δλ(y)2 Z Z where the the inequality comes from applying strong subadditivity to the off-diagonal second derivatives [36]. The two-dimensional energy momentum tensor is defined in terms of the normal ordered product of the two-dimensional fields, T2D =: ∂Φ2D∂Φ2D :. However, using Wick’s theorem one can easily check that T2D acts on the dimensionally reduced theory in the same way as the integrated D-dimensional Tkk:

T (w)Φ (z ) Φ (z ) = dy T (w, y)Φ (z ) Φ (z ) . (5.91) h 2D 2D 1 ··· 2D n i h kk 2D 1 ··· 2D n i Z Therefore the QNEC holds for a free scalar field in two dimensions: 1 δ2S 1 T = dy T (y) dy out S00 . (5.92) h 2Di h kk i ≥ 2π δλ(y)2 ≥ 2π 2D Z Z The extension to bosonic fields with spin is trivial, as these simply reduce on N to multiple copies of the 1+1 chiral scalar CFT, one for each polarization. These facts are reviewed in [164]. Similarly, fermionic fields reduce to the chiral 1+1 fermion CFT; we expect that there is a similar proof in this case. CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 104

Astute readers may have noticed that the mass term of the higher dimensional field theory plays no role in our analysis. This phenomena occurs because the mass term contribute to neither the commutation relations on N nor to the null-null component of the stress tensor Tkk. Regardless of whether the D dimensional theory has a mass, the 1 + 1 chiral theory is massless. In a sense, null surface quantization is a UV limit of the field theory. One might therefore expect that the addition of interactions with positive mass dimension (superrenormalizable couplings) will also not change the algebra of observables on N. So long as this is the case, the extension to theories with superrenormalizable interactions is trivial. One argument that superrenormalizable interactions are innocuous proceeds in two stages [164]. First, one considers the direct effects of adding interaction terms to the Lagrangian; for example a scalar field potential V (φ). So long as these interaction terms contain no derivatives (or are Yang-Mills couplings), they do not contribute to the commutation relations of fields restricted to the null surface, or to Tkk. (So far, the interaction could be of any scaling dimension, so long as one avoids derivative couplings.) Next, one considers loop corrections due to renormalization. In the case of a marginally renormalizable, or nonrenormalizable theory, these loop corrections normally require the addition of counterterms containing derivatives (for example, field strength renormalization), spoiling the null surface formulation. On the other hand, in a superrenormalizable theory, only couplings with positive mass dimension require counterterms. For a standard QFT consisting of scalars, spinors, and/or gauge fields, none of these superrenormalizable interactions include the possibility derivative couplings. Thus one expects that loop corrections do not spoil the algebra of observables on the null surface. However, superrenormalizable theories are difficult to construct except when D < 4. (For example, the φ3 theory is superenormal- izable in D < 6, but is unstable.) It is an open question whether the QNEC is valid for non-Gaussian D = 2 CFT’s in states besides conformal vacua, or more generally for QFT’s in any dimension which flow to a nontrivial UV fixed point.9 Nor have we carefully considered the effects of making the scalar field noncompact. QCD in D = 4 is a borderline case; the coupling flows to zero, but slowly enough that there is an infinite field strength renormalization. Strictly speaking this makes null surface quantization invalid, yet it is still a useful numerical technique for studying hadron physics [41]. However, we conjecture that the QNEC will be true in every QFT satisfying reasonable axioms.

9In more than 2 dimensions, interacting CFTs appear to have no nontrivial observables on the horizon[164, 37], so the current proof cannot be extended to this situation. 105

Appendix

5.A Correlation Functions

Scalar field The chiral scalar operator ∂Φ(z) is a conformal primary of dimension (h, h¯) = (1, 0). Its two point function on the Euclidean plane is fixed by conformal symmetry up to an overall constant. We will take the following normalization: 1 ∂Φ(z)∂Φ(w) = − . (5.93) h i (z w)2 − The two point function on the n-sheeted replicated manifold is obtained by application of the conformal transformation z zn: → 1 (zw)1/n ∂Φ(z)∂Φ(w) = − . (5.94) h in n2zw (z1/n w1/n)2 − The second-derivative of this two point function under translations of the holomorphic coordinate, evaluated at λ = 0, is defined by ∂Φ(z λ)∂Φ(w λ) 00 = ∂3Φ(z)∂Φ(w) + ∂Φ(z)∂3Φ(w) + 2 ∂2Φ(z)∂2Φ(w) . h − − in n n n (5.95)

One can show that this combination of correlation functions can be written as 1 w q sign(q)q(q2 1) , (5.96) n(zw)2 − z |Xq|<1 where q is an integer divided by n. Notice that this implies that the sum vanishes for n = 1, as required by translation invariance. Our convention for the only nonzero component of the stress tensor for the holomorphic sector of the theory is 1 T (z) = 2πT (z) = : ∂Φ(z)∂Φ(z): , (5.97) − zz −2 where : AB : denotes the normal-ordered product. Thus using Wick’s theorem we have 1 ∂Φ(z)∂Φ(w)T (0) = − . (5.98) h i (zw)2 CHAPTER 5. PROOF OF THE QUANTUM NULL ENERGY CONDITION 106

Auxiliary system In this appendix we will evaluate the θ-ordered correlation functions of the auxiliary system,

−2πnKaux 0 0 tr e [Eij(θ)Ei0j0 (θ )] Eij(θ)Ei0j0 (θ ) = T . (5.99) n −2πnKaux h i tr [e ] First, consider the case θ > θ0:

0 −2πnKaux 0 −2πnKi (θ−θ )αij tr e Eij(θ)Ei0j0 (θ ) = e e δij0 δji0 , (5.100)

0 where αij Ki Kj is the diﬀerence in two of the eigenvalues of Kaux. For θ < θ , we have the opposite≡ ordering− inside the expectation value, which gives

0 −2πnKaux 0 −2πnKi (θ−θ +2πn)αij tr e Ei0j0 (θ )Eij(θ) = e e δij0 δji0 (5.101)

0 We will ﬁnd it convenient to use the following complex exponential representation of e(θ−θ )αij , valid for θ θ0 (0, 2πn): − ∈

0 1 0 sinh nπα e(θ−θ )αij = e−ip(θ−θ ) ij enπαij . (5.102) πn ip + α p ij X Here p is being summed over all rational numbers which are integers divided by n. This can be substituted directly into equation (5.100). For the expectation value when θ < θ0 given by equation (5.101), we can take θ θ0 + 2πn as our Fourier series variable instead of θ θ0, which also lies in (0, 2πn) in this− case. This means we can substitute this into equation− (5.102), giving the same complex exponential representation:

0 1 0 sinh nπα e(θ−θ +2πn)αij = e−ip(θ−θ ) ij enπαij . (5.103) πn ip + α p ij X Collecting these results, the θ-ordered correlation function in the auxiliary system is simply

0 0 −2πnKi 1 −ip(θ−θ ) sinh nπαij nπαij Eij(θ)Ei0j0 (θ ) = δij0 δji0 e e e , (5.104) h in ãux ip + α πnZn p ij X where Zãux tr e−2πnKaux . (5.105) n ≡ ãux Note that Z1 = 1. 107

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