Entropy Bounds and Entanglement - University of California ......Illan Halpern, Adam Levine, Arvin Moghaddam, Mudassir Moosa, Vladimir Rosenhaus and Claire Zukowski. Finally, I would

Entropy Bounds and Entanglement

by

Zachary Fisher

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Physics

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor Raphael Bousso, ChairProfessor Yasunori Nomura

Professor Nicolai Reshetikhin

Spring 2017


Copyright 2017by

Zachary Fisher

1

Abstract


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 onthe entropy of a region of space in a gravitational theory. It bounds the entropy passingthrough certain null surfaces. The bound remains nontrivial in the weak-gravity limit, andprovides non-trivial constraints on the entropy of ordinary quantum states even in a regimewhere gravity is negligible.

In the first half of this thesis, we present a proof of the Bousso bound in the weak-gravityregime within the framework of quantum field theory. The bound uses techniques fromquantum information theory which relate the energy and entropy of quantum states. Wepresent 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 quantumfocussing conjecture. Our conjecture is a bound on the rate of entropy generation in a quan-tum field theory coupled semiclassically to gravity. The conjecture unifies and generalizesseveral ideas in holography. In particular, the quantum focussing conjecture implies a boundon entropies which is similar to, but subtly different from, the Bousso bound proven in thefirst 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-nullcomponent of the stress tensor of quantum matter. We give a proof of this bound in thecontext of free and superrenormalizable bosonic quantum field theory.

i

For Melanie, Dennis, Jeremy and Laura.

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Contents

Contents ii

List of Figures iv

1 Introduction 11.1 The Holographic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Bousso Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Holography and Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . 51.4 Entropy, Energy and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 The Bousso Bound in Free Quantum Field Theory 92.1 Regulated Entropy ∆S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Proof that ∆S ≤ ∆ 〈K〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Proof that ∆ 〈K〉 ≤ ∆A/4GN~ . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.A Monotonicity of ∆A(c,b)

4GN~−∆S . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 The Bousso Bound in Interacting Quantum Field Theory 223.1 Entropies for Null Intervals in Interacting Theories . . . . . . . . . . . . . . 253.2 Bousso Bound Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Holographic Computation of ∆S for Light-Sheets . . . . . . . . . . . . . . . 333.4 Why is ∆S = ∆ 〈K〉 on Null Surfaces? . . . . . . . . . . . . . . . . . . . . . 373.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.A Extremal Surfaces and Phase Transitions on a Black Brane Background . . . 433.B Toy Model with ∆ 〈K〉 = ∆S 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . 49

4 The Quantum Focussing Conjecture 534.1 Classical Focussing and Bousso Bound . . . . . . . . . . . . . . . . . . . . . 564.2 Quantum Expansion and Focussing Conjecture . . . . . . . . . . . . . . . . . 584.3 Quantum Bousso Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.4 Quantum Null Energy Condition . . . . . . . . . . . . . . . . . . . . . . . . 664.5 Relationship to Other Works . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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

5 Proof of the Quantum Null Energy Condition 825.1 Statement of the Quantum Null Energy Condition . . . . . . . . . . . . . . . 865.2 Reduction to a 1+1 CFT and Auxiliary System . . . . . . . . . . . . . . . . 875.3 Calculation of the Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.4 Extension to D = 2, Higher Spin, and Interactions . . . . . . . . . . . . . . . 1035.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 pointswith b(x⊥) ≤ x+ ≤ c(x⊥). (b) The light-sheet can be viewed as the disjoint unionof small transverse neighborhoods of its null generators with infinitesimal areas{Ai}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Operator algebras associated to various regions. (a) Operator algebra associatedto the domain of dependence (yellow) of a space-like interval. (b) The domain ofdependence of a boosted interval. (c) In the null limit, the domain of dependencedegenerates 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 comparedto the affine distance over which the area changes by a factor of order unity. Theentropy 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 defectoperators D inserted at the endpoints of the interval. An operator in the ith CFTbecomes 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 nullslab, for conformal field theories with a bulk dual. Here d = 2, 3, 4, 8,∞ frombottom to top. Near the boundaries (v → 0, v → 1), we find g → 0, g′ → ±1, inagreement with the modular Hamiltonian of a Rindler wedge. We also note thatthe functions are concave. In particular, we see that |g′| ≤ 1, in agreeement withour general argument of section 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Operator algebras associated to various regions. (a) Operator algebra associatedto the domain of dependence (yellow) of a space-like interval. (b) The domain ofdependence of a boosted interval. (c) In the null limit, the domain of dependencedegenerates to the interval itself. . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.A.1The maximum value Emax(p) of E for getting a surface that returns to the bound-ary (solid line). For comparison, the line E = p − 1 is plotted (the dashedline). 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 thelogarithmic 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 thiscontour, the boundary interval is time-like. If ∆x+ & 15 and we follow a contourof constant ∆x+, we find two solutions with exact ∆x− = 0. For all contours offixed ∆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 demonstratesthat, for sufficiently large ∆x+ (in d = 3, the condition is ∆x+ & 15), there existsa phase transition at finite ∆x− to a different, perturbative class of solutions. Atsmaller ∆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. Thegeneralized 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 quantumexpansion Θ at σ, we erect an orthogonal null hypersurface N , and we considerthe response of Sgen to deformations of σ along N . (b) More precisely, N canbe divided into pencils of width A around its null generators; the surface σ isdeformed an affine parameter length � along one of the generators, shown in green. 60

4.2 (a) For an unentangled isolated matter system localized to N , the quantumBousso bound reduces to the original bound. (b) With the opposite choice of“exterior,” one can also recover the original entropy bound, by adding a distantauxiliary system that purifies the state. . . . . . . . . . . . . . . . . . . . . . . . 64

4.3 (a) A portion of the null surface N , which we have chosen to coincide with Σoutin the vicinity of the diagram. The horizontal line at the bottom is the surfaceV (y), and the orange and blue lines represent deformations at the transverselocations y1 and y2. The region above both deformations is the region outside ofV�1,�2(y) and is shaded beige and labeled B. The region between V (y) and V�1(y)is labeled A and shaded lighter orange. The region between V (y) and V�2(y)islabeled C and shaded lighter blue. Strong subadditivity applied to these threeregions proves the off-diagonal QFC. (b) A similar construction for the diagonalpart of the QFC. In this case, the sign of the second derivative with respect tothe 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 Soutof the yellow region. The quantum expansion Θ at one point of Σ is the rateat which Sgen changes under a small variation dλ of Σ, per cross-sectional areaA of the variation. The quantum focussing conjecture states that the quantumexpansion cannot increase under a second variation in the same direction. If theclassical expansion and shear vanish (as they do for the green null surface in thefigure), the quantum null energy condition is implied as a limiting case. Ourproof involves quantization on the null surface; the entropy of the state on theyellow space-like slice is related to the entropy of the null quantized state on thefuture (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 Euclideanplane. 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) ofthe näıve bracketed digamma expression in equation (5.74) and the one in equa-tion (5.78) obtained from analytic continuation with z = −m−iαij for m = 3 andvarious values of αij. The oscillating curves are equation (5.74), while the smoothcurves are the result of applying the specified analytic continuation prescriptionto that expression, resulting in equation (5.78). . . . . . . . . . . . . . . . . . . 101

vii

Acknowledgments

A journey of this magnitude cannot be undertaken alone. First and foremost, I wouldlike to thank my advisor, Raphael Bousso. His brilliance and leadership made my fiveyears at Berkeley some of the most intellectually challenging and fulfilling of my life. I oweanother 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 theopportunity to work closely with these outstanding scientists.

Of course, I cannot forget to acknowledge the central role played by my instructors andmentors through the years. I would especially like to thank Miles Chen, Isaac Chuang, TomHenning, Petr Hořava, Holger Müller, Yasunori Nomura, Nicolai Reshetikhin, Andrew Shawand Barton Zwiebach. I could not have reached this point without their leadership andencouragement.

I would furthermore like to thank the many people who have encouraged me, laughed withme and taught me over these five years, my dear friends and colleagues. Foremost amongthem I would like to thank Netta Engelhardt, Chris Mogni, Ben Ponedel, Fabio Sanchez, SeanJason Weinberg and Ziqi Yan. I would also like to acknowledge the influence and supportof the other members of the Bousso group: Christopher Akers, Venkatesh Chandrasekaran,Illan Halpern, Adam Levine, Arvin Moghaddam, Mudassir Moosa, Vladimir Rosenhaus andClaire 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 gratefulto them all.

1

Chapter 1

Introduction

In the last century, much time and effort has been expended on the problem of quantizinggravity. There is now widespread agreement in the community that the problem of quan-tum gravity will require radically new physical ideas and principles. Indeed, a completeunderstanding of quantum gravity still eludes us today.

However, an excellent candidate for such a framework is string theory. String theory isa beautiful and self-consistent theory from which gravity arises naturally. It confirms manyof our expectations about how quantum gravity should work. Unfortunately, even with thispowerful tool at our disposal, many of the most important questions about quantum gravityremain open. One of the remaining questions is to understand what features of quantumgravity are visible at low energies, where effective field theories agree with experiment toexcellent precision.

Therefore, we must study the qualitatively new features of quantum gravity. One of themost 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 blackhole thermodynamics. Therefore, we begin this thesis with a brief tour of the holographicprinciple.

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 thehorizon of a black hole is a point of no return, a black hole is an entropy sink for anythingthat falls inside it. However, the second law of thermodynamics prohibits entropy fromdecreasing in a closed system, such as the exterior of a black hole. For example, a scrambledegg can surely never unscramble, but merely by throwing the egg into a black hole, the eggis no longer accessible and the entropy goes to zero! Bekenstein posited that the way toavoid 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:

SBH ∝ A . (1.1)We can now understand heuristically how the paradox might be resolved: when the egg istossed 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 thereforeconceivably 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 areaof a black hole horizon can only increase with time:

dA

dt≥ 0 . (1.2)

For example, the area of a black hole formed from the merger of two black holes is greaterthan the sum of the areas before the merger. By comparing equations (1.1) and (1.2), wesee that the second law of thermodynamics automatically holds in a universe consisting ofjust black holes.

Soon thereafter, in a groundbreaking paper [88], Hawking fixed the proportionality con-stant in equation (1.1). Hawking’s calculation used the framework of quantum field theoryon 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 entropyby the first law of thermodynamics, dS = dE/T . Thus the constant in equation (1.1) wasfixed1:

SBH =A

4GN~. (1.3)

Black holes radiate away their energy in the same way as any other thermal object. SinceTBH ∝ GN~, the effect is a prediction of quantum gravity, which disappears in the classicallimit ~→ 0.

Collecting these results, it is possible to write down a well-motivated definition of thetotal 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 theblack hole. The resulting quantity is called the generalized entropy:

Sgen =A

4GN~+ Smatter outside . (1.4)

The conjecture that Sgen is non-decreasing with time in a semiclassical theory is called thegeneralized 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 remainexplicit unless otherwise specified.

2The name is something of a misnomer, since the ordinary second law of thermodynamics only holdsif one includes every physical source of entropy; the only generalization made is assuming that black holescontribute some entropy.


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

1.2 The Bousso Bound

A particularly important holographic bound was conjectured by Bousso [25]. The covariantentropy bound relates matter entropy to the area of arbitrary surfaces, not just black holehorizons. The bound is formulated in terms of light-sheets. A light-sheet is a null surfacewhose null generators are everywhere converging. We will now introduce some importantterminology. Denote the infinitesimal area element between null generators by A, and definean affine parameter λ for the congruence. The (classical) expansion scalar is defined as thelogarithmic derivative of A with respect to λ:

θ ≡ 1AdAdλ

. (1.5)

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 terminatethe null generator there. For example, the past lightcone of a point in Minkowski space is alight-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 boundaryA = ∂B. Some of these congruences will be light sheets. Allow the light-sheet to terminatewhen the generators reach caustics. The Bousso bound states that the entropy S whichcrosses through the light-sheet is bounded by the area of the boundary A:

S ≤ Area[A]4GN~

. (1.6)

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,


i.e. before reaching a caustic, landing on a codimension-2 surface A′. Then the bound saysthat the entropy crossing through the prematurely-terminated light-sheet is bounded by thedifference of the areas ∆A = Area[A]− Area[A′]:

S ≤ Area[A]− Area[A′]

4GN~(1.7)

Formally, this bound is called the generalized covariant entropy bound. Following commonparlance, we will take the stronger statement in equation (1.7) to be our working definitionof the Bousso bound.

Fundamentally, the Bousso bound is a conjecture. It might capture aspects of howspacetime and matter arise from a more fundamental theory [29, 31]. A general proof maynot become available until such a theory is found. Nevertheless, it is of interest to provethe bound at least in certain regimes, or subject to assumptions that hold in a large class ofexamples.

In this spirit, the Bousso bound in equation (1.7) has been shown to hold in settingswhere the entropy S can be approximated hydrodynamically, as the integral of an entropyflux over the light-sheet; and where certain assumptions constrain the entropy and energyfluxes [72, 35]. These assumptions apply to a large class of spacetimes, such as cosmology orthe gravitational collapse of a star. Thus they establish validity of the bound in some broadregimes.

However, the underlying assumptions in these earlier proofs have no fundamental status.Unlike the stress tensor, entropy is not local, so the hydrodynamic approximation breaksdown 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 witha Gaussian profile propagating through otherwise empty flat space. In order to obtain thetightest bound, we may take the light-sheet to have initially vanishing expansion. Thedifference 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 findsthat 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 satisfiedfor 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 capturedon the light-sheet, but how can this be quantified? (In field theory, the entropy in a finiteregion would be dominated by vacuum entropy across the initial and final surface, and hencelargely unrelated to the photon.) Worse, for short light-sheets, there is no intuitive notionof 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 thegeneralized covariant bound in the weak-gravity limit [30]: precisely in the regime where the bound becomestight, one lacks a sharp definition of entropy.


This issue is resolved in chapters 2 and 3 in a novel way. We use tools from quantuminformation theory and quantum field theory to prove the Bousso bound in a weak-gravitylimit. We will now expand further on how these tools can be used to prove holographicbounds of this type.

1.3 Holography and Quantum Field Theory

The generalized second law and the Bousso bound are physically reasonable expectations ofa theory of quantum gravity. However, these entropy bounds can be reexpressed and un-derstood within ordinary (non-gravitational) quantum field theory. This is possible becausethese 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 matrixρ. Often, we will consider the density matrix for the degrees of freedom localized inside somespatial region A and we will denote the state as ρA for clarity. The state ρA is related to theglobal state ρ by tracing out the degrees of freedom localized outside A: ρA = trH−A ρ.

The entropy we will bound is the von Neumann entropy4 SA associated to the region A.It is given in terms of ρA via the formula

SA = − tr[ρA log ρA] . (1.8)In any quantum theory, the von Neumann entropy satisfies a number of important equal-

ities and inequalities. The most important among these is strong subadditivity, which saysthat given density matrices with support on three disjoint regions A,B,C,

S(ρABC) + S(ρB) ≥ S(ρAB) + S(ρBC) . (1.9)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 entropyin an � expansion scales like the area5:

SA =k(d−2)�d−2

+k(d−4)�d−4

+ · · ·+ finite, where k(d−2) ∝ Area[∂A] . (1.10)

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 sameHilbert 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 mis-

leading. There can be contributions to the von Neumann entropy that arise from classical uncertainty, forexample 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.


where σ is some fiducial state which one usually takes to be the vacuum state. Relativeentropy is an asymmetric measure of the distance between the two density matrices in theHilbert 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)

where

K ≡ − log σ (1.13)

is called the modular Hamiltonian and

∆ 〈K〉 = − tr[ρ log σ − σ log σ] = 〈K〉ρ − 〈K〉σ (1.14)∆S = − tr[ρ log ρ− σ log σ] = Sρ − Sσ (1.15)

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, wehave 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 modularHamiltonian of a half-space takes a simple universal form [19]. The modular Hamiltonian isproportional to the generator of spacetime boosts which leaves the boundary invariant. Forexample, the modular Hamiltonian of the region A = {x|x0 = 0, x1 > 0} is7

KA = 2π

∫ ∞

0

dx1∫dd−2x⊥ x

1 T00 . (1.16)

This expression is remarkable for many reasons: it is universal for any field theory; it involvesonly local operators, in fact only the stress tensor; and it provides a direct connection betweenenergy and entropy.

Relative entropy also obeys a number of important properties. For example, a simplecalculation shows that relative entropy is always positive. A more involved calculation isrequired to show that relative entropy is monotonic under inclusion; that is, given disjointregions A,B,

S(ρAB||σAB) ≥ S(ρA||σA) . (1.17)The meaning of this inequality is that more operators are available in the region AB todistinguish 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 themodular Hamiltonian expectation value and the von Neumann entropy [124]. Such examples do not applyto the von Neumann entropy of null surfaces in the interacting proof. More generally, the state-dependentdivergences will contribute equally to the modular Hamiltonian and the entropy. We can then circumventthe issue of divergences by modifying the regularization scheme.

7This expression is valid up to a constant (divergent) factor which drops out of ∆ 〈KA〉.


These properties of relative entropy, positivity and monotonicity, are remarkably power-ful. They interrelate the energy content of a region of spacetime with its entropy, providingconstraints. One may ask whether these bounds are related to the holographic bounds of theprevious section. Indeed, in a beautiful 2008 paper, Casini [51] showed that a holographicbound 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 ofthe generalized second law as a statement about monotonicity [164]. That proof applies forany 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 technologyas well. Chapters 2 and 3 of this thesis will prove the Bousso bound in weakly gravitatingsystems, using relative entropy and properties of quantum field theory. These proofs werefirst presented in [38, 37]. Chapter 2 presents the proof in free and superrenormalizable fieldtheory, where the technique of null quantization is employed to simplify the analysis. Theproof is highly nontrivial and implies counterintuitive properties of entropies on null surfaces.Chapter 3 presents the proof in field theories with nontrivial interactions. The von Neumannentropy exhibits some counterintuitive properties in this context which we will explore anduse 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 boundon entropy. This arises from the connection between energy and entropy comes from blackhole thermodynamics, and the connection between geometry and energy from Einstein’sequation. Entropy, energy and geometry are intimately related by the holographic principle.

We will explore these connections further in the second half of this thesis. Chapter4 presents a novel conjecture for quantum fields and semiclassical gravity: the quantumfocussing conjecture (QFC). This conjecture is a strengthening of the Bousso bound into aform similar to the generalized second law. In short, the generalized second law states thatthe first derivative of the generalized entropy is positive; the quantum focussing conjecturestates that the second derivative of generalized entropy is positive. We conjecture that theQFC holds even when the generalized entropy is evaluated not just for black holes, but forany arbitrary surface in the spacetime that divides a Cauchy surface into an interior and anexterior. The QFC was first presented in [36].

Intriguingly, there is a close relationship between the QFC and the positivity of energydensities in classical physics. In classical physics, one typically assumes the null energycondition (NEC). The null energy condition states that Tkk ≡ Tabkakb ≥ 0, where Tab is thestress tensor and ka is a null vector. This condition is satisfied by physically realistic classicalmatter 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].


However, quantum fields can potentially violate all local energy conditions, including theNEC [66]. The energy density 〈Tkk〉 at any point can be made negative, with magnitudeas large as we wish, by an appropriate choice of quantum state. An example of a regionwhere the null energy condition is violated is the horizon of an evaporating black hole. In astable 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, butmust be nonlocal. For example, a total energy may be obtained by integrating an energydensity over all of space; an “averaged null energy” is defined by integrating 〈Tkk〉 alonga null geodesic [24, 163, 113, 159, 84, 93]. Some field theories have been shown to satisfyquantum 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 earlierproofs of the Bousso bound [72, 35]. There are realistic quantum states to which these proofsdo not apply. Indeed, a desirable feature of the proof of the Bousso bound in chapters 2 and3 is that it does not assume the null energy condition. The quantum focussing conjecturefurther develops the connection between energy positivity and entropy inequalities. TheQFC implies a novel energy condition called the quantum null energy condition (QNEC). Itis a generalization of the null energy condition, and reduces to the null energy condition inthe limit ~→ 0. The QNEC is a bound on the value of the stress tensor at a point in termsof the second derivative of a particular von Neumann entropy. In chapter 5, we prove thequantum null energy condition in free and superrenormalizable bosonic field theory. Thisproof was first presented in [39].

9

Chapter 2

The Bousso Bound in Free QuantumField Theory

The Bousso bound, as described in section 1.1, states that the entropy ∆S of matter on alight-sheet cannot exceed the difference between its initial and final areas ∆A:

∆A

4GN~≥ ∆S . (2.1)

A light-sheet is a null hypersurface whose cross-sectional area is decreasing or staying con-stant, in the direction away from A.

In this chapter, we will present a proof of this bound for the case that matter consists offree fields, in the limit of weak gravitational backreaction. We will provide a sharp definitionof 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 expectedentropy flux in obvious settings. Using this definition, we will prove the Bousso bound. Wewill 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 theentropy 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-nian. This property 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,in section 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 explic-itly the modular Hamiltonian in terms of the stress tensor. The expression is essentially thesame as the result we would obtain for a null interval in a 1+1 dimensional CFT. Finally, insection 2.3, we use the Raychaudhuri equation to compute the area difference ∆A. The areadifference 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, ifthe null energy condition is violated, it can become necessary to choose a negative initialexpansion, in order to keep the expansion nonpositive along the entire interval in questionand evade premature termination of the light-sheet. We find that the two contributionstogether ensure that ∆A/4GN~ ≥ ∆ 〈K〉. Combining the two inequalities, we obtain theBousso bound, ∆A/4GN~ ≥ ∆S.

In section 2.4, we discuss possible generalizations of our result to the cases of interactingfields and large backreaction. We comment on the relation of our work to Casini’s proofof Bekenstein’s bound from the positivity of relative entropy [51], to Wall’s proof of thegeneralized second law [164], and to an earlier proposal for incorporating quantum effects inthe Bousso bound [152].

In the Appendix, we prove monotonicity of ∆A/(4GN~) −∆S under inclusion, a resultstronger 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 Minkowskispace is a good approximation to any spacetime at sufficiently short distances, our finalresult should apply in arbitrary spacetimes, if the transverse and longitudinal size of thelight-sheet is small compared to curvature invariants. For definiteness, we work in 3+1spacetime 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)

where dx2⊥ = dy2 + dz2. 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 becharacterized by two piecewise continuous functions b(x⊥) and c(x⊥) with −∞ < b ≤ c


H

� = x+

A

A0

x?

Lb(x?)

c(x?)

(a)

H

� = x+

x?

LA1 A2

Ai

(b)

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

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

A(H) =∏

i

A(Hi) . (2.4)

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)

and p+(x⊥)|0〉x⊥ = 0 defines a vacuum state independently for each generator. By ultralocal-ity, the vacuum state on H is a tensor product of these states. (In terms of small transverseneighborhoods of each generator, Hi, one can write |0〉H =

∏i |0〉i.)

It will be convenient to write the vacuum state on H as a density operator,

σH ≡ |0〉HH〈0| . (2.6)

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

σL ≡ trH−L σH (2.7)ρL ≡ trH−L ρH (2.8)

Correlators of φ with no derivatives are non-zero at space-like distances. However, they do not lead to welldefined operators along the light front since we cannot control the UV divergences by smearing it along thelight front directions. For this reason we do not consider φ as part of the algebra A(H). The canonicalstress tensor component T++ ∝ (∂+φ)2 depends only on such derivatives of the field in the null direction.For further details, see reference [164].


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

∆S ≡ S(ρL)− S(σL) = − tr ρL log ρL + trσL log σL . (2.9)

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

(a) (c)(b)

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

An important feature is that we are computing these entropies for null segments. It ismore common to consider entropies for spatial segments, see figure 2.2. In that case, thealgebra of operators includes all the local operators in the domain of dependence of thesegment, 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 domainof 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 dependencedegenerates. Despite these issues, we find that the entropy difference between any state andthe vacuum, (2.9), is finite and well defined. In the free theory case, the limiting operatoralgebra has the ultralocal structure described above.


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

σL =e−KL

tr e−KL, (2.10)

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

∆ 〈K〉 ≡ trKLρL − trKLσL . (2.11)

For any two quantum states ρ, σ, in an arbitrary setting, one can show that the relativeentropy,

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)

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

2.3 Proof that ∆ 〈K〉 ≤ ∆A/4GN~We can think of the null hypersurface H as the disjoint union of small neighborhoods Hiof a large discrete set of null generators; see figure 2.1(b). By ultralocality of the operatoralgebra, 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 σi in equations (2.10) and (2.11), a modular energy ∆ 〈K〉i can be defined for eachneighborhood, which is additive by ultralocality: ∆ 〈K〉 = ∑i ∆ 〈K〉i. Strictly, we shouldtake the limit as the cross-sectional area of each neighborhood becomes the infinitesimal areaelement orthogonal to each light-ray, Ai → d2x⊥. However, we find it more convenient tothink of Ai as finite but small, compared to the scale on which the light-sheet boundaries band c vary.

Since both the modular energy and the area are additive,3 it will be sufficient to show that∆ 〈K〉i ≤ ∆Ai/4GN~, where ∆Ai is the change in the cross-sectional area Ai produced at

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)4GN~ −∆Sdecreases monotonically to zero if the boundaries b and c are moved towards each other. This is shown inthe Appendix.

3By contrast, the entropy ∆S is subadditive over the transverse neighborhoods. In equation (2.9), thevacuum state σL factorizes, but the general state ρL can have entanglement across different neighborhoodsHi. This does not affect our argument since we have already shown directly that ∆S ≤ ∆ 〈K〉.


first order in GN~ by gravitational focussing. We will demonstrate this by evaluating ∆ 〈K〉iand bounding ∆Ai. For any given neighborhood Hi, we may take the affine parameter λi torun 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〉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 < λ′


Focussing and Non-expansion Bound ∆A

Generally, the expansion of a null congruence is defined as [162]

θ(λ) ≡ ∇̂aka =d log δA

dλ(2.18)

where δA is an infinitesimal cross-sectional area element. Recall that in the present contextwe consider the transverse neighborhood of one null geodesic, with small cross section Ai, sowe may replace δA ≈ Ai. Our task is to compute the change ∆Ai of this small cross-section,from one end of Li to the other, by integrating equation (2.18). We will drop the index i, asit 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 inMinkowski space, and so has vanishing expansion θ and vanishing shear σab everywhere. Onemay compute the expansion at first order in GN~ by integrating the Raychaudhuri equation

dθ

dλ= −1

2θ2 − σabσab − 8πGNTλλ , (2.19)

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 iseverywhere satisfied: the cross-sectional area must be nonexpanding away from the initialsurface, everywhere on L. As we shall see, this implies that at first order in GN~, we mustallow for a nonzero initial expansion θ0 at λ = 0. The required initial expansion can beaccomplished 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 integrate inthe opposite direction. For any given state, both ∆A and the initial expansion will dependon the choice of direction. But we will demonstrate that ∆ 〈K〉 ≤ ∆A for all states onfuture-directed light-sheets beginning at λ = 0. By symmetry of KL under λ → 1 − λ, thesame result immediately follows for past-directed light-sheets beginning at λ = 1.)

From equation (2.19) we obtain at first order in GN~:

θ(λ) = θ0 − 8πGN∫ λ

0

Tλ̂λ̂dλ̂ . (2.20)

The non-expansion condition is

θ(λ) ≤ 0, for all λ ∈ [0, 1] . (2.21)

If the null energy condition holds, Tλλ ≥ 0, then this condition reduces to θ0 ≤ 0. Moregenerally, however, we may have to choose θ0 < 0 to ensure that antifocussing due to negativeenergy densities does not cause the expansion to become positive, and thus the light-sheet toterminate, before λ = 1 is reached. However, it is always sufficient to take θ0 to be of orderGN~, so it was self-consistent to drop the quadratic terms ∝ θ2, σabσab, in the focussing


equation. Note that, in the semiclassical quantization scheme, the σ2 term can be viewed asarising from the stress tensor of the gravitons and can be explicitly included as part of thetotal 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 betweeninitial and final cross-sectional area:

∆A

A= −

∫ 1

0

dλθ(λ) = −θ0 + 8πGN∫ 1

0

dλ(1− λ)Tλλ , (2.22)

where we have used equation (2.20) and exchanged the order of integration. In order toeliminate θ0 we now use the non-expansion condition: let F (λ) be a function obeying F (0) =

0, F (1) = 1 and F ′(λ) ≥ 0 for 0 ≤ λ ≤ 1. From equation (2.21), we have 0 ≥∫ 1

0F ′θdλ, and

thus from equation (2.20) and integration by parts we find

θ0 ≤ 8πGN∫dλ[1− F (λ)]Tλλ . (2.23)

With the specific choice F (λ) = 2λ− λ2, we find from equations (2.22) and (2.23) that thearea difference is bounded from below by the modular Hamiltonian:

∆A ≥ A× 8πGN∫ 1

0

dλ λ(1− λ)Tλλ . (2.24)

Comparison with equation (2.17) shows that ∆ 〈K〉 ≤ ∆A/4GN~, as claimed.Combined with the earlier result ∆S ≤ ∆ 〈K〉, this completes the proof of the Bousso

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 microscopicrelation between energy and entropy. We did have to assume that we had a local quantumfield theory at short distances. Therefore the necessary relation between entropy and energyis the one automatically present in quantum field theory, i.e., given by the explicit expressionof the modular Hamiltonian in terms of the stress tensor. Our discussion required a carefuldefinition of the entropy that appeared in the bound. In that sense it is very similar to theCasini version [51] of the Bekenstein bound (see also [123, 94]), and also to Wall’s proof ofthe generalized second law [165, 164].

All these developments underscore the interesting interplay between local Lorentz in-variance of the quantum field theory, Einstein’s equations, and information. It has oftenbeen speculated that the validity of these entropy bounds would require extra constraintson the matter that is coupled to Einstein’s equations. Here we see that the only constraintis that matter obeys the standard rules of local quantum field theory. (Conversely, it maybe possible to view these rules as a consequence of entropy bounds [27].)


Relation to other work In [121] a possible counterexample to the Bousso bound wasproposed. The idea is to feed matter so slowly into an evaporating black hole that thehorizon area remains static or slowly decreases during the process. Hence the horizon is afuture-directed light-sheet, to which the bound applies. Yet, it would appear that one canpass a very large amount of entropy through the horizon in this way. How is this consistentwith 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, one musthave 〈T++〉 ≤ 0. By equation (2.17), this implies ∆ 〈K〉 ≤ 0,4 and positivity of the relativeentropy requires ∆S ≤ ∆ 〈K〉. Hence, in this case, ∆S ≤ 0. Thus we find that with ourdefinitions, the entropy is negative for an evaporating black hole, even with the addition ofsome positive, partially compensating flux; and the entropy is at least nonpositive in thestatic 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 Boussobound. Their proposal is analogous to the definition of generalized entropy, in that oneadds to the area the von Neumann entropy of quantum fields that are outside the horizonand distinct from the matter crossing the light-sheet. In contrast, we have given a definitionwhich only involves properties of the quantum fields on the light-sheet L, i.e., on the relevantportion of the horizon.

A similar distinction must be made when comparing our result to Wall’s proof of thegeneralized 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, andSm(A) is the matter entropy on the portion the horizon to the future of A (which is closelyrelated to the matter entropy on spatial slices exterior to A). Given two horizon slices withA2 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 theGSL:

0 ≤ Sgen(A1)− Sgen(A2) . (2.25)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 assumethe non-expansion condition. This is as it should be, since the GSL does not require anysuch condition. Suppose, for example, that the expansion is not monotonic between A1 andA2, 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- orfuture-directed light-rays. Yet, the GSL must hold. On the other hand, our proof applies toall 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 smallcompared to the curvature scale. This is not the case for the horizon of a black hole. However, the vacuumstates σH and σL can be defined directly on the black hole background; σH is the Hartle-Hawking vacuum.


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. Ingeneral the difference in the matter entropy outside A1 and A2 is distinct from the entropythat we have defined directly on the interval stretching from A1 to A2:

DS ≡ Sm(A1)− Sm(A2) 6= ∆S . (2.26)

Because DS − ∆S is not of definite sign (and because of the different assumptions aboutnon-expansion), our result does not imply Wall’s, and his does not imply ours even in thespecial case where a horizon segment coincides with a light-sheet. Instead, this case givesrise to two nontrivial constraints on two different entropies: one from the GSL and one fromthe 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 gravityregime. At the time, a sharp definition of entropy for either bound was lacking [26, 28]. Adifferential definition of entropy was later applied to the right Rindler wedge, and positivityof the relative entropy was shown to reduce to the Bekenstein bound on this differentialentropy, in settings where the linear size and the energy of an object are approximatelywell-defined [51].

Our present work offers two additional routes to the Bekenstein bound, in the senseof providing precise statements that reduce to Bekenstein’s bound in the special settingswhere the entropy, energy, and radius of a system are intuitively well-defined. Combiningour result with [30] proves a Bekenstein bound, while supplementing a definition of entropyfor 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-localizedto the light-sheet. Alternatively, we may regard our section 2.2 alone as a direct proof ofBekenstein’s bound. Again the bound is on the differential entropy, but now in terms of themodular energy ∆ 〈K〉 on a finite light-sheet. For a system of rest energy E that is welllocalized to the center of a light-sheet of width 2R in the rest frame, one has ∆ 〈K〉 ≈ 2πER,so [7] is recovered.

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 forspatial intervals, in the limit where the spatial interval becomes null. In order to explore theproperties of the entropy defined in this way one can consider strongly coupled field theoriesthat have a holographic gravity dual. We have followed the recipe of [21] to obtain themodular 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. Thatis, the relative entropy for every state is zero. This means that in the light-like limit, the


Ainitial

Afinal

matter

�Ai

�Si

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 affinedistance over which the area changes by a factor of order unity. The entropy is defined asthe sum of the differential entropies on each segment.

operator algebra on the null interval becomes trivial, and all states on the null intervalbecome 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 equivalentto exploring the theories at large energies, since we want to localize the measurements atx− = 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 expectthe same equation ∆S = ∆ 〈K〉 to hold for non-superrenormalizable theories because, incontrast to the free theories we have discussed in this paper, these do not have operatorslocalizable on a finite null surface [142, 150]. We will discuss these issues further in chapter 3.

Another question is how to extend our definition of entropy, and our proof, to the moregeneral situation of a rapidly evolving light-sheet in a general spacetime. One approachis to divide the light-sheet into small segments along the affine direction in such a waythat the change in area is small and then do an approximately flat space analysis for eachpiece. 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 inthe QFT in order to define the modular Hamiltonian and to compute ∆S. We assume thatthis is possible. Then, for the original region we end up with a bound of the type

∆A

4GN~=

∑i ∆Ai

4GN~≥∑

i

∆Si (2.27)

where ∆Si are the entropies differences, as in equation (2.9), for each of the consecutive nullsegments. We can take the right hand side of equation (2.27) as the definition of the totalentropy flux.5 It would be desirable to have a definition of the right hand side which involves

5We thank D. Marolf for this suggestion.


the whole null interval. Nevertheless, already equation (2.27) is a nontrivial bound. In theregime where we have a clear entropy flux, such as a star or a bucket of water, it reduces tothe expected entropy flux if one takes the intervals to be large enough to capture many ofthe infalling particles.

21

Appendix

2.A Monotonicity of ∆A(c,b)4GN~−∆S

In sections 2.2 and 2.3, we showed that 0 ≤ ∆A(c, b)/4GN~ − ∆S. In fact, this differencedecreases monotonically to zero as the boundaries b and c are moved together. To establishthis 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 isrestricted. This follows immediately from the monotonicity property of relative entropyS(ρ||σ) = ∆ 〈K〉 − ∆S under restriction to a subspace (via a partial trace operation), ormore generally under 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 decreasemonotonically under restriction. We will now prove this for the modular Hamiltonian of afree scalar field.

Equation (2.22) for the area difference and equation (2.17) for the modular Hamiltoniancan easily be generalized to an interval of length c. Their difference is

δ(c) =

∫d2x⊥

[−θ0(c)

4GN+ 2π

∫ c

0

dλ(c− λ)2

cTkk(λ)

]. (2.28)

As we vary c, we always choose the initial expansion to be the largest value compatible withthe light-sheet condition:

θ0 = 8πGN inf0≤λ≤c

∫ λ

0

dλTkk(λ) . (2.29)

The monotonicity of δ(c) is established by

dδ

dc=

∫d2x⊥

[− c

4GN

∂θ0∂c− θ0

4GN+ 2π

∫ c

0

dλ

(1− λ

2

c2

)Tkk(λ)

]. (2.30)

The first term is non-negative, since increasing c broadens the range of the infimum inequation (2.29). The latter two terms are together non-negative. This follows from thenon-expansion condition by integrating

∫ c0dη ηθ(η) ≤ 0. It follows that δ is monotonically

decreasing under restriction (and monotonically increasing under extension) of the light-sheet. This proves our claim.

22

Chapter 3

The Bousso Bound in InteractingQuantum Field Theory

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

∆S ≤ A− A′

4GN~, (3.1)

for light-sheets with initial area A and final area A′. The proof applies to free fields, in thelimit where gravitational back-reaction is small, GN~→ 0, that the change in the area is offirst order in GN .

Though this regime is limited, the proof had some interesting features. We made no as-sumption about the relation between the entropy and energy of quantum states beyond whatquantum field theory already supplies. This suggests that quantum gravity may determinesome properties of local field theory in the weak gravity limit.

In this chapter, we will generalize our proof to interacting theories. We will continue towork in the weakly gravitating regime. In the course of this analysis, we will establish anumber 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 en-tropies [51, 123]. The relevant states are the reduced density operators of an arbitraryquantum 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, thepositivity of the relative entropy [120], implies that ∆S ≤ ∆ 〈K〉, where ∆ 〈K〉 is the vacuum-subtracted expectation value of the modular Hamiltonian operator1 [51]. For free theories,

1For any state ρ1, the modular energy is ∆ 〈K〉 ≡ tr (Kρ1) − tr (Kρ0). The modular Hamiltonian K isthe logarithm of the vacuum density matrix K = − log ρ0. K is defined up to an additive constant, whichcan be fixed by requiring that the vacuum expectation value of K is zero, such that ∆ 〈K〉 = 〈K〉. Similarly,∆S = − tr(ρ1 log ρ1) + tr(ρ0 log ρ0) is the difference between the entropy for the state ρ1 under considerationand the vacuum ρ0.

CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELDTHEORY 23

the modular energy is found to be given by an integral over the stress tensor,

∆ 〈K〉 = 2π~

∫dd−2y

∫ 1

0

dx+ g(x+)〈T++(x

+, y)〉. (3.2)

Here x+ is an affine parameter along the null generators, which can be scaled so that thenull 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 higherdimensions it will not.)

Using Einstein’s equation, the area difference ∆A = A − A′ can be written by a localintegral over the stress tensor, plus a term that depends on the initial expansion of thelight-rays. The latter must be chosen so that the expansion remains nonpositive everywhereon the null interval. This is the “non-expansion condition” that determines whether a nullhypersurface is a light-sheet. equations (3.2) and (3.3), combined with Einstein’s equationand the non-expansion condition, imply that ∆ 〈K〉 ≤ ∆A/4GN~.

To generalize this proof to interacting theories, a number of difficulties must be addressed.Wall’s results do not apply, so the entropy and modular Hamiltonian cannot be defineddirectly on the light-sheet. Instead, we must consider spatial regions that approach thelight-sheet. The positivity of the relative entropy, ∆ 〈K〉 −∆S ≥ 0, holds for every spatialregion [51], so it could still be invoked. But it is no longer useful: for spatial regions, ∆ 〈K〉is highly nonlocal, and we are unable to compute it before taking the null limit.

Instead, we benefit from a new simplification, which happens to arise precisely in the caseto which our previous proof did not apply: for interacting theories in d > 2.2 In this case, theentropy ∆S must be equal to the modular energy ∆ 〈K〉 in the null limit. To show this, werecall that the von Neumann entropy is analytically determined by the Rényi entropies. Then-th Rényi entropy is given by the expectation value of twist operators inserted at the twoboundaries of the spatial slab. The approach to the null limit can thus be organized as anoperator product expansion. We argue that, in the limit, the only operators that contributeto ∆S have twist d − 2; and that for interacting theories in d > 2, there is only one suchoperator. This implies that ∆S becomes linear in the density operator, and hence [21]

∆ 〈K〉 −∆S → 0 (3.4)

in the null limit.

2Our original proof applies to theories for which the algebra of observables is nontrivial and factorizesbetween null generators. This includes free theories but also interacting theories in d = 2 [164]. For d = 2, thearea is the expectation value of the dilaton-like field Φ that appears in the action as 116πGN

∫d2xΦ(x)R+ · · · .

If the d = 2 theory arises from a Kaluza-Klein reduction of a higher dimensional theory, then Φ is the volumeof the compact manifold.


The unique twist-2 operator is the stress tensor. This implies a second key result:

∆S =2π

~

∫dd−2y

∫ 1

0

dx+ g(x+)〈T++(x

+, y)〉. (3.5)

Together with equation (3.4), this extends the validity of equation (3.2) to the interactingcase: 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 interactingconformal field theories with a gravity dual [122], we are able to compute g(x) explicitlyfrom the area of extremal bulk surfaces [140, 99].3 For d > 2 we find that g differs from thefree 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 propertieshold in the interacting case. In particular, we will show that the key property,

∣∣∣∣dg

dx+

∣∣∣∣ ≤ 1 , (3.6)

can be established by considering highly localized excitations and exploiting strong subad-ditivity. This will be sufficient to establish the extension of our free proof to the interactingcase.

Outline This chapter is organized as follows. Sections 3.1 and 3.2 contain the new resultssufficient 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 operatorsthat compute the Rényi entropies. We derive equations (3.4) and (3.5), thus recovering akey step in the free-field proof: the local form of the modular energy, equation (3.2). Wefurther constrain the modular energy in section 3.2, where we establish equation (3.6) forinteracting 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 modularenergy 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+) forthese theories. For d > 2, we find that g(x+) differs from the free field result. The approachto 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,whereas no operators can be localized on the null slab. Any fixed operator is eliminated inthe limit and thus cannot be used to discriminate between states. Appendix 3.B illustratesthis 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 coupledto gravity, not light-sheets in the dual bulk geometry.


�i

�i+1

D DA

Figure 3.1: The Rényi entropies for an interval A involve the two point function of defectoperators D inserted at the endpoints of the interval. An operator in the ith CFT becomesan 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 aspatial slab in the limit where the finite dimension of the slab becomes light-like (null). Weconsider free and interacting conformal field theories with d ≥ 2 spatial dimensions. (We willcomment on the non-conformal case at the end.) For interacting theories in d > 2, we willfind that the entropy is equal to the modular Hamiltonian, and that both can be expressedas a local integral over the stress tensor.

It is convenient to consider the Rényi entropies first. The nth Rényi entropy associatedwith a spatial region A

Sn(A) = (1− n)−1 log tr ρnA (3.7)can be computed by taking the expectation value of a defect operator in a theory, which wedenote by CFTn, obtained from taking n copies of a single CFT. The operator in questionis a codimension 2 defect operator localized on the boundary ∂A of a spatial region A inthe full Euclidean theory. In other words, the second orthogonal direction to the operator isEuclidean time. The defect operator is such that when we go around it, the various copiesof the original CFT are cyclically permuted. In other words, an operator φk(x) defined onthe kth CFT is mapped to φk+1(x) on the (k+ 1)

th CFT, and φn(x) is mapped to φ1(x); seefigure 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 thenanalytically continue them to Lorentzian time. Finally, we take the light-like limit. In thislimit, we expect to have an operator product expansion. This expansion differs from thestandard Euclidean operator product expansion in two respects. First, we are approachingthe light-like separation, where the operators have zero metric distance but do not coincide,

4These defect operators are oriented: there is a D+ which maps φi → φi+1 and a D− which mapsφi → φi−1. For an interval, we have the insertion of D+ and one end and of D− at the other end. We willnot explicitly discuss this distinction.


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 apoint. Despite these differences, we expect that there is a kind of operator product expansionthat is applicable in this case.

To our knowledge, the systematics of operator product expansions of extended operatorsin the light-like limit has not been explored. For the remainder of this section, we willmake 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 scalaroperators has the form

O(x)O(0) ∼∑

k

|x|−2τO+τk(x+)skOk,sk . (3.8)

In this equation, the operator Ok,sk has spin sk, scaling dimension ∆k and twist τk ≡ ∆k−sk;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.These operators contain two free fields, each with twist 1

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 morerecent discussion). The only operator with spin 2 and twist d− 2 is the stress tensor, unlesswe have two decoupled theories. Operators with spin 1 include conserved currents. Scalaroperators and operators with spin 1/2 can have twist τ ≥ 1

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 inthis 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 theeffective action for the resulting defect operator. In general, new light degrees of freedomcould emerge when the two defect operators coincide. However, in our case the two twistoperators annihilate each other, leaving only terms that can be written in terms of operatorsof the original theory. In other words, we expect

D(x)D(0) ∼∑

exp

{∫dd−2y

[∑

k

1

|x|d−2−∆kOk(x = 0, y)]}

(3.9)

where y denotes the transverse dimensions and Ok denotes local operators on the defect atx = 0. Thus the expansion is local in y. We can view this equation as an expansion ofthe effective action for the combined defect (consisting of both defects close together) byintegrating out objects with a mass scale of order 1/|x|.


The leading term in equation (3.9) is given by the identity operator and contributes afactor of Ay/|x|d−2 in the exponent (with a coefficient that depends on n), where Ay is thetransverse area. This is the expected form of tr ρn0 = e

−(n−1)Sn , which gives the vacuum Rényientropies for the interval. In the vacuum case, all other operators have vanishing expectationvalues. This contribution cancels when we compute the difference ∆S of the von Neumannentropies 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 anexpansion which looks both like equation (3.8) and like equation (3.9). In other words, weexpect the expression to be local along the y direction as in equation (3.9), but with termsthat 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(x)D(0)|light-like ∼ exp{∫

dd−2y

[∑

k

|x|−(d−2)+τk(x+)skOk,sk

]}. (3.10)

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 = O1O2 · · ·On , (3.11)

where Ok is an operator on the kth copy of the original CFT. Some of the factors in equa-

tion (3.11) could be the identity, and the simplest operators we consider have only one factorwhich is not the identity. Performing the replica trick, the operators with a single factorthat appear in the OPE of the two defect operators contribute to the entropy proportionallyto an operator in the original CFT. Specifically, we find

Ssingle = 〈OS〉 . (3.12)

Such contributions are linear in the density matrix, and therefore do not give rise to anon-zero value of ∆ 〈K〉 − ∆S. The reason is that the operator on the right hand side isnecessarily equal to K, since K is the only operator localized to the region whose expectationvalue coincides with ∆S to linear order for any deviation from the vacuum state [21] (seealso [171]).

The d > 2 interacting case

We will now argue that for interacting theories in d > 2, all operators that contribute toequation (3.10) are of this simple type: they all have only one nontrivial factor. In fact, onlythe stress tensor contributes.

Clearly, operators with τ > d−2 will not contribute; this includes all higher spin operatorsin an interacting theory. Conserved spin 1 currents have twist τ = d− 2, but cannot appearbecause the defect operators are uncharged. Next, consider possible contributions from


operators with twist 12(d− 2) < τ ≤ d− 2. These operators could appear in representations

which are not symmetric and traceless5. However, since the twist operator is invariant undertransverse rotations, these operators must appear in pairs; their combined twist would bebigger than d− 2.

Thus we can focus on the operators with spin zero. An operator of CFTn consisting ofa single-copy scalar operator with twist in the range 1

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 contributionto ∆ 〈K〉, so these operators do not contribute to ∆ 〈K〉 −∆S.6 On the other hand, if wehad two operators in the range 1

2(d− 2) < τ ≤ d− 2 on different CFT copies inside CFTn,

the total twist will be higher than d− 2 and we will not get a contribution in the light-likelimit.

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.

Notice that throughout this discussion, we have taken the coupling fixed and then takenthe null limit. In particular, if we have a weakly coupled theory, we will get corrections tothe result from free field theory which at each fixed order in perturbation theory will containlogs. One must resum the logarithms first, before taking the null limit, to recover the resultthat only the stress tensor survives.

Returning to the Rényi entropyies, we conclude that in interacting conformal theories, theonly 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 expansionaround x+ = 0. Discarding the contribution from the identity operator, which will drop outof ∆S, we get

Dn(x)Dn(0)|light-like ∼ exp{−(n− 1)2π

∫dd−2y

∫ 1

0

dx+ gn(x+)〈T++(x

− = 0, x+, y)〉]

}.

(3.13)In this expression, we have set the size of the interval ∆x+ = 1 and extracted an overallfactor of n − 1 from the exponent. This factor accounts for the vanishing of the exponentfor trivial Rényi operators when n = 1. We have also replaced the sum over descendantsby 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

Wilson-Fisher fixed point at small � = 4− d. In this case, the dimension of φ is 12 (d− 2) + O(�). However,due to the φ→ −φ symmetry, this operator does not appear in the OPE of the defect operators involved inthe replica trick. Another example is the Klebanov-Witten theory [112]. These are four dimensional theorieswith operators of dimension 3/2 < 2. However, these operators carry a U(1) charge and cannot appear inthis OPE. A relevant question here is whether there are theories with scalars with twists in this range whichare not charged under any symmetry. If these operators are present, then our definition for ∆S will becomedivergent and will need to be modified.


operator T . The integral is restricted inside the null interval because operators outside thisrange 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 givenby analytic continuation:

∆S = limn→1

1

1− n log〈Dn(x)Dn(0)〉

= 2π

∫dd−2y

∫ 1

0

dx+g(x+)〈T++(x

− = 0, x+, y)〉

= ∆ 〈K〉 . (3.14)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 Khas the same general form in terms of the stress tensor, whether ∆ 〈K〉 = ∆S or not wouldgenerically depend on further details. For relevant deformations of a free UV fixed pointwe expect to have ∆ 〈K〉 ≥ ∆S as in the free theories, while we expect ∆ 〈K〉 = ∆S forasymptotically free theories.7

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 above argument for ∆S = ∆ 〈K〉 does notapply in these cases.

If the operator (3.11) which appears in equation (3.10) contains more than one nontrivialfactor, it can give rise to a contribution to the entropy which is not equal to the expectationvalue of any operator in the original CFT. These contributions are interesting because theymake ∆S < ∆ 〈K〉 possible. In a free field theory, such operators arise from insertions ofthe fundamental field φ in one copy and another field φ in another copy. They have twistτ = d− 2 and can contribute in the light-like limit.

In an interacting theory, all such operators gain a non-zero anomalous dimension. Inparticular, in a unitary theory, the field φ gains a positive anomalous dimension and so willnot contribute in the null limit8. However, in a d = 2 interacting theory, multiple copies ofthe stress tensor can appear. Since τ = d− 2 = 0, the total twist will remain equal to d− 2no matter how many times the stress tensor appears in (3.11). Thus, in d = 2, we can have∆S < ∆ 〈K〉 even for interacting theories.

7In asymptotically free theories, the coupling runs as g2 ∝ 1/ logµ as a function of the scale µ. TheOPE is not given by a simple power behaviour but we need to integrate the anomalous dimensions of arange of scales as exp[−

∫dµµ γ(µ)]. Since γ(µ) ∼ g2(µ) ∝ 1/ logµ, this integral diverges at short distances.

Therefore, operators with non-zero anomalous dimensions do not contribute in the null limit, which involvesgoing to very high scales. 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 thedefect.


3.2 Bousso Bound Proof

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

∆ 〈K〉 = 2π∫d2x⊥

∫ c

0

dλ g(λ, c)Tλλ(λ) . (3.15)

Here we have rescaled the affine parameter and emphasized the dependence of the functiong 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 impliesthat

g(λ, c) = cḡ(λ̄) , (3.16)

where λ̄ = λ/c. We will now show that monotonicity of ∆A − ∆ 〈K〉 is guaranteed if thefunction g satisfies a small number of other simple properties of g(λ), including concavity.

We havedδ

dc= −cdθ0

dc+

[−θ0 +

∫ c

0

dλ

(1− ∂g

∂c

)Tλλ(λ)

](3.17)

The first term is nonnegative

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