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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
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Entropy Bounds and Entanglement
Copyright 2017by
Zachary Fisher
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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 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.
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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
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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
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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
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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
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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.
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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
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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.
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CHAPTER 1. INTRODUCTION 3
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,
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CHAPTER 1. INTRODUCTION 4
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.
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CHAPTER 1. INTRODUCTION 5
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.
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CHAPTER 1. INTRODUCTION 6
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〉.
-
CHAPTER 1. INTRODUCTION 7
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].
-
CHAPTER 1. INTRODUCTION 8
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
-
CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 11
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].
-
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 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.
-
CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 13
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〉.
-
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〉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′
-
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]
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
-
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 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].)
-
CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 17
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.
-
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. 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
-
CHAPTER 2. THE BOUSSO BOUND IN FREE QUANTUM FIELD THEORY 19
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.
-
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 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.
-
CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELDTHEORY
24
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.
-
CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELDTHEORY
25
�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.
-
CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELDTHEORY
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 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|.
-
CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELDTHEORY
27
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
-
CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELDTHEORY
28
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.
-
CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELDTHEORY
29
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.
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CHAPTER 3. THE BOUSSO BOUND IN INTERACTING QUANTUM FIELDTHEORY
30
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