#### The gravity duals of modular Hamiltonians

HJE
The gravity duals of modular Hamiltonians
Daniel L. Ja eris 1 2 3 4
S. Josephine Suh 0 2 3 4
Cambridge 2 3 4
Massachusetts 2 3 4
U.S.A. 2 3 4
Cambridge 2 3 4
Massachusetts 2 3 4
U.S.A. 2 3 4
0 Center for Theoretical Physics, Massachusetts Institute of Technology , USA
1 Center for Fundamental Laws of Nature, Harvard University
2 Using (2.4) , we have @
3 and switching derivatives using @ G
4 and region R , @ h(x)
5 For example consider the response @ C =
In this work, we investigate modular Hamiltonians de ned with respect to arbitrary spatial regions in quantum eld theory states which have semi-classical gravity duals. We nd prescriptions in the gravity dual for calculating the action of the modular Hamiltonian on its de ning state, including its dual metric, and also on small excitations around the state. Curiously, use of the covariant holographic entanglement entropy formula leads us to the conclusion that the modular Hamiltonian, which in the quantum acts only in the causal completion of the region, does not commute with bulk operators whose entire gauge-invariant description is space-like to the causal completion of the region.
AdS-CFT Correspondence; Field Theories in Higher Dimensions
1 Introduction 2
Modular response
2.1
2.2
2.3
2.4
Deformation of the metric
Deformation of general expectation values
Action of the modular Hamiltonian on excitations
A special symmetric case
3
4
Modular Hamiltonians as precursors?
Conclusions A
Metric and Weyl response in Poincare AdS
A.1
Metric-metric and metric-Weyl propagators
A.2 Causality of modular response from extremal surface
A.2.1
Extremal surface condition
T i
A.2.3
Causality of bulk response @ Cabcd
A.3 Response from interior of extremal surface
B Two-point function of energy-momentum tensor in momentum space
1=N , it is given by the area of the bulk surface with minimal area ending on the boundary of
the spatial region [1]. In addition to considerable evidence, a derivation has been provided
in [2], based on analytic extrapolation of classical replica geometries in the bulk.
A generalization to time-dependent states was proposed in [3], in which a bulk surface
with extremal area replaces the minimal surface. Primary evidence in favor of the general
validity of this proposal are its consistency with strong subadditivity [4{6] and boundary
causality [7]. However, the proposal is not yet as well established as in the static case.
{ 1 {
Given that the entanglement entropy of a spatial region is encoded in the gravity dual
as the area of the minimal surface, one can ask if there is a simple interpretation in the
bulk of the corresponding modular Hamiltonian operator
H =
log :
Note that the entanglement entropy su ers from UV divergences associated to
shortdistance entanglement across the boundary of the region. However, the claim that the
relative entropy Tr( 2 log 2)
Tr( 2 log 1) is regulator-independent [8{10] is equivalent to
the claim that the divergences of H are state-independent, or proportional to the identity
operator. We discuss the nature of these divergences below.
Very few characterizations of a general modular Hamiltonian are known other than
those which follow directly from its de nition. For
associated to the causal completion1
C of some general spacetime region, H is a Hermitian and possibly unbounded operator
acting on the Hilbert space of states in C. The conjugation by H
is an automorphism on A(C), the algebra of bounded operators in C [11], and is a symmetry
of the expectation value of all operators in C,
O( ) = ei H
Oe i H
Tr ( O( )) = Tr ( O) :
(1.2)
(1.3)
(1.4)
(1.5)
(1.6)
De ned thus as an operator on the full Hilbert space, H is a non-smooth operator
due to a kink at the boundary of R. However, integrating it over the location of R with
respect to a smooth test function should result in a smooth operator. Furthermore, we will
sometimes consider the operator K = HR
HR, which we conjecture is a smooth operator.
1We de ne the causal completion of a set of points S in spacetime as the set of all points C(S) such that
all causal curves passing through each point also passes through S. In the literature C(S) is also called the
domain of dependence or the causal development of S.
{ 2 {
In the special cases of the region being half-space in the Minkowski vacuum of any quantum
eld theory [12, 13] or conformally related con gurations [14], and the case of the region
being a null slab in the Minkowski vacuum of a CFT [15], H has been obtained explicitly
and is a linear smearing of components of the energy-momentum tensor over the region.
For a general region and state, however, little is known about H and one merely has the
expectation that it cannot be written as a spacetime integral of local operators.
In this paper, we consider the modular evolution of a quantum
eld theory density
matrix
which has a semi-classical gravity dual,
e i H ei H ;
where H is the modular Hamiltonian associated to the reduced density matrix of an
arbitrary but xed spatial region R,
H =
log R
I :
The logic is that the action of H very close to the boundary of R is very similar to that of
the modular Hamiltonian associated to a half space in the Minkowski vacuum, for which
K is a smooth operator.
The holographic duals of density matrices are not fully understood, and there is a
related ongoing investigation of the possibility of formulating AdS/CFT for subregions [6,
7, 16{19]. Moreover, the relation between entanglement and topology proposed in [20]
would imply that knowledge of a density matrix is insu cient even to make probabilistic
predictions for general bulk observables.2 Thus although none of our results depend on
being pure, we will take
to be that of a pure state j i,
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after which
is again a pure state,
= j i h j ;
j i
e i H
j i :
Note that H is an example of a state-dependent operator. Somewhat analogously
to [21, 22], we will nd that there is a useful holographic interpretation of H when acting
on states close to the reference state, in the sense that they are given by a small number
of single-trace operators acting on j i.
We rst show that at linear order in , one can construct the classical metric g in the
gravity dual of
to leading order in 1=N , using the rst law of entanglement entropy [23]
= j i h j
hHi = S
and the minimal (RT) and extremal (HRT) surface prescriptions for calculating spatial
entanglement entropy. Beyond linear order in , we can use the fact that H
= H to
interpret the following expression for the metric perturbation (1.10) as a non-linear di erential
equation in
for the metric g . The metric perturbation takes the form
1
4GN
i DhA^h; h^iE
where h^ is a metric perturbation operator constructed by smearing boundary single-trace
operators using bulk equations of motion,3 and A^h is the change due to a metric
perturbation h in the area of the extremal surface corresponding to R, obtained by elevating h
to the operator h^. This equation should be interpreted in the linearized theory around the
background dual to , and the expectation values in (1.10) are taken with respect to .
Given that expectation values of operators inserted solely in C(R) or in C(R) are
invariant under modular evolution, if there is a bulk region B(R) in g
g0 that is dual
to R in the sense that the metric in B(R) is determined by
R, and similarly B(R) for
R, the metric in those regions will be unchanged in g up to di eomorphisms. Assuming
the HRT prescription, the form of the `modular response' @ h in (1.10) implies that its
2This is because information about entanglement is lost when a general probability distribution on the
Hilbert space of quantum states is replaced by an associated density matrix
3Such a construction of the metric perturbation operator in Poincare AdS appeared in [24]. Also see [25].
= Pi pi j ii h ij.
(1.7)
(1.8)
(1.9)
(1.10)
di eomorphism-invariant support is causal from the extremal surface of R | in other
words, one can choose coordinates, at least patch-wise, such that the response vanishes
at spacelike separation from the extremal surface.
Thus the support is indeed absent
from the `entanglement wedge' advocated in [7] to be B(R),4 the causal completion of
the codimension-1 bulk region which interpolates between R and its extremal surface on a
Cauchy slice. Parallel statements hold for R.
Proceeding further and explicitly computing the di eomorphism-invariant support of
@ h in simple examples, we nd the following: except when R is a half-space or a sphere
and
is the Minkowski vacuum of a CFT, in which case the support is causal from the
boundary of the extremal surface,5 generically there is support on interior points of the
extremal surface and thus at space-like separation from C(R) [6, 7]. Since H as a boundary
operator is localized in C(R), this implies that generically the modular Hamiltonian is a
`precursor' [26] in the sense of being a boundary operator that is sensitive to bulk processes
at space-like separation. Alternatively, the HRT prescription may need to be modi ed.
Moving beyond the metric, we discuss two methods of obtaining the deformation
@ hO1 : : : Oni of n-point functions and other expectation values in general. The rst is
to utilize (1.9) and perform bulk computations of the change in entanglement entropy,
staying in Lorentzian signature. The second is to analytically continue from Euclidean
path integrals de ned on replica sheets. In certain instances the Euclidean calculation
simpli es further as we are able to use geometries with continuous conical de cit
following [2]. The knowledge of n-point functions allows us to recover the action of the modular
Hamiltonian on excitations about its de ning state.
This paper is organized as follows. In section 2, we study the modular response, or
deformations of the metric and correlation functions in the linearized
state. We give
an explicit construction of the metric deformation. We discuss methods of computing the
deformation of general expectation values, and show that one can recover the action of the
modular Hamiltonian on nearby excited states. We also examine the special symmetric case
that R is a half-space and
is the Minkowski vacuum of a CFT. In section 3, we show that
generically the modular response of the metric as computed using the HRT prescription
violates bulk causality, and discuss the known resolution of a similar conundrum for Wilson
loops and geodesics. In section 4, we present conclusions and open questions. In the
appendices we present computations of the metric response for an arbitrary region R when
the gravity dual to
is the Poincare AdS vacuum.
2
2.1
Modular response
Deformation of the metric
Let us x a quantum
eld theory density matrix
and a spatial region R understood to
be lying at some xed time tR. This de nes a modular Hamiltonian
H ;R =
log R
I ;
R = TrR :
(2.1)
4For other papers that have discussed how large B(R) should be, see [6, 17{19].
5The same is true for conformally related con gurations.
{ 4 {
respectively. The action of H ;R extends in the obvious way to the whole Hilbert space H.
From here on we omit the subscripts on H ;R.
We start by making a simple observation as follows. Consider the unitary evolution of
by some Hermitian operator O,
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
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Working to linear order in ,
operator O,
Then deforming alternatively by the modular Hamiltonian H and another Hermitian
! e i O ei O :
O
= i [ ; O] :
H hOi =
O hHi = TrR ( O R H) =
O
S
where S =
TrR ( R log R) is the entanglement entropy of region R. In the second
equality we have used that H acts trivially on HR and and the last equality is the rst law
of entanglement entropy.
The statements so far do not rely on gauge/gravity duality. Now let us assume
has
a semi-classical gravity dual with classical metric g. Then we note that if the deformation
O
again gives a semi-classical state, O
S in (2.4) can be computed at leading order in
1=N using the HRT prescription. In particular, from the knowledge of H hOi for single
trace operators fOg, we can construct the modular response h
H g as
1
H hh^abi =
1
h^ab S =
i
4GN
DhA^h; h^abiE
for h^ together with other bulk elds.
at leading order in 1=N or O(1). Here h^ is a metric perturbation operator written as a
smearing of boundary single trace operators fOg, by solving linearized equations of motion
about the gravity dual of the original state
at leading order in 1=N . When is the vacuum,
or dual to a matter-free solution of Einstein's equations, only the expectation value of the
boundary energy-momentum tensor H hT i is non-vanishing at O(N ), so h^ decouples from
other bulk elds and is a smearing of T only. For generic states , H hOi is non-vanishing
at O(N ) for O 6= T as well, and one has to solve a system of coupled di erential equations
Meanwhile,
A^h =
1 Z
operator. By causality in the bulk eld theory, an operator ^ whose entire di
eomorphisminvariant description, or `framing', is space-like to E, commutes with A^h. Thus for instance,
in the linearized theory, if one forms a curvature combination of the metric response @ hab
{ 5 {
which transforms homogeneously under di eomorphisms, its support must be restricted to
J~(E),6 the causal future and past of E. Similarly, the modular response on the boundary
corresponding to the leading fall-o of @ hab will be restricted to the intersection of J~(E)
with the boundary, or J (@R) [7], as required by (1.4) and triviality of the action of H on
C(R). We are able to check this explicitly for general regions R when g is Poincare AdS,
as shown in appendix A.2.
Note that in the above the HRT prescription is put to use even when g is a static
metric | we used extremal surfaces in the bulk to compute the entanglement entropy in
the presence of time-dependent perturbations on top of g. Furthermore, the r.h.s. of (2.5)
can only give the piece of the response at absolute leading order in 1=N , as it was derived
using the leading order expression for S. In it h^ab can be replaced with any boundary
operator, but the leading order piece it yields will be zero for instance for multi-trace
operators, and in the vacuum, single-trace operators other than T as well.
2.2
Deformation of general expectation values
In order to compute the deformation of general expectation values H hOi =
O can be any operator, for instance a Wilson loop or a string of single-trace operators) which
are O(1) or smaller using the r.h.s. of (2.4), one has to reckon with quantum corrections
to the RT/HRT prescriptions such as were considered in [27, 28].
However, if j i is a time-symmetric state that can be obtained from a real Euclidean
path integral with a corresponding classical gravity dual, one can derive the deformation
of expectation values of operators in j i in another way.
Consider a Euclidean QFT path integral on a space with boundary at t = 0, with
sources for single trace operators turned on. This de nes a quantum state at t = 0 whose
gravity dual is the analytic continuation to Lorentzian signature, of the Euclidean bulk
eld and metric con guration with AdS boundary conditions determined by the sources.
The trace Zk
Tr( kR) is given by the normalized QFT partition function on the
k-sheeted covering space branched over @R. In the bulk, the leading classical saddle is
smooth in the interior and asymptotes to the k-sheeted AdS boundary geometry. As an
operator, kR
and glues in the k sheeted region over R, see gure 1.
I is given by the Euclidean path integral from t = 0 that does nothing in R
Therefore h jO(x)( kR
I) j i for any operator O is given by the associated Euclidean
expectation value, Zk+1(O(x))=Zk+1. The operator ordering is determined by the
Euclidean time of x relative to
= it = 0. Analytically extrapolating in n = k + 1 and
continuing x to Lorentzian signature, one nds that7
Zn(O(x))
Zn
=
hO(x)Hi :
(2.7)
6We distinguish a causal domain in the bulk as opposed the boundary by placing a tilde above the
character.
7The same equation but formulated using a twist operator appeared in [29].
{ 6 {
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k
(
I,
left, and R, right.
The commutator response is given by the di erence between the two operator orderings,
"!0 n!1
Zn(O(~x; t
i"))
Zn(O(~x; t + i"))
Zn
Zn
Note that the r.h.s. is continued from Euclidean signature and only non-analyticities in
Zn(O(~x; )) contribute, which only exist for ~x 2 R,
= 0. Also note that for a
timesymmetric state j i, one could in principle compare the value of some @ hOi that is
O(N ) as computed from the above with that from the Lorentzian method, and this would
constitute a check of the HRT prescription.
As explained in [2], one can quotient the bulk saddle dual to Zn by its replica symmetry,
to obtain a geometry which asymptotes to the original single-sheeted AdS boundary but
has a conical opening angle of 2 =n along a bulk defect. If the Euclidean expectation value
in (2.7) is dominated by the classical bulk geometry, it can be extracted from the quotient
space, and the analytic extrapolation in n is straightforward. This is the case when O is a
single-trace operator or some other operator whose expectation value in the large-N limit
is given by a minimal geodesic or surface in the Euclidean bulk saddle.8
Then immediately from (2.8) we have
:
(2.8)
"!0
Z (O(~x; t
i"))
Z (O(~x; t + i"))
Z
Z
=0
;
(2.9)
where the r.h.s. is continued from Euclidean signature with Z being the partition function
over conical defect geometries with opening angle 2 (1
). This is particularly useful in
computing the deformation of 2-point functions of high-dimension scalar operators, since
the expectation values involved are given at leading order in terms of lengths of geodesics
in classical saddles g of Z .
8[30] suggested that general correlation functions of H with operators in Minkowski space are equivalent
to correlation functions of just the operators on spacetimes with conical defects.
{ 7 {
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Finally, we note that the two methods outlined above translate to respective arguments
that j i with
in nitesimal is indeed a semi-classical state with a gravity dual if j i is. One
needs to check that corrections to an n-point function of single-trace operators hO1 : : : Oni
besides the classical value hO1i : : : hOni
are subleading in 1=N . Below we also
nd it
instructive to explicitly identify the subleading corrections in the Lorentzian method.
1
O1O2 S. For small , the state ei O1O2 j i
is described by the same classical metric in the bulk dual as j i to leading order in 1=N ,
and thus
S vanishes at leading order. At subleading order, there are two sources of
contributions to
S | corrections to the bulk metric, and subleading corrections to S
besides the extremal area. The former arises from the bulk tree-level diagram of two
scalars and the metric, and is calculable from the bulk Lagrangian | in CFT language,
the T OO 3-point function gives a nonzero VEV to T in the state, and there is also an
explicit OO correction to the boundary expression for the bulk metric operator [31]. The
latter corrections include the bulk scalar eld entanglement entropy [27], so is more di cult
to determine.
In any case, contributions from both are suppressed by 1=N compared to the classical,
factorized 2-point function. The same reasoning applies to general n-point functions, and
we have the desired conclusion.
Using (2.8), we reach an identical conclusion by noting that Zn(O1 : : : On) obeys
largeN factorization, due to the fact that a semi-classical bulk con guration dominates its
gravity dual and n-point functions on the boundary are limits of bulk n-point functions.
2.3
Action of the modular Hamiltonian on excitations
It is interesting to determine if the action of the modular Hamiltonian on states other
than its de ning state also has a useful holographic description. For states that are dual
to completely di erent geometries, we do not expect the action to be simple. However,
for states that are made by acting with a small (compared to N ) number of single-trace
operators on j i, one can make some progress.
We would like to characterize states of the form HO j i where now O again
denotes a single-trace operator.
This can be done by computing the inner products
h jO(x)[H; O(y)] j i. For general position x, this is di cult to determine. However in
the special case that x 2 C(R), one can move the insertion of H such that it always acts
on j i, and use the methods in the previous section to compute the result.
We know that H is not a smooth operator because of a kink at @R, but we conjecture
that K = HR
HR is smooth. This is because the action of H on operators inserted very
close to @R should be well approximated by the half-space result (2.11), for which K is
explicitly a smooth operator. One can see that this approximation is valid from the Euclidean
expressions (2.7) and (2.8). At su ciently short distances @R may be approximated by a
at plane. Moreover, the expectation values of operators inserted su ciently close to @R
in the n-sheeted covering spaces are governed by short-distance physics and thus do not
depend on the state. Finally, at least in our situation where the analytic continuation in
n is simple in the gravitational dual, it is clear that the same is true with H replacing the
branch point of the covering space.
{ 8 {
For y 2 C(R), we have [K; O(y)] = [H; O(y)] and thus ih jO(x)[K; O(y)] j i =
@ h jO(x)O(y) j i. Then assuming that K is a smooth operator, we may determine the
entire action of K on such states by analytic continuation of x from C(R) to the entire
spacetime.
In computing @ h jO(x)O(y)j i, we saw in the previous section that there is a
contribution that is di cult to determine in the Lorentzian method, the change in the bulk
entanglement entropy. Thus here we restrict to j i such that the Euclidean method can
be applied, and consider the special case of high-dimension single-trace scalar operators,
dual to heavy elds in the bulk (with mass parametrically larger than the AdS scale, but
smaller than the Planck scale).
Then using (2.9) and the geodesic approximation hO(x)O(y)i / e ml(x;y; ), we have
lim
"!0
l(x; y ; ))j =0 hO(x)O(y)i
(2.10)
where l(x; y; ) is the length of the geodesic of minimal length connecting points x and y
in g , and y are obtained by the replacements ty ! ty
i". The discontinuity in l(x; y; )
For a complete characterization of the action of H on excited states near j i, one
needs to compute more general inner products h jO(x1) : : : O(xn)[H; O(y1) : : : O(ym)] j i.
Then modulo analytic continuation in xi to the entire space-time, one could again take
xi 2 C(R), yi 2 C(R), converting the problem to that of obtaining modular deformations
of n-point functions. The latter can be computed in principle with the methods we have
discussed, although the computations will be more di cult than in the simplest instance
of (2.10).
2.4
A special symmetric case
Here we examine the modular response @ hab as given by (2.5) in the simple case that is
the Minkowski vacuum of a d-dimensional CFT, and the spatial region R is half-space. The
modular Hamiltonian H is known in this con guration [12, 13], and taking the half-space
to be x1 > 0 at tR = 0, can be written as
H = 2
Z 1 d 2
d
1
x
?
dx1 x1Ttt(t = 0; ~x) :
(2.11)
Given that H is a smearing of local operators on R, bulk-causality implies the
di eomorphism-invariant support of the modular response @ hab must be causal from R.
As we have argued above it is also causal from E, so in fact it should be causal from
9A related result that appeared previously in the literature is the rst law of black hole mechanics that
the perturbed area of a stationary black hole horizon reduces to an integral of energy density over the
boundary of the horizon [32]. In [33] the authors used the HRT prescription to translate the
rst law of
entanglement entropy for balls in a CFT vacuum to a sub statement of the rst law of black hole mechanics,
and derived from it linearized Einstein equations.
{ 9 {
HJEP09(216)8
x
z
HJEP09(216)8
Cartoon of the di eomorphism-invariant support of the modular response
g)= , as computed using the HRT prescription. The entanglement wedge of R, whose
intersection with the AdS boundary is C(R), is delineated in pink. Left: for con gurations ( ; R)
with special symmetry, the response is causal from @R. Right: for generic ( ; R) the response is
causal from the entire extremal surface associated with R. To avoid clutter here we have only drawn
the upper half of time evolution.
It is easy to check that indeed Einstein equations in Poincare AdS conspire with the
geometry of E in this case to make the integral over E
/
(2.12)
into a boundary term at @E = @R, where in the integrand a sum over p is implied. Here
we are working in transverse-traceless and Fe erman-Graham gauge for h and using (A.10)
and (A.30) derived in appendix A. For independent components @ hti and @ hij where i; j
are spatial indices, we have the propagator components
Gtipp =
((d
Gijpp = ((d
1) ip jp
(2.13)
x0p, and terms without a @p integrate by parts over z0 using the equation of motion (A.31)
metric-Weyl propagator Wabcd; (z; x; z0; x0) in (A.12) are causal for z
z0.
From the metric response as computed in (2.12), one can reproduce the modular
evolution of space-like two-point functions as e ected by (2.11), as follows.
Consider the two point function of a primary scalar operator O of dimension D, which
in the CFT vacuum is up to a constant
(2.14)
(2.15)
hO(x)O(y)i =
(x
1
y)2D
:
The modular Hamiltonian (2.12) acts on any operator O(y) in the Rindler region C(R) as
ei H
O(y)e i H = O(y( )) ;
and trivially on operators localized in the complementary Rindler region C(R). Thus if we
choose x 2 C(R) and y 2 C(R),
y0(x1
y1)
(x
y1(x0
y)2
y0)
hO(x)O(y)i :
Now, if the gravity dual of a boundary quantum
eld theory has metric g, and if g
is the analytic continuation of a Euclidean geometry and possesses a natural vacuum, we
expect that generically we will be able to approximate space-like two-point functions of
scalar operators of large dimension in the boundary theory using geodesics in g [34, 35]. In
the case at hand, the metric evolved in modular time g of Poincare AdS g is a topological
black hole up to an isometry in g [14]. Thus we expect the geodesic approximation to be
valid, and for D
1, to have up to a constant
hO(x)O(y)i
e DL 1l(x;y; )
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
(2.21)
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where L is the AdS radius and l(x; y; ) is the length of the geodesic of minimal length
connecting space-like boundary points x and y in the metric g . It follows that (cf. (2.10))
D
L
@ l(x; y; ) hO(x)O(y)i
and restricting ourselves to equal-time two-point functions without loss of generality |
space-like two-point functions can be rotated in the t x
? dimensions to be brought to equal
time without breaking the symmetry of our con guration | and comparing with (2.18),
we would like to verify
Since the length of a geodesic is invariant under linear deformations, @ l is given by the
change in length of the original geodesic in Poincare AdS. The most general such geodesic
at equal time is a semi-circle
z(w) = pr2
w2 ;
xk(w) =
yk + xk
2
+
w
r
y
k
x
k
2
;
k = 1; 2 ;
(2.22)
of radius r = p(y1
x1)2 + (y2
x2)2=2 parametrized by
r
w
r. One can check
that enforcing x 2 C(R) and y 2 C(R), the geodesic never enters the light-cone from @R
inside which the Weyl response @ Cabcd is non-zero. See gure 3. A metric perturbation
@ hab whose Weyl response vanishes is locally a pure di eomorphism, and such a
perturbation, if smooth, cannot contribute to @ l which is a di eomorphism-invariant quantity much
like (2.6). Thus @ l is entirely due to the singular kink in the metric perturbation which
exists on the boundary light-cone of @R in transverse-traceless and Fe erman-Graham gauge.
equal-time two-point function hO(x)O(y)i, with x and y in Rindler regions C(R) and C(R), where R
is the half-space x1
in the paper are in d
0. (Note we have d = 2 in the gure for ease of drawing, but all considerations
3 for which there are local gravitational excitations in the bulk.) The
lightcone and boundary light-cone from @R are shown in solid beige and transparent purple, respectively.
The solid purple line is the extremal surface E. All contributions to @ hO(x)O(y)i come from the
intersection of the geodesic with the boundary light-cone from @R.
Integrating the metric perturbation (2.12) over the geodesic given in (2.22), we have
where
Z r
r
x
k
CT 2d 2dLd 1
64GN
dw
Z 1
0
r3 I ;
Z
dz0
d
d 2x0 z01 d
?
x
xk. Thus verifying (2.21) is equivalent to checking
I / t(y1
x1)r
where the constant of proportionality only
xes the O(N ) constant CT in the
energymomentum tensor two-point function appearing in (A.9). The integral I simpli es after
integrating by parts and using the equation of motion (2.14), but we still need to integrate
numerically at a
nite cuto
" on light-cone singularities appearing in the bulk-to-bulk
kernels (A.21), as we have not isolated the analytic form of the singularities. We nd
precise agreement with (2.24) as shown for example in d = 3 in gure 4.
3
Modular Hamiltonians as precursors?
We have seen that although the modular response of the metric @ h (2.5) is naively an
integral over the extremal surface E, and propagates in the bulk space-time g from the
entirety of E, in the special case that
is the CFT vacuum in Minkowski space and R is
half-space, it reduces to a boundary term at @E = @R. The same is true for conformally
related con guration. As a consequence, the modular Hamiltonian H, which by de nition
is localized to the boundary spacetime region C(R), is seen to act locally in the bulk
spacet
each plot are as follows | left: y1 =
1 and extending in nitely in remaining boundary spatial
coordinate x2. Left: the point on E at which we measure the Weyl response. We show a t = tR,
x2 = const: cross section of the bulk space-time g which is Poincare AdS. Right: the component
@ Cz2t2 (A.68) on the point speci ed as a function of the light-cone cuto ". It diverges as " ! 0.
However, for a generic state
[H; h^(x)] is non-zero at
spacetime points x of g at which the entire operator h^(x) in its bulk-local form, including its
framing, is space-like to C(R).10 This can be seen, for instance, by considering general
regions R in the case that g is Poincare AdS. As we show in appendix section A.3, one can
use the extremality of E to integrate by parts the integrand in @ h, but there is a genuine
bulk integrand remaining that does not integrate to a boundary term. The same is true
after acting with derivatives to obtain the Weyl response @ C, which measures the
gaugeinvariant support of @ h. In particular, the Weyl response is non-vanishing on points in the
interior of E, which is guaranteed to be space-like from C(R) [6, 7]. In
gure 5 we plot a
component of the Weyl response on an interior point of E for an R which is a slab in d = 3.
The upshot is that assuming the HRT prescription of using extremal surfaces in
Lorentzian spacetimes to calculate entanglement entropy is valid, H for a generic state
and region R cannot be a simple operator in the form of an integral of the product
an O(1) number of single-trace operators with support on C(R). We have seen that the
expectation value of H, which for linear perturbations measures the entanglement entropy
of R with respect to R in the boundary eld theory, is sensitive to bulk perturbations that
10The need to consider the full di eomorphism-invariant description of a bulk operator when considering
issues of bulk locality was pointed out in [36].
log "
are space-like to the region on which H is supported. This could not be the case if H
were a simple operator in the sense described above, expected to obey the causality of an
ordinary quantum
eld theory on a xed background.
Starting in [26], operators in the boundary theory whose expectation values are
sensitive to space-like bulk perturbations so as to ensure that the boundary theory encodes all of
the information | in the gravity dual in the context of gauge/gravity duality | have been
called precursors. Using the naive saddle approximation and computing expectation values
of Wilson loops with areas of extremal world-sheets in perturbed backgrounds, it would
seem that Wilson loops are precursors [37] in complete analogy to the case of modular
Hamiltonians that we have investigated, yet in [38] it was pointed out that such reasoning
fails because the naive saddle is incorrect. In fact the same
awed reasoning would lead
one to conclude that the two-point function of space-like operators in ordinary quantum
eld theory are sensitive to perturbations at space-like separation from both operators, in
violation of locality. Similarly, Wilson loops are dual to extended string states, hence the
locality and causality of perturbative string theory implies that such commutators must
vanish. One can also understand that the extremal world-sheet approximation for expectation
values of Wilson loops and the geodesic approximation for space-like two-point functions
cannot be generally valid in Lorentzian signature, from the fact that the one-point function
of the metric does not uniquely specify the state of the bulk quantum
eld theory [35].11
Here we simply point to two possibilities. The rst is that the HRT proposal is correct
in which case we have shown that modular Hamiltonians associated to spatial regions
are genuine precursors which di er qualitatively from possible precursors as previously
characterized in [40]. There it was proposed that short-distance properties of Wilson loops
may be sensitive to bulk processes at space-like separation. However, the value of a modular
Hamiltonian, for example at linear order, measures spatial entanglement which generically
includes long-range entanglement. It follows that, again generically, at least part of the
bulk information which is encoded acausally via the modular Hamiltonian is encoded in
long-distance properties of the boundary state.12 The second possibility is that the HRT
proposal needs to be modi ed in a manner in which the state of the bulk quantum
eld
theory is explicitly taken into account, and that with the correct computation of spatial
entanglement entropy, corresponding modular Hamiltonians may not be precursors as seen
in the large N limit.
4
Conclusions
In this paper, we have obtained descriptions in the gravity dual of the action of the modular
Hamiltonian | associated to an arbitrary spatial region in a quantum
eld theory state
that has a gravity dual | on its de ning state and nearby states. This was possible after
obtaining the deformation of expectation values, in particular n-point functions of
singletrace operators, in the original state evolved unitarily with the modular Hamiltonian.
11In [39] it was shown that even in static black hole backgrounds one needs to take into account non-trivial
complexi ed geodesics.
C^ hHi =
C^S for g Poincare AdS and R an arbitrary
region. For even d, the non-local part of the response sensitive to perturbations space-like to C(R) includes
a piece that is nite in the limit that the light-cone cuto
" goes to zero. See appendix section A.2.3.
The Lorentzian method of obtaining deformed expectation values relies on the rst law
of entanglement entropy, and at leading order in 1=N , the RT and HRT prescriptions for
computing entanglement entropy. We also discussed a Euclidean method that applies only
in the case of certain time-symmetric states but does not rely on the HRT prescription,
and thus could potentially be used to cross-check it.
Focusing on the metric deformation obtained using the Lorentzian method, we found
that in special symmetric cases in which the modular Hamiltonian generates a di
eomorphism on a subset of the bulk space-time, the deformation propagates causally from the
boundary of the spatial region in the quantum
eld theory. That this reduction to the
boundary does not occur generically, however, led us to the surprising statement that the
action of a generic modular Hamiltonian does not respect causality in bulk perturbation
theory, or in other words, cannot be written as an integral of the product of an O(1) number
of single-trace operators with support on C(R).
That the interior points of generic HRT surfaces are not in causal contact with C(R)
is actually crucial for it to be compatible with causality in the boundary quantum
eld
theory [7]. Yet we found that it is precisely this feature that puts it at tension with causality
in the low energy bulk quantum
eld theory. It is an interesting future direction to think
about whether and how the HRT prescription could be modi ed to resolve this tension.
We pointed out that any modi cation will likely have to do with incorporating the state
of the bulk quantum
eld theory more explicitly.
It will also be interesting to explore whether we can obtain further characterizations of
the action of generic modular Hamiltonians with the aid of methods we began developing
in this paper.
Acknowledgments
The authors would like to thank Ben Freivogel, Guido Festuccia, Veronika Hubeny, Juan
Maldacena, Mukund Rangamani, and Steve Shenker for useful discussions. J.S. would also
like to thank Ian Moult and Anand Natarajan for helpful conversations. D.J. is supported in
part by NSF CAREER Award PHY-1352084. J.S. was supported by the U.S. Department
of Energy under cooperative research agreement Contract Number DE-SC00012567.
A
Metric and Weyl response in Poincare AdS
Here we consider the metric response (2.5) for the simplest state
given by the
d-dimensional Minkowski vacuum of a CFT. The dual metric g is (d + 1)-dimensional
Poincare AdS,
L2
z2
ds2 =
dz2 +
dx dx
:
We use Greek indices ; ; : : : 2 f0; 1; : : : ; d
1g on boundary coordinates.
In order to solve for the metric perturbation operator h^, we impose transverse-traceless
and Fe erman-Graham gauge conditions, after which perturbations take the form
ds2 =
L2
z2 dz2 +
L2
z2
+ h (z; x) dx dx :
(A.1)
(A.2)
In the absence of any sources, the normalizable modes only depend on the expectation
value of the conserved and traceless boundary energy-momentum tensor T
naturally be constructed as14
We proceed to note [43]
h (z; k) =
L2
z2 d hT (k)i d (z; jkj) ;
d =
16 GN
dLd 1
d(z; jkj) = d
d=2
Jd=2(jkjz) ;
d = 2d=2
d
2
+ 1
h^ (z; x) = d z2
ddk
(
2
)
d ( k2) T (k)
d(z; jkj)eix k :
Z
h (z; k) =
ddx e ik xh (z; x) :
(A.5)
as z ! 0. It follows that the bulk metric perturbation operator can
and
= 0 ;
2)
h
= 0
do so after Fourier-transforming along boundary coordinates,
h T (k); T (k0) i = CT N
(k) sgn(!) ( k2)jkjd d(k + k0)
where CT is proportional to N and N
(k) is the tensor structure computed in (B.12).
Then the metric response for spatial region R with corresponding minimal surface E is
given by
d
=
1
8GN
Z
E
i Dhh^ (z0; x0); h^ (z; x)
iE
d
L4
G
(z; x; z0; x0) ;
(A.10)
where we have de ned a metric-metric propagator
G
(z; x; z0; x0) / i Dhz2h^ (z; x); z02h^ (z0; x0)iE
We then solve for normalizable modes satisfying linearized Einstein equations
h
d
(A.3)
(A.4)
(A.6)
(A.7)
(A.8)
(A.9)
(A.11)
i(zz0)d=2 Z
ddk
(
2
)
d
eik (x x0)N
(k) ( k2)sgn(!)Jd=2(jkjz)Jd=2(jkjz0) :
13 k
2 = !2
k2 > 0 in order for the perturbation h
to be nite at the Poincare horizon.
14A scalar eld operator in the bulk was constructed analogously in [43].
In (A.10) w are coordinates on E and z0(w); x0(w) give the embedding of E in the
unperturbed metric (A.1). To measure the gauge-invariant support of the metric response one
can compute its Weyl tensor
L4
where Wabcd;
is the metric-Weyl tensor propagator obtained from (A.11). The linearized
Weyl tensor about anti-de Stter space transforms homogeneously under di eomorphisms.
Note the Ricci tensor can at most convey information about the bulk energy-momentum
tensor, which is zero for
at linear order in .
A.1
Metric-metric and metric-Weyl propagators
Here we derive explicit expressions for G
(z; x; z0; x0) and Wabcd; (z; x; z0; x0), and see
the extent to which they are causal.15 Below we sometimes drop the arguments of the
propagators.
The metric propagator (A.11) can be expressed as a sum of terms in which an s number
Gs(z; x; z0; x0)
i zz0 d=2Z ddk
; ( k2)sgn(!)eik (x x0) Jd=2(jkjz)Jd=2(jkjz0)
; s = 0; 2; 4
=
(zz0)d=2 Z 1
2
0
(
2
)
d
dm2 G(m)(x
jkjs
x0) Jd=2(mz)Jd=2(mz0)
:
ms
(A.13)
We have expressed Gs as a weighted integral of the causal propagator of a real scalar eld
in the boundary eld theory [44],
Z
ddk
(
2
)
d
h (x); (x0) i
G(m)(x
x0) = 2 i
k2 + m2 sgn(!)eik (x x0)
(A.14)
where
and m is the mass of .16
(x) =
Z
d
d 1k
1
(
2
)
d 1 p2Ek
akeix k + ake ix k ;
Ek = pm2 + k2
(A.15)
15Commutation relations between metric perturbations and between the Weyl tensor and the boundary
energy-momentum tensor in AdS were studied previously in [25] using slightly di erent smearing functions.
16Note G(m) has been normalized such that
= d 1(x
x0) :
t=t0
(A.16)
= lim
1
where our notation
Gm and consequently Gs can be regulated by inserting i"'s in (A.14) which x the
ordering of operators as in (B.4),17
G(m)(x x0) = lim 2 i
d (k2 +m2)
(!)e i!(t t0 i")
( !)e i!(t t0+i") ei~k (~x ~x0)
is such that
+ creates the `forward' Wightman function h (x) (x0)i. Substituting (A.17)
in (A.13) and rewriting [45]
Z 1
0
dm
=
K ( im )J (mz)J (mz0)
where
we arrive at
$
pz2 + z02
2zz0 cos
~
Gs
where in the complex plane the function
Z
1
2d s 0
(A.17)
(A.19)
(A.20)
(A.21)
(A.22)
(A.23)
(A.24)
K (i m)
(i )
+
!
+ c.c. ;
d
2
1
2
(t
t0)2
(~x
~x0)2 ;
p(t
t0
i")2
(~x
~x)2
(A.18)
$2
2
d
2
Gs(z; x; z0; x0) = "li!m0 cd zz0 d ( 1)s=2
2s
1
2 i
G~s( 2; z; z0)
; cd
2
+
2
( 1)d+1
d+1
2
d+1
2
Fd;s ( )
s
2
d
s
2 2F1
2
; d
;
s d
> 1 on the real axis, and a pole at
= 1 if d
s > 0.
In order to obtain the singularities of Gs on the boundary light cone
2 = 0 and bulk
light cone
2
(z
z0)2 = 0 as well as its nite parts, it is necessary to evaluate the full
Wightman propagators ~
Gs
~
Gs
;
G~s+ =
~
Gs
:
Note that the regulated G~s+ is analytic as a function of space-time coordinates (x; z), (x0; z0).
Thus we can choose to perform the integral (A.22) on a convenient open set. Except for
the marginal case d = 4 s = 4, such an open set is
2 < 0, for which Fd;s is evaluated away
17In (A.17)
is used to denote a fraction and should not be confused with the used to index a boundary
coordinate. Similarly in (A.19) the Greek letters ; ; denote fractions.
from its branch cut and the integral is easily obtained. For brevity we do not write down
the propagators explicitly.
To test the causality of Gs, it is useful to isolate its nite parts in the limit " ! 0,
Gs;f (z; x; z0; x0)
cd sgn(t
t0) (
2
) (zz0)d ( 1)s=2
2d s
G~s;f ( 2; z; z0) ;
G~s;f
lim
"!0 2 i 0
G~s;f can be evaluated by drawing its contour in the
in gure 6 and accounting for pole and branch cut contributions,18
= tan( =2) plane shown as shown
+
(z + z0)2
(z
z0)2
2d+1 d
Res
= 0 (1 + 2)d+1 Fd;s
$2
2
0
d sind
F~d;s
$2
2
where F~d;s(x)
(2 i) 1 Fd;s(x)jxx+ii"" is the jump across the branch cut
F~d;s(x) = ( 1)d s
s
2
d
s
2
d2 + 1 + 2s
s
d
2
2
; d
s
2
Due to the relevant F~d;s vanishing, one nds that for space-like separation
2 (z z0)2 < 0,
G0;f vanishes for all d while G2;f and G4;f vanish for even d and even d > 4, respectively.
Also accounting for singularities at
2 = 0, only G0 is causal. However, restricting to
(d
2)z 1@z)G4 are causal for all d.
Next we derive explicit expressions for G
and Wabcd; . Noting
; d + 1
s; 1
x :
one can write
G
=
d
2
(
d 1
2
D
1
(
G4 :
+
$
)
Here G
/ hz2h^ ; z02h^
i
structure of derivatives encodes (A.3), while G4 satis es the equation following from (A.4),
satis es Einstein equations, and in the above expression the
z
d 1
18For d = 3, s = 4 and d = 4, s = 4 there is no pole at
= 0 and no contribution from the contour
around that point.
(A.25)
(A.26)
+
(A.27)
(A.28)
(A.29)
!
G4
(A.30)
(A.31)
z2Cz
z2C
1
2
1
d
2
d
W
d
2
1
(
where
z2h , we also obtain
2Wz
and
where
and
2W (1)
2W (
2
)
2zz0
2
= tan( =2) plane. Generically there is a pole at
and a branch cut along
> 0 on the real axis.
Note some components of G
This is allowed because the metric operator h^
(z; x; z0; x0) are acausal as G4 and G2 = @2G4 are acausal.
is gauge-dependent. Given (A.10), (A.30),
and expressions for independent components of the Weyl tensor linearized around AdS [25]
2W (3)
d
2
+
d
z
(
)
2
) :
;
) ;
(
)
$
(
$
1
)
(
(A.32)
(A.33)
(A.34)
(A.35)
$
) ;
(A.36)
(A.37)
propagator Wabcd;
completely causal.
2)z 1@z G4 are only causal for z > z0, the metric-Weyl
is causal only for z > z0. We expect the Weyl-Weyl propagator to be
A.2
Causality of modular response from extremal surface
Given the formulas (A.10) and (A.12), we can check that @ h
gauge-invariant and gauge-homogeonous, are causal from the extremal surface as argued
near (2.6). In order to do so, we express the extremal surface condition in a form that
can used to integrate by parts over the surface. For summaries of the di erential geometry
used below, see e.g. [46, 47].
A.2.1
Extremal surface condition
For an arbitrary space-like codimension-2 surface embedded in a (d + 1)-dimensional
background as xa(w ), we can introduce tangential vectors
HJEP09(216)8
and unit normal vectors
satisfying
The geometry of the surface is characterized by the induced metric
e a
= 1; : : : ; d
1
nAa ;
A = t; r
gabnAanBb = AB ;
gabe anAb = 0 :
whose associated Christo el symbols are
and the extrinsic curvatures
= gabe ae b
= e be arae b
KA
= e ae branbA :
e arantb = Kt
completeness relation
Specializing to the static case where there exists a Killing vector @t, the entire surface
lies at constant t so et = 0 and the time-like normal vector19 is nta =
taz=L for which
= 0. Denoting nta
na, nra
ra, and Kr simply as K
, there is a
gab =
nanb + rarb +
e ae b
and the Gauss and Weingarten equations decomposing the derivatives of tangential and
normal vectors along the surface are
19We choose it to point in the direction of increasing t.
r e a +
abce be c =
abce brc = K
K
ra ;
e a :
(A.38)
(A.39)
(A.40)
(A.41)
(A.42)
(A.43)
(A.44)
(A.45)
(A.46)
Using (A.45), the extremal surface condition is
We record the Christo el symbols in Poincare AdS (A.1),
K =
K
= 0
()
1=2
e a
+ 1=2
e be c abc = 0 :
(A.47)
zzz =
tzt =
izi =
1
ttz =
iiz =
1
ztt =
1
;
zii =
1
z
;
(A.48)
1 are spatial boundary coordinates.
A.2.2
i
The boundary energy-momentum tensor response is given by taking z ! 0 in (A.10),20
Z
E
d 1w 1=2
e0 e0 z0 2D
K4(x
x0; z0)
where D
is as in (A.30) and we have de ned bulk-to-boundary kernels
Ks(x
x0; z0)
lim
"!0 z!0
lim z dGs;"(z; x; z0; x0)
where Gs;" are the kernels in (A.21) before the " ! 0 limit is taken. For consistency, the
limit " ! 0 must be taken after z ! 0.
Next we note the causality properties of Ks. All Ks are boundary-casual, i.e. vanish
for
2 < 0, but K0 is also causal, i.e. vanishes for
2
z2 < 0. Below we integrate by parts
using the extremal surface condition the derivatives in D
K4 which are contracted in
the (A.49), and show that the remaining integrand is causal. Then the boundary response
is either boundary-causal from @R or causal from E, and since J~(E) intersects with the
AdS boundary at J (@R), it is restricted to J (@R). The combinations of K2 and K4 which
will appear in the remaining integrand and are causal for all d are
K^4 ;
^
Ks
1 +
z0
2 z
K^2 ;
^
K2
2(d
1
2 z
K^4 ;
z0 2Ks ;
s = 2; 4 :
and
where
z0
2 z
spatial indices i; j. From (A.30),
Z
E
20In this subsection and the next all equalities are up to constant factors.
d
e0 e0
(A.49)
(A.50)
(A.51)
(A.52)
(A.53)
(A.54)
Using the extremal surface condition (A.47) to integrate by parts and the Christo el
symbols (A.48), the rst term in (A.54) is proportional to
Z
E
Z
E
Z
E
Z
E
d
where we can identify (A.51). Similarly, integrating twice by parts the third term in (A.54),
(boundary term)
Z
E
and we see that between the second and third terms in (A.54) the bulk integrand only
depends on the combinations in (A.52). Thus the entire expression (A.54) reduces to a
boundary term at @R and a bulk integral which is causal from E. It is easy to check the
same is true for @ hTij i by repeatedly integrating by parts as in (A.55) and (A.56).
A.2.3
Causality of bulk response @ Cabcd
Entirely parallel considerations as in the previous section allow us to prove that the
Weyl response is causal from the extremal surface.
Recall that kernels appearing in
Wabcd; (z; x; z0; x0), in addition to being boundary-causal, are half-causal i.e. causal for
z > z0 or the Weyl tensor inserted below the metric. After integrating by parts using
the extremal surface condition, we are left with a bulk integrand which is causal, while the
boundary term at z0 = 0 is also causal due to the kernels in Wabcd;
being boundary-causal
and half-causal. The combinations which will remain after integration by parts and which
are causal for all d, are besides G^0,
1 +
z0
z0
2 z
G^4 ;
1
2
z02
1+
z0
2 z
2
d 2
z
G^4 :
e0ze0 1
z0
1 +
z0
2 z
^
K2
1
z02
z0 z
1 +
z0
2 z
(A.55)
(A.56)
^
K4
(A.57)
(A.58)
(A.59)
by parts in (A.12) to produce a bulk integrand depending on (A.57), while the remaining
combination of terms
Z
E
d 1w 1=2e0 e0
z
2
G4
1
will similarly produce a bulk integrand depending on the combinations (A.58). One needs to
integrate once and twice by parts as in (A.55) and (A.56), respectively. All boundary terms
2
(d
{ 23 {
Finally, we note that in odd d, the bulk integrand left after integrating by parts as
above is not only causal, but vanishes identically except on the light cone
This follows from the fact that inside the light cone (z
z0)2 <
2 < (z + z0)2,
z
2
Z
E
and
1
2
z02
1 +
z0
2 z
2
^
G4 / sgn(t
have trivial dependence on boundary coordinates x , while (A.57) and (A.58) vanish. Thus
causal propagation of metric perturbations from extremal surfaces in Poincare AdS follows
the Huygens principle that the light-front does not disperse in odd spatial dimensions.
A.3
Response from interior of extremal surface
The extremal surface condition restricts the Weyl response to be causal from the extremal
surface, but it does not further reduce the response to be causal from the boundary of the
extremal surface, except in the case that the spatial region R is half-space or a sphere.
Thus generically, the modular response is non-zero on and propagates from interior points
of the extremal surface. This is explicitly seen as follows.
After using the extremal condition to integrate by parts as in (A.55) and (A.56),
(A.60)
(A.61)
(A.62)
(A.63)
(A.64)
(A.65)
G0 / sgn(t
Z
E
Z
E
G^2 ;
e0 e0
e0ze0i 1
d
+ 2(d
d 1w 1=2
2(d
d 1w 1=2
e0ze0
e0ze0z 1
z0 F2
1
i z0 F1
z0
2 z
G^4 ;
1
2 z
G^4 + G^2
where
satisfy the relations
F1 =
F3 =
1 +
2(d
z0
2 z
Z
E
Z
E
d
2
d
2
F3 ;
F2 =
z0
2
F3
from the equation of motion (A.30). Then neglecting boundary terms,
d 1w 1=2
(d
2)e0ze0z F3
2(d
where using the completeness relation (A.44) in the rst line, and the Christo el
symbols (A.48) in the second, we have up to a constant factor,
Z
E
d 1w 1=2 rzri 2
z0 F1
Z
E
and carrying out a similar procedure,
d
z0
1 rzrz ij
2
It is easy to check that (A.66) and (A.67) reduce to a boundary integral when E is the
extremal surface for half-space or a sphere. However, this reduction does not occur for
general spatial regions R. This is true even after acting with derivatives to obtain the
Weyl response. For example we have
d
z0 F1 + rirj G^0
for spatial indices i; j in the case that R is a strip in d = 3.
B
Two-point function of energy-momentum tensor in momentum space
Here we obtain the Fourier transform of the Wightman two-point function of a symmetric
and traceless tensor operator of dimension D,
where
and
h0jO
(x) ;
I
(x) = I (x)I (x) + I (x)I (x)
2
d
C
x2D I
I (x) =
x2 =
2x x
x2
We are in Lorentzian at space R1;d 1. Note (B.1) is determined by conformal
symmetry [48] and the "'s in (B.4) have been placed to
x the ordering of operators on the l.h.s.
of (B.1). The result has appeared in the literature previously [49], but we include it here
to make our presentation self-contained.
First note the scalar operator result21
(A.66)
rirj G^0 :
(A.67)
(A.68)
(B.1)
(B.2)
(B.3)
(B.4)
Gd;D(k)
Z
ddx e ik x
1
i")2
(d=2
(D)
x2
D)
D
d=2 (2i)d 2D+1
(!) ( k2)( k )
2 D d=2 :
(B.5)
21The integral below converges only for D < 1, but we assume the result can be continued to D
Now we Fourier-transform
Z
ddx e ik x x x
(x2)D+1 =
1
4(D
1)D
and similarly,
to
to
Z
x x
(x2)D+1 =
1
4(D
1)D
1
(x2)D 1
+ 2(D
1)
(B.6)
( k k Gd;D 1(k) + 2(D
1)
Gd;D(k))
(B.7)
x x x x
(x2)D+2
=
1
16(D
2)(D
1)D(D + 1)
4(D
2)(D
1)
+ 8(D
2)(D
1)D
x x
+ (perm.) !
(x2)D+1
ddxe ik x x x x x
(x2)D+2
k k k k
16(D
2)(D
k k
8(D
Gd;D 1(k)+
Putting (B.5), (B.7) and (B.9) together, we have
Z
ddx e ik x I
+ (d 2D)(D
1)
+ 2(d
2D + 2)(d
2D)
k k
k2
k k k k !
D(D + 1)
d
1
2(d
2D)
k k
k2
(x2)D
1
(x2)D 2
k k k k
k4
2k k
k2
:
(B.8)
(B.9)
Gd;D(k):
k k
k2
(B.10)
(B.12)
(B.13)
HJEP09(216)8
where CT is some constant proportional to N , and
with
N
(k) =
d
1
4
I
(k) +
(d
2)
I
(k) = I (k)I (k) +
I (k) =
which agrees when d = 4 with (3.4) in [50].
For the energy-momentum tensor with D = d, we may write
h0jT (k)T (k0)j0i = CT d(k + k0) (!) ( k2)( k2)d=2N
(B.11)
1
2
I
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