Mutual Information of Generalized Free Fields Valent n Benedetti Horacio Casini and Pedro J. Martinez Centro At omico Bariloche and CONICET

2025-05-02 0 0 2.68MB 28 页 10玖币
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Mutual Information of Generalized Free Fields
Valent´ın Benedetti∗
, Horacio Casini†
, and Pedro J. Martinez‡
Centro At´omico Bariloche and CONICET
S. C. de Bariloche, R´ıo Negro, R8402AGP, Argentina
Abstract
We study generalized free fields (GFF) from the point of view of information measures. We first
review conformal GFF, their holographic representation, and the ambiguities in the assignation of
algebras to regions that arise in these theories. Then we study the mutual information (MI) in
several geometric configurations. The MI displays unusual features at the short distance limit: a
leading volume term rather than an area term, and a logarithmic term in any dimensions rather
than only for even dimensions as in ordinary CFT’s. We find the dependence of some subleading
terms on the conformal dimension ∆ of the GFF. We study the long distance limit of the MI for
regions with boundary in the null cone. The pinching limit of these surfaces show the GFF behaves
as an interacting model from the MI point of view. The pinching exponents depend on the choice
of algebra. The entanglement wedge algebra choice allows these models to “fake” causality, giving
results consistent with its role in the description of large Nmodels.
∗valentin.benedetti@ib.edu.ar
†horacio.casini@cab.cnea.gov.ar
‡pedro.martinez@cab.cnea.gov.ar
arXiv:2210.00013v1 [hep-th] 30 Sep 2022
Contents
1 Introduction 1
2 Conformal GFF and local algebras 3
2.1 Algebras............................................ 4
2.2 Mutualinformation...................................... 7
3 Finiteness of the MI 8
4 Short distance mutual information 10
4.1 The conformal bulk case ∆ = (d±1)/2 .......................... 10
4.2 Volumetermforany∆ ................................... 11
4.3 Firstsubleadingterm .................................... 12
4.4 General form of the short distance expansion . . . . . . . . . . . . . . . . . . . . . . . 13
5 Long distance mutual information 14
5.1 Twospheres.......................................... 14
5.2 Long distance MI under pinching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2.1 Pinching the entanglement wedge in AdS4/CFT3................ 19
5.2.2 Pinching the causal wedge in AdS4/CFT3.................... 21
5.2.3 Comparing the entanglement and causal wedges under pinching . . . . . . . . . 22
6 Final remarks 23
1 Introduction
Generalized free fields (GFF) are the simplest models of quantum field theories (QFT) satisfying
Wightman’s axioms [1]. They are defined by having Gaussian correlations, that is, satisfying Wick’s
theorem for the n-point correlation functions. The theory is then completely specified by a two point
function satisfying positivity, spectral condition, and Poincare covariance. For a scalar field, the most
general two point function has the Kall´en Lehmann form
hφ(x)φ(y)i=Z∞
0
ds ρ(s)W0(x−y, s),(1.1)
with W0(x−y, s) the two point function of a free scalar field of square mass m2=s≥0. The spectral
density ρ(s) is a positive measure for s≥0 with at most polynomial increase in s.
GFF appear naturally in some formal results in axiomatic QFT [2, 3, 4, 5]. Due to the simplicity
of the theory they have also been used in the mathematical literature as a source of examples to
test different conjectures or analyze the independence or consistency of different properties, see for
example [6, 7, 8]. From the physical point of view, they appear naturally as limits in large Nvector
or matrix models [9]. The large Nlimit suppresses higher truncated point functions with respect to
the two point functions for the symmetric fields. A notable example are holographic theories where
generalized free fields describe the low energy sector of the theory in the large Napproximation, and
are equivalent to ordinary free fields living in AdS space [10, 11, 12].
In this paper we study the entanglement entropy of GFF or, more precisely, we analyze the behavior
of the mutual information in several cases of interest. Mutual information has the advantage of being
regularization independent. However, the setup of the problem needs some distinctions to be made.
To expose the peculiarities that appear in entropic quantities for GFF as opposed to the case of
more ordinary QFT let us consider a simple case first. When the GFF is a free field of mass m,
the spectral density consists in a single delta function ρ(s) = δ(s−m2). In this case, an algebra of
operators can be assigned to a spatial region Vat x0= 0. This algebra is generated by φand ˙
φin
V. Because of the hyperbolic equations of motion of the field, (+m2)φ= 0, this algebra coincides
with the algebra generated by the field in the causal development of the spatial region V. In this
1
case the entropy can be computed by the usual formulas for Gaussian states in terms of the field and
momentum correlator at x0= 0, see for example [13].
If we now take ρ(s) = δ(s−m2
1) + δ(s−m2
2), corresponding to the sum of two independent free
fields φ(x) = φ1(x) + φ2(x), we could still apply the same formula in terms of correlators of φand ˙
φ
at x0= 0. The algebra generated by φ, ˙
φstill closes in itself because of the numerical commutator
of the GFF. However, notice that φ(x) now obeys an equation of motion with higher number of time
derivatives (+m2
1)(+m2
2)φ= 0, so that ¨
φand φare independent operators. The inclusion or
not of the operator ¨
φleads to different algebras with different entropies. Considering just φand
˙
φat x0= 0 will give us an entropy increasing like the volume of the region because it measures
translational invariant local entanglement in field degree of freedom between φ=φ1+φ2and φ1−φ2.
An analogous calculation can be found in [14]. If we now include ¨
φ, ...
φin the algebra the result turns
out to be exactly the algebra of two independent free fields of masses m1and m2. This follows from
+m2
2
m2
2−m2
1
φ=φ1,+m2
1
m2
1−m2
2
φ=φ2,(1.2)
from which we can reconstruct the two independent field and momentum operators. Hence, this new
algebra containing higher derivatives of φis equal to the algebra of the two fields φ1, φ2in the causal
development of V, and we get an area law rather than a volume law for the entropy.
For a spectral density with any finite number nof delta functions we have an analogous situation.
We can take algebras of the field and less than 2n−1 time derivatives at x0= 0 and get a volume
term for the entropy, or, provided we include 2n−1 time derivatives, the algebra and entropy will be
the same as the one of nindependent free fields. In this last sense the nindependent free fields are
encoded in a single GFF.
In relativistic QFT it is natural to define the algebras taking a spacetime rather than a spatial
region. If we take a finite time span around the spatial region Vthere is no difference between the GFF
defined with a finite number of delta functions in the spectral density and a theory of independent free
fields. However, this discussion anticipates us the problems we can find when considering a continuous
measure ρ(s). In this case the theory has quite unusual properties. It does not satisfy the time slice
axiom [8], meaning that the algebra generated by field operators in a finite time slice around x0= 0
does not exhaust all operators of the theory. This is another way to say that the field does not obey
any local equation of motion, with any finite number of time derivatives. By the same reason it does
not contain a stress tensor. Otherwise we could use it to construct the Hamiltonian in the algebra of
a time slice. With the Hamiltonian we can then move operators in time to generate all operators in
the theory. The Hamiltonian for a GFF with spectral measure having support in a non discrete set
still exists but is rather non local.
Then, it is clear that a spacial region does not determine uniquely an algebra for these theories and
we must choose a spacetime region instead. A natural choice is to focus on causally complete regions.
These are the domain of dependence of spatial regions. However, even for a causally complete region
there is in general an ambiguity in the algebra that can be associated to it for a GFF. Ambiguities on
the assignation of algebras to regions appear also in ordinary QFT with generalized symmetries such
as the Maxwell field [15], but they are much more severe for the GFF.
A great simplification in the understanding of the nature of these ambiguities and the allowed
algebras appear with the holographic realization of (a class) of these GFF as ordinary free fields in the
bulk of a spacetime of one more dimension. We will focus on holographic GFF, specially in conformal
GFF, and profit from the dual description to define the algebras for a given region and compute the
entropy. In fact, independent computation of the entropies of the GFF in the boundary theory by
standard methods without using the holographic description run into difficulties precisely because the
nature of the algebras remain unspecified. For example, it is unclear how to apply the replica method
because there is no action for the GFF. There is an action in the holographic bulk description that
allows to apply the replica method there but the region/algebra in the bulk is not uniquely specified
by the boundary region in the GFF. Large Nholographic theories choose automatically this bulk
region through the gravity equations [16]. Another boundary way of computing the entropy would be
through formulas for Gaussian states in terms of correlation functions. This seems a complicated task.
2
One should use correlation functions in the chosen space-time region using the methods of [17]. This,
however, has only access to one specific algebra for the region which is selected from the correlator.
We will not attempt this calculation here.
An outline of the contents of the paper is as follows. We first review GFF and their holographic
description in the next section, and describe possible assignations of algebras to regions. Two choices
of algebras are specially relevant. One of them, the causal wedge algebra, is more natural from the
point of view of the GFF itself, while the other, the entanglement wedge algebra, is more relevant
from the point of view of the limits of holographic large Ntheories.
In section 3 we show why the MI can be expected to be finite even is the AdS dual space is of
infinite volume. In section 4 we explore the short distance limit of the mutual information (MI).
Interestingly, we find that the GFF has a volume law in this regime. This is in contrast to the case of
ordinary theories where we have an area law in the short distance limit, even for theories coming from
higher dimensions by Kaluza Klein dimensional reduction. The coefficient of the volume law can be
computed and is universal in the sense that it does not depend on particular details of the GFF such
as the conformal dimension. Other peculiarities include the existence of a logarithmic term for odd
spacetime dimensions. We compute this logarithmic term in d= 3 as a function of the GFF conformal
dimension.
In section 5 we study the long distance limit of the MI. For spheres in CFT’s this long distance
limit is fixed by symmetry reasons and apply as well for conformal GFF [18, 19, 20]. For GFF the
universality of this result can be understood by two reasons. The first is that there is a unique choice
of algebras for spheres. The second is that spheres have a universal modular flow in CFT’s. We show
that the fact that general results for spheres apply to GFF imply certain holographic relations for
the coefficients of the MI in general theories for different specific dimensions and spins whose reason
would be rather mysterious otherwise. Taking non spherical regions, we study the case of regions with
arbitrary boundaries in the light cone. This is useful to study the pinching limit of the MI in these
theories by taking out a pencil of null generators from the null horizon of the region [20, 21]. Two
particular pinching limits are relevant. One of them is a discriminator between free and interacting
CFT. Free here is used in the sense of having a linear equation of motion rather than having Gaussian
correlators. This gives us, as expected, vanishing MI in the pinching limit for all possible algebra
choices of the GFF. The other limit is an indicator of violations of causality in the sense of the time
slice axiom. For the causal wedge algebra we find causality violations while the entanglement wedge
algebras avoids detection of causality violations in the GFF. This is a necessary condition for this
algebras to come from the large Nlimit of a theory with stress tensor. In both cases we compute the
relevant pinching exponents for specific conformal dimensions. Here we make use of the results for the
MI of free fields derived in [20]. We end with a discussion of the results.
2 Conformal GFF and local algebras
We first introduce conformal GFF fields. We will follow the description of [12]. These have a spectral
density given by a power law
ρ(s) = s∆−d
2,(2.1)
and conformal dimension ∆. Eq. (2.1) gives a measure provided ∆ obeys the unitarity bound ∆ >
(d−2)/2. For any such ∆ the GFF defines a CFT. The case ∆ = (d−2)/2 is excluded because ρ(s)
becomes non integrable around s= 0. The free massless field has ρ(s) = δ(s) instead.
The holographic description is in AdS space. In the Poincare patch we write the metric
ds2=z−2(dz2+dx2),(2.2)
with dx2the Minkoswki metric in dspacetime dimensions and z∈(0,∞). The dual field ϕof the
GFF is a free massive field in AdS with equation of motion
z2∂2
z+z2d+ (1 −d)z∂z−m2ϕ= 0 ,(2.3)
where
m2= ∆(∆ −d) (2.4)
3
∆
m2
BF m2
BF +1/4m2
BF +1 m2
d/2
(d−2)/2
(d−1)/2
(d+ 1)/2
Figure 1: The plot shows the relation (2.4) and highlights some important points in the curve. The blue and
green colored segments correspond to the standard and alternative quantization with Dirichlet and Neumann
boundary conditions respectively. From bottom to top, the red dot is the end of the curve, where the CFTd
reaches the unitarity bound, precisely at m2=m2
BF + 1. The green dot shows the point where the massive AdS
field is conformally coupled. The yellow and blue point is the BF mass bound, the lowest possible mass in AdS
consistent with unitarity m2
BF =−d2/4. The blue point highlights the conformally coupled AdS field with the
other boundary condition. The conformal dimension ∆ at this point does not match the one of a free field in
flat d+ 1 space.
can be negative. The minimal possible mass square is given by the Breitenlohner-Freedman bound
m2≥m2
BF =−d2/4 [23]. The field ϕcan be canonically quantized with an AdS symmetric vacuum.
For −d2/4≤m2<−d2/4 + 1 there are are two inequivalent quantizations corresponding to the two
roots of (2.4). These are defined by different boundary conditions for the field at the boundary z= 0 of
AdS. Dirichlet boundary condition corresponds to ∆ = 1
2d+√d2+ 4m2and Neumann boundary
condition to ∆ = 1
2d−√d2+ 4m2. See Fig. 1. For m2≥ −d2/4 + 1 only the Dirichlet boundary
condition is allowed. The limit m2→ −d2/4 + 1 of the Neumann branch hits the unitarity bound
∆→(d−2)/2. There is no holographic description of this point. There are also notable points at
m2=−d2/4+1/4, ∆ = (d±1)/2, in which the bulk is a massless conformally coupled scalar, and hence
a conformal field. These particular bulk theories can be conformally mapped to half d+ 1-dimensional
Minkowski space with metric ds2= (dz2+dx2), where we have the two possible conformal boundary
conditions at z= 0. The Neumann branch at this point has ∆ = (d−1)/2 corresponding to a free
massless d+ 1 dimensional free field, and the Dirichlet branch has a different boundary dimension
∆=(d+ 1)/2 due to the boundary condition.
The relation between the boundary and bulk fields can be described as follows. The GFF is the
boundary limit of the bulk field,
lim
z→0z−∆ϕ(x, z) = 2−α−1/2
Γ[α+ 1] φ∆(x),(2.5)
where α= ∆ −d/2, while the bulk field has a non local expression in terms of the boundary one
ϕ(x, z) = 1
√2z∆(z2)−α/2Jα(z√)φ∆(x).(2.6)
2.1 Algebras
A more illuminating relation between bulk and boundary theories is given in terms of local algebras.
If Wis a region in AdS let us call W0to the set of points spatially separated from Win the bulk. The
causal completion of Wis W00 and a causally complete region satisfies W=W00. Causally complete
4
摘要：

MutualInformationofGeneralizedFreeFieldsValentnBenedetti*,HoracioCasini„,andPedroJ.Martinez…CentroAtomicoBarilocheandCONICETS.C.deBariloche,RoNegro,R8402AGP,ArgentinaAbstractWestudygeneralizedfree elds(GFF)fromthepointofviewofinformationmeasures.We rstreviewconformalGFF,theirholographicrepresen...

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