Reformulating our recent result (arXiv:1007.1246 [hep-th]) in coordinate
space we point out that no matter how regular is short-distance behavior of
Green's function the entanglement entropy in the corresponding quantum field
theory is always UV divergent. In particular, we discuss a recent example by
Padmanabhan (arXiv:1007.5066 [gr-qc]) of a regular Green's function and show
that provided this function arises in a field theory the entanglement entropy
in this theory is UV divergent and calculate the leading divergent term.Comment: LaTeX, 6 page

Dmitry Nesterov

DV Vassilevich

H Casini

L Bombelli

M Srednicki

Sergey N. Solodukhin

T Padmanabhan

English

arXiv

Crossref

Journal of High Energy Physics

Short-distance regularity of Green’s function and UV divergences in entanglement entropy

Springer - Publisher Connector

J
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P09(2010)041
Published for SISSA by Springer
Received: August 13, 2010
Revised: August 23, 2010
Accepted: August 24, 2010
Published: September 13, 2010
Short-distance regularity of Green’s function and UV
divergences in entanglement entropy
Dmitry Nesterova,b and Sergey N. Solodukhina
aLaboratoire de Mathe´matiques et Physique The´orique,
Universite´ Franc¸ois-Rabelais Tours Fe´de´ration Denis Poisson - CNRS,
Parc de Grandmont, 37200 Tours, France
bTheory Department, Lebedev Physical Institute,
Leninsky Prospect 53, 119991 Moscow, Russia
E-mail: Dmitry.Nesterov@lmpt.univ-tours.fr,
Sergey.Solodukhin@lmpt.univ-tours.fr
Abstract: Reformulating our recent result (arXiv:1007.1246 [hep-th]) in coordinate space
we point out that no matter how regular is short-distance behavior of Green’s function the
entanglement entropy in the corresponding quantum field theory is always UV divergent.
In particular, we discuss a recent example by Padmanabhan (arXiv:1007.5066 [gr-qc]) of
a regular Green’s function and show that provided this function arises in a field theory
the entanglement entropy in this theory is UV divergent and calculate the leading diver-
gent term.
Keywords: Field Theories in Higher Dimensions, Renormalization Regularization and
Renormalons
ArXiv ePrint: 1008.0777
Open Access doi:10.1007/JHEP09(2010)041
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Contents
1 Introduction 1
2 Heat kernel, Green’s function and entanglement entropy 2
3 Padmanabhan’s example 4
4 Conclusions 5
1 Introduction
Entanglement entropy [1, 2] remains a fascinating subject of current research (for a recent
review see [3]). It is defined by tracing degrees of freedom residing inside a surface Σ and,
to some degree, measures the short-distance correlations across the surface. Its geometrical
feature (the proportionality to the area A of the surface) makes it a very attractive candi-
date to provide a statistical explication to the gravitational entropy associated to horizons.
A major difficulty on this way, however, is the fact that entanglement entropy calculated
for a quantum free field is UV divergent. In d space-time dimensions one has that
Sent ∼ A
ǫd−2
, (1.1)
where ǫ is an UV cut-off. A simple (and perhaps somewhat naive) way to understand the
origin of the divergence in the entropy is to relate it to the short-distance divergence of the
2-point function, in d space-time dimensions one has in a standard field theory
< φ(x), φ(y) >= G(x, y) =
Ωd
|x− y|d−2 , (1.2)
where Ωd =
Γ(d−2
2
)
4π
d
2
andG(x, y) is Green’s function. At first sight this relation seems natural:
two sub-systems separated by surface Σ know about each other due to the short-distance
correlations that exist between the modes residing on different sides of the surface. As a
result of this correlation the entropy is non-vanishing and is determined by geometry of the
surface, to leading order by the area. Since the short-distance correlations are divergent,
as in (1.2), this divergence seems to manifest in the UV divergences in the entropy (1.1).
Therefore, one may think that there is a one-to-one correspondence between diver-
gences in (1.1) and (1.2) so that one may expect that in a theory in which 2-point functions
are regular in the coincidence limit the entanglement entropy would be automatically UV
finite. A simple example of this sort is the following modification of (1.2)
GL(x, y) =
Ωd
((x− y)2 + L2) d−22
, (1.3)
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where the short-distance divergences are now regularized by parameter L. This example
was recently considered by Padmanabhan [4] who argued that in a theory with Green’s
function (1.3) entanglement entropy is UV finite. In his approach the parameter L in-
corporates some fundamental, possibly stringy, effects so that GL, as it appears in [4],
is seemingly not a field theoretical Green’s function. Nevertheless, since (1.3) may well
appear in some field theory, one may have impression from reading [4] that in the field
theory the regularity of Green’s function in the coincidence limit implies the UV finiteness
of entanglement entropy. The latter is not the case as we show in this note.
In a recent paper [5] we have proved a sort of “no-go theorem” by showing that no
matter how well is UV behavior of the propagator of a quantum field theory the entangle-
ment entropy calculated in this theory is always UV divergent. We have showed this by
using momentum representation of the heat kernel for a quantum field satisfying a rather
general (Lorentz invariant or non-Lorentz invariant) field equation. Obviously a field the-
ory, in which (1.3) appears as Green’s function, belongs to the class of theories we have
considered and hence entanglement entropy in this theory is still UV divergent even though
the short-distance correlations in (1.3) are regular.
In this note we first reformulate the statement made in [5] in terms of Green’s function
and then discuss a theory with Green’s function of the type (1.3). For simplicity we only
consider Lorentz invariant field theories.
2 Heat kernel, Green’s function and entanglement entropy
We consider a quantum field that satisfies a rather general Lorentz invariant field equation
Dφ = F ()φ = 0 , (2.1)
where F () is an arbitrary function of the Laplace operator  = −∂µ∂µ. Many important
quantities that characterize the quantum field can be expressed in terms of the heat kernel
K(s,X,X ′) =< X|e−sD|X ′ > (for a standard review on the heat kernel method see [6]).
The latter is defined as a solution to the heat equation{
(∂s +D)K(s,X, Y ) = 0 ,
K(s=0,X, Y ) = δ(X,Y ) .
(2.2)
In particular, the effective action is defined as
W = −1
2
∫ ∞
ǫ2
ds
s
TrK(s) , (2.3)
where parameter ǫ is an UV cutoff. In flat spacetime one can use the Fourier transform in
order to solve the heat equation (2.2). In d spacetime dimensions one has
K(s,X, Y ) =
1
(2π)d
∫
ddp eipµ(X
µ−Y µ) e−sF (p
2) . (2.4)
Note that we consider Euclidean theory so that p2 ≥ 0. The Green’s function is a solution
to the field equation with a delta-like source
DG(X,Y ) = δ(X,Y ) (2.5)
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and can be expressed in terms of the heat kernel as follows
G(X,Y ) =
∫ ∞
0
dsK(s,X, Y ) . (2.6)
Obviously, Green’s function can be represented in terms of the Fourier transform in a
manner similar to (2.4),
G(X,Y ) =
1
(2π)d
∫
ddp eipµ(X
µ−Y µ) G(p2) . (2.7)
Using (2.6) or Fourier transformed (2.5) we find that
G(p2) = 1/F (p2) . (2.8)
In a Lorentz invariant theory described by equation (2.1) Green’s function (2.7) is a function
of the space-time interval σ = (X − Y )2. On the other hand, any given Green’s function
G(σ) can be Fourier decomposed as in (2.7). Then using relation (2.8) we can restore the
field equation this Green’s function satisfies. In the coincidence limit, X = Y , we find
from (2.7)
G(X,X) =
2
Γ(d2)
1
(4π)
d
2
∫ ∞
0
dp pd−1G(p2) . (2.9)
This limit is finite if function G(p2) is decaying faster than 1/pd for large p.
As was shown in [5] in a theory described by the field equation (2.1) entanglement
entropy takes the form (for simplicity we take Σ to be a (d− 2)-dimensional plane)
S =
A(Σ)
12 · (4π)(d−2)/2
∫ ∞
ǫ2
ds
s
Pd−2(s) , (2.10)
where function Pn(s) is defined as follows
Pn(s) =
2
Γ(n2 )
∫ ∞
0
dp pn−1 e−sF (p
2) . (2.11)
Clearly, the function Pd−2(s) is divergent in the limit s→ 0 for any function F (p2) growing
at large p.1 This can be seen by taking literally this limit in the integral (2.11) the
integration over p then is divergent in the upper limit. For small but finite s this divergence
is translated into divergence in variable s. Since arbitrary F (p2) means arbitrary Green’s
functions (2.7), including those which are regular at X = Y (i.e. integral in (2.9) is finite),
we conclude that no matter how well Green’s function behaves in the coincidence limit the
entanglement entropy remains UV divergent. This is a slightly different formulation of the
statement made in [5].
1Our consideration is valid for the class of functions F (p2) for which integral representation (2.4) for the
heat kernel is well defined. This class includes functions unboundedly growing for large p. One may suggest
to interchange the integrations over p and s in (2.10) and (2.11). Then naivley the integral over momentaR
∞
0
dp pd−3 lnF (p2) may seem to be finite if lnF (p2) is smaller than 1/pd−2. However, this function F (p2)
does not belong to the class of functions specified above and should be excluded from consideration. In
fact the integral (2.11) is divergent in this case and hence the interchange of integration over s and p is not
allowed. We thank the referee for bringing to us this example.
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3 Padmanabhan’s example
For simplicity in this section we consider the case of space-time dimension d = 4. Green’s
function (1.3) gives us an example of Green’s function which is regular in coincidence limit
X = Y . As was noted in a recent paper [4] function (1.3) can be represented in a form
similar to (2.6)
GL(X,Y ) =
∫ ∞
0
ds H(s,X, Y ) ,
H(s,X, Y ) =
1
(4πs)2
e−|X−Y |
2/4s−L2/4s . (3.1)
If functionH(s,X, Y ) was a heat kernel then the L-dependent term in (3.1) would regularize
all UV divergences so that the effective action (2.3) and entanglement entropy (2.10) would
appear UV finite. This is the point of view advocated in [4]. However, despite the apparent
similarity of (3.1) and (2.6) the functionH(s,X, Y ) is NOT a heat kernel: it does not satisfy
neither the field equation (2.2) for any differential operator D nor the “initial condition”
at s = 0, i.e. it does not reproduce a delta-function.
On the other hand, the Fourier transform for Green’s function GL(X,Y ) (1.3) is well
defined. We find that [7]
G(p2) =
L
p
K1(pL) . (3.2)
The corresponding heat kernel is defined by (2.4) taking into account relation (2.8). Hence
we can reconstruct the relevant field equation F ()φ = 0 and obtain
F () =
√

L
1
K1(L
√
)
. (3.3)
For small values of p function F (p2) = G−1(p2) behaves as p2 while for large p it
grows exponentially
F (p2) ≃
√
2
πL
p3/2 epL . (3.4)
For a function F (p2) with asymptotic behavior (3.4) we have that
P2(s) =
1
L2
(
ln
s
L2
)2
+O
(
ln
s
L2
)
. (3.5)
Hence entanglement entropy (2.10) is UV divergent and the leading divergence
S =
A
48πL2
1
3
(
ln
ǫ2
L2
)3
(3.6)
is logarithmic. This is despite the fact that Green’s function (1.3) is completely regular in
the coincidence limit.
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4 Conclusions
In this note we have emphasized that, perhaps contrary to intuition, the short distance
regularity of Green’s function does not imply the UV finiteness of entanglement entropy.
In fact our statement is rather general: in any theory characterized by Lorentz invariant
Green’s function the corresponding entanglement entropy is UV divergent, the degree of
divergence is determined by the behavior of Fourier transform G(p2) (propagator) for large
values of p. The requirement of the Lorentz symmetry is not essential. As we show in [5]
entanglement entropy remains UV divergent even if the Lorentz symmetry is violated by
a generic term in the field operator.
Comparing our results with approach of Padmanabhan [4] we should note that from
the field theoretical point of view his prescription to use the function H(s,X, Y ) instead
of heat kernel can be viewed as a new method of regularization of the UV divergences
similar to the Pauli-Villars regularization. In his regularization parameter L plays the role
of the UV cut-off similar to the parameter ǫ in the proper time regularization which we
use. Then the divergences in ǫ of effective action or entropy are just replaced by equivalent
divergences when L is taken to zero.
The postdoctoral position of D.N. is financed by the University of Tours. D.N. was
also supported by the RFBR grant 08-02-00725.
Open Access. This article is distributed under the terms of the Creative Commons
Attribution Noncommercial License which permits any noncommercial use, distribution,
and reproduction in any medium, provided the original author(s) and source are credited.
References
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[3] H. Casini and M. Huerta, Entanglement entropy in free quantum field theory,
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[4] T. Padmanabhan, Finite entanglement entropy from the zero-point-area of spacetime,
[arXiv:1007.5066] [SPIRES].
[5] D. Nesterov and S.N. Solodukhin, Gravitational effective action and entanglement entropy
in UV modified theories with and without Lorentz symmetry, arXiv:1007.1246 [SPIRES] .
[6] D.V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rept. 388 (2003) 279
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[7] T. Padmanabhan, Duality and zero-point length of spacetime,
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Short-distance regularity of Green's function and UV divergences in
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Short-distance regularity of Green's function and UV divergences in entanglement entropy

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