Debye mass in de Sitter space
Accepted: May
mass in de Sitter space
Fedor K. Popov 0 1
Time Symmetries
0 Princeton , NJ 08544 , U.S.A
1 Department of Physics, Princeton University
We calculate the oneloop contributions to the polarization operator for scalar quantum electrodynamics in di erent external electromagnetic and gravitational elds. In the case of gravity, de Sitter space and its di erent patches were considered. It is shown that the Debye mass appears only in the case of alphavacuum in the Expanding Poincare Patch. It can be shown either by direct computations or by using analytical and causal properties of the de Sitter space. Also, the case of constant electric eld is considered and the Debye mass is calculated.
Field Theories in Higher Dimensions; Classical Theories of Gravity; Space

Debye
1 Introduction
2
3
4
5
6
1
General properties of magnetic and Debye masses in the curved
backOne of the problems of the modern theoretical physics is the understanding of quantum
eld theory in the external gravitational or electromagnetic elds. These two problems at
the rst glance seem to be very di erent, but they share a lot of similarities and common
properties. For instance, particle production occurs in both situations [
1, 2
]. Moreover, it is
wellknown that the constant electric eld and de Sitter spacetime share a lot of common
infrared properties. Thus, both of them acquire large secularly growing contributions [3{
9, 9{16], that distort the treelevel picture by loop e ects and can possibly destroy the
initial background.
In the case of gravity it is extremely di cult even to solve a free eld theory for the
general gravity background, but in the case of highly symmetric spacetimes like de Sitter
calculations simplify and in principle can be done. Also, because the de Sitter spacetime
perhaps well describes the universe immediately after the Big Bang, quantum
eld theory in
such a background is a very interesting topic for physics community. One of the e ects that
makes de Sitter spacetime di erent from the Minkowski is the appearance of temperature
that is proportional to the Hubble constant of de Sitter spacetime TdS = 2H . For example,
if we release a detector in the de Sitter spacetime that has some energy levels, eventually
we get an excited detector with a energy distribution like thermal with a temperature
TdS [2]. It gives rise to some questions concerning the physical origin of this temperature.
For example, it is wellknown that in plasma the temperature gives photon a mass, that is
{ 1 {
mass determined by the temperature TdS , does it depend on the choice of the vacuum state?
Also, the value of the photon mass can indicate di erent divergences that are emergent
in the de Sitter space. For example, if the square mass is negative, we have a direct sign
of instability  any initial perturbation will be ampli ed. Or the mass can become very
large or even UV divergent, it means that any charge will be eventually screened. This
question can be projected to the problem of understanding of the stability of de Sitter
spacetime itself. Indeed, if the photon aquires a mass, a graviton can also get it. In the case of
thermal state in a at spacetime this mass is negative that leads to the Jeans' gravitational
instability [17{19]. The same scenario can happen also in the case of the impulse or eternal
electric eld. Due to the particle production, photon in such a medium can acquire a
nonzero mass. If this mass is negative or very large, the initial electric eld will be destroyed.
Below we will discuss how to de ne the Debye and magnetic masses for a curved
gravitational background and perform calculations for de Sitter spacetime for di erent
types of invariant vacua. We will get that there is no Debye mass for the case of
BunchDavies vacuum, but there is a nonzero photon mass for any alphavacuum or for the
de Sitter broken phase, while magnetic mass is equal zero for any chosen initial state.
The analogous calculations will be done for the case of the constant electric eld and the
impulse, it will be shown that the Debye mass diverges linearly with the growth of time
in the both situations. The last statement is similar to the linear growth of the electric
current in the Schwinger e ect [
3, 9, 12
]
2
Polarization operator in curved spacetime
In this section we will de ne Debyemass for scalar quantum electrodynamics on a some
gravitational background, that we specify later.
S =
Z
d x
4 p
g
4
2
m2 j j2 ;
g
= a2(t)
:
(2.1)
To simplify the calculations we assume that the gravitational eld doesn't depend on
spatial coordinates and the metric is diagonal. Also, because the system is not at the
equilibrium (for example the external gravitational eld may depend on time) we will
use SchwingerKeldysh diagrammatic technique to calculate the polarization operator for
photon. In this technique the number of elds is doubled: for every eld we have a classical
eld, that can be thought as its value, and the quantum part, that can be considered as a
uctuation and gives the information about the spectrum of the system [20, 21]. Despite the
fact that this technique is computationally di cult, it gives some advantages in comparison
to the Feynman one, e.g. that we can write down the classical equations of motion deformed
by quantum e ects for the classical eld [22]:
Z
t
1
xA (x; t) +
dt0d3yp
g(t0)
0 (x; tjy; t0)g 0 (t0)A (y; t0) = 0;
(2.2)
{ 2 {
where
x is the corresponding wave operator for the vector eld under some gauge
conditions in the given spacetime, for simplicity we consider Coloumb gauge @ A
= 0, that
explicitly breaks the internal coordinate reparamtrization invariance.
The polarization operator in the eq. (2.2) has the following form:
xA (x; t) + A (x; t)
dt0dd 1yp
g(t0)
0 (x; tjy; t0)g 0 (t0) = 0;
(2.5)
Z
(q; t; t0) =
d3x e iqy
(x; tjx + y; t0)
The equations of motions (2.2) can be thought also as the application of Kubo
formula [23]. Indeed, the polarization operator (2.4) has a form of a current susceptibility of
the media to the presence of external electromagnetic eld A (x; t). And we try to take
into account the in uence of the induced currents on the external electromagnetic eld.
If we assume that the
eld A (x; t) changes slowly in comparison with the scalar
eld uctuations or with the polarization operator
(x; tjy; t0) then we can assume that
A (x; t)
A (y; t0) to obtain
Where ; indexes run over the spatial directions, dots stands for the covariant di
erentiation with respect to the time f_(t) = @t a 2(t)f (t) . Substiting these expressions into
the equations of motion (2.2) with the use of the Coloumb gauge we get
where
p is an image of spatial Fourier transformation of the wave operator
x, that was
introduced above.
pA0(q; t) + A0(q; t)
gg00CE(q; t; t0) = 0;
and
pA (q; t) + A (q; t)
gg00CM (q; t; t0) = 0;
(x; tjy; t0) = i (t
t0) h j J (x; t); J (y; t0) j i
;
where J (x; t) is a current operator for the given quantum eld theory and
is an arbitrary
state of the system. If the system is invariant under spatial translations we make a Fourier
transformation of the polarization operator (2.3)
HJEP06(218)3
The last term in this equation has the form similar to the mass term, but because it
has a tensor structure, this mass will depend on the polarization. To understand better its
tensor structure we can make the Fourier transformation of (2.4) in spatial directions and
constraint it with the use of SO(3) symmetry and Ward identities. Then we get
00(q; t; t0) = CE(q; t; t0) = M E(q; t; t0);
0 = M_ E(q; t; t0) q2
;
q
=
CM (q; t; t0)
q q
q2
+ ME(q; t; t0)
q q
q4 :
Z
t
1
Z
t
1
t
Z
1
dt0p
dt0p
{ 3 {
(2.3)
(2.4)
(2.6)
(2.7)
to the scalar eld. The solid ones are photon propagators.
So we can separate equations for A
and A0 elds. Therefore we can introduce two
notions of mass like we did above
m2Deb(t) = lim
q!0
Z
t
1
dt0p
gg00CE (q; t; t0);
m2mag(t) = lim
dt0p
gg00CM (q; t; t0) (2.8)
q!0
Z
t
1
If there is a limit at t ! 1 for these functions we introduce
m2Deb;mag = lim m2Deb;mag(t)
t!1
And the equations (2.7) will be rewritten as
qA0(q; t) + m2DebA0(q; t) = 0;
qA (q; t) + m2magA (q; t) = 0:
(2.9)
That indicates the appearance of the mass for the electromagnetic eld (because the
dispersion relation for the photon has a gap). The rst one corresponds to the electric, or just
a Debye mass, while the second one corresponds to the appearance of magnetic mass. If the
magnetic mass is not equal to zero, it leads to the superconductivity [24]. If the mass (2.8)
vanishes, we can expand it at small q the next term will proportional to the momentum q
and corresponds to the electric, magnetic susceptibility and the conductivity of the state.
If we get, e.g. that m2Deb is negative or very big, it means that the system either in the
state of instability or any initial electromagnetic eld will be e ectively screened.
3
General properties of magnetic and Debye masses in the curved
background
In this section we consider the general case of a gravitational background with SO(3) spatial
symmetry and discuss the general features of the magnetic and debye mass in the presence
of an external gravitational eld. Let us choose some de nition of the vacuum state and
make the decomposition of the eld
through creation and annihilation operators
=
d3p h
(2 )3 apeipxfp(t) + bypfp (t)e ipxi ; [ap; ayq] = [bp; byq] = (2 )3 (p
(3.2)
Further we will consider only the vacuum state, that is nulli ed by the annihilation
operators ap jvaci = bp jvaci = 0. Then with the use of KeldyshSchwinger diagrammatic
technique we can calculate explicitly the magnetic and electric mass. To do it we need to
sum up the contributions from two diagrams [22] (see the gure 1). One can show that the
Debye mass doesn't get any UV divergences. Eventaully, we get the following expressions
for masses according to the formulas (2.8)
and for the case of magnetic mass we have
; (3.3)
HJEP06(218)3
Z
t
1
m2mag
= 8e2
Z
d3p
Z
d3p
(2 )3 jfp(t)j2 :
Note that sometimes we can take the limit t ! 1 to understand the behaviour of the
system after the interaction, also this limit should be taken before taking the limit q ! 0,
otherwise we get zero due to the charge conservation. For instance, in the case of quantum
electrodynamics in the at space in an excited state we have [25]:
00(q; !) /
00(0; !) = 0;
Z
d3k
nk+q
(2 )3 !k+q
nk
!k + !
lim
q!0
00(q; 0) /
;
Z
From where it follows that:
So formally,
00(0; !) = 0, but the Debye mass is not equal to zero, if nq is nontrivial.
In general case we are not able to take the limit t ! 1 because of the presence of the
external elds.
We can show that the magnetic mass is identical equal to zero. To do this we will
use the equations of motions for the modes. Then with the use of quantum mechanical
perturbation theory one can prove the following identity
Z
t
1
{ 5 {
(3.4)
(3.5)
(3.6)
(3.7)
then the equation (3.4) can be rewritten as
dt0a2(t0)Im fp2(t)fp 2(t0) ;
m2mag
= 2e2
Z
Z
d3p
(2 )3 jfp(t)j2 =
d3p
(2 )3 jfp(t)j2
2e2
Z
d3p
(2 )3 jfp(t)j2 =
Z
d3p
(2 )3 jfp(t)j2 = 0
Where in the last line we have integrated by parts the rst term and used the relation
=
. It would be interesting to consider the general case (no spatial SO(3) and
translational invariance, with o diagonal components of metric) and investigate the
conditions under which magnetic mass appears. Also it would be useful to prove that magnetic
mass is equal to zero at all orders of perturbation theory. Due to the di cultity of this
problem we will leave it for the future research.
Now let us consider the Debye mass. For the general case we can't prove that it is
equal to zero. Nevertheless as it was shown above it can appear only due to the some
singularities if we take the lower limit of the integration over t0 in (3.3) to
us rewrite the formula for the Debye mass as a commutator
1. Indeed, let
pe i!pt + pei!pt ;
Z
d3p j pj2j p+qj2
q!0
2e2
Z
2
m2Deb / qli!m0
m2Deb / qli!m0
Z
t
1
Z
~
t
1
(3.8)
(3.9)
(3.11)
dt0p
gg00
h j J0(t; q); J0(t0; q) j i
dt0p
gg00
h j J0(t; q); J0(t0; q) j i
conservation we can easily shift the upper limit of the integration t ! t~:
Because when we take the limit q ! 0 the operators under integrals become just total
charges qli!m0 J0(t; q) / Q, that trivially commutes with any operator due to the charge
So if the integral converges it means that there is no Debye mass in the system. For
example, one can check that for Minkowski spacetime and vacuum state harmonics behave
like fp(t)
e i!pt with some dispersion relation ! = !p. Such a behavior doesn't lead to
singularities at t !
1. While if there is a linear combination of exponents it can lead to
the photon mass [
22, 26
]. Indeed, consider
It is known that j pj2 can be interpreted as the number of the quasiparticles np at the
given level p. Then we obtain the classical formula for Debye mass (compare it to (3.5))
:
some divergences and instabilities of the system.
4
DeSitter spacetime
In this section we consider di erent patches of de Sitter spacetime such as Expanding
Poincare Patch (EPP), Contracting Poincare Patch, Global de Sitter and one that was
obtained from Minkowski spacetime by switching on the eternal exponential expansion.
{ 6 {
In the expanding patch of de Sitter spacetime the metric has the following form [11]
+1 to 0 from the past to future in nity. Also we will only consider the case of massive
elds in the de Sitter m > D2 1 .The harmonics in such a case are linear combinations of
Hankel's functions
HJEP06(218)3
Davies [27]. This vacuum has some nice properties when the interactions are switched o .
It can be thought as an analog of usual Minkowski vacuum in de Sitter spacetime [3, 11].
It means that in the distant past ( = 1) modes are just plane waves eip . If 6= 0 we will
name such a state as an alpha vacuum [28]. The last states are usually considered to be
unphysical due to the incorrect UV behavior, nevertheless they are function of a geodesic
distance Z on a deSitter space and therefore could respect de Sitter invariance (at least
on a tree level). Therefore, it is interesting to consider alphavacuums and compare to the
BunchDavies(BD) state.
= 0 then this vacuum is named after Bunch and
In terms of this modes the Debye mass is
! 0 one can notice that the l.h.s. of the equation (4.3) depends
only on the physical momentum, q , m2(q ); m2Deb = lim m2(q ). Indeed, one can rescale
p ! p ; 0 !
distant future
momentum q
0 and get that the only left parameter is q . In this case the limit of the
! 0 and small momentum q ! 0 coincide with the limit of small physical
! 0. As we will see, in the case of CPP there could arise some problems
q!0
with de ning of the Debye mass.
One can check that for BunchDavies vacuum there are no divergences as we integrate
over 0, therefore there is no Debye mass m2Deb; BD = 0. In contrast, alphavacuums (4.2)
in the limit of
! 1 are combinations of the positive and negative frequency harmonics
(partially, due to this fact they do not have a proper UV behavior) that leads to some
divergences
chosen alphavacuums. This indicates, that the alphavacuum can e ectively screen any
charge. The same e ect can happen for the gravitational mass. In such a case, large mass
can drastically change the de Sitter solution or even make it unstable.
The di erence between alpha vacuums and the Bunch Davies vacuum can be shown
in the following way. In the BD vacuum propagators have a good analytic properties
G^(X1; X2) =
G
G+
G
G
G +
G++
!
=
1 2
( 1 2+i ( 1 2))2 ~x2 A :
as polarization operators will have the same i prescriptions for time, even after Fourier
transformation over the spatial coordinates. Now let us use this property in the formula
for Debye mass in Kubo representation
m2Deb = lim
k!0
Z d 0
1
02 h[J0(k; ); J0( k; 0)]i :
Because of the analytic properties discussed above this integral can be seen as the integral
of the analytic function
goes from +1
00( 0) = hJ0(k; )J0( k; 0)i of the variable 0, where the contour
i to
i turns up and goes from
+i to +1+i . This allows us to make
the integral (4.5) convergent by slightly changing the contour
can interchange the integral and the limit to get
! (1 + i ) . Therefore we
Z d 0
1
m2Deb;BD =
02 h[J0(0; ); J0(0; 0)]i = 0:
This proof will not work for other vacuums, because the functions will not be analytic on
the whole Riemann surface, for example,
= 1 will have an essential singularity. Namely,
the product of two Green functions for an alphavacuum contains two cuts and contour
lies between them. Also the fact that the Debye mass for electric eld is equal to zero for
BunchDavies vacuum can be shown in the following way. The BunchDavies vacuum is
the only state that respects the de Sitter isometry at any level of perturbation theory, while
any alpha vacuum does break the global de Sitter symmetry even in the rst loop [3, 11].
The proof will work for our situation, because of this if we get nonzero photon mass m2Deb
to respect de Sitter isometry group and gauge invariance the equation for the classical
electromagnetic eld (2.5) should have the following form
A + m2Deb
r
1
r
A = 0
Therefore we should get the nonzero magnetic mass, but as it was shown above it is equal
to zero for any choice of gravitational background.
A similar calculation was done in a locally de Sitter spacetime for a nearly minimal
coupled, light scalar eld [29{31]. In these papers the Debye mass was investigated by
{ 8 {
Static deSitter
A
HJEP06(218)3
gray region. This gray region lays both within static de Sitter and EPP. Hence, for a proper choice
of the state in static de Sitter the calculations should give the same answers.
expanding a propagator as a power series in a mass and studying the leading contributions
in the limit of vanishing mass.
Because of the di erent behavior of a scalar
eld for
a di erent spectra of mass, usually two sectors are considered: the case of light mass
2
m < D 1 and heavy mass m > D 1
2 . These limits are drastically di erent, because of
di erent mode behavior, like in the case of heavy elds only the Keldysh propagator gets
secularly growing loop corrections and modes thermalize [3], while in the case of light elds
even the higher correlators start growing and there is no thermalization of harmonics [32].
The case of vanishing mass corresponds to the light masses and makes the physics di erent
from the heavy one that was considered in this paper, as it was indicated in the beginning
of this section. That can cause a discrepancy between the results.
4.2
Static coordinates
In this section we consider the static patch of EPP and calculate the Debye mass for the
vacuum state (see
gure 2). Because the KeldyshSchwinger diagrammatic technique is
causal, the calculation in static de Sitter and EPP should coincide for a proper choice of
the state. We do not think, that the BD state in the EPP corresponds to the vacuum
state in the static coordinates. In some sense the static de Sitter is similar to Rindler
space. In the latter case the thermal propagator in Rindler space corresponds to the usual
Minkowski propagator [33], therefore one would except the emergence of photon mass in
Rindler space, while in the Minkowski space photon mass of the vacuum state must be equal
to zero. Therefore it is interesting to investigate the di erence between this situations also
in the case of the de Sitter space. Because the thermal propagator for the static deSitter
is unknown, we consider only the case of the vacuum state of static de Sitter.
The metric of static de Sitter is
ds2 = 1
r
2 dt2
r
2 d 2 + sin2 d 2
(4.8)
1
1
r2 dr2
{ 9 {
1
1
r2
!
2
1
l(l + 1)
r2
h!;l;m(r) = 0;
(4.11)
a linear combinations we get modes for alphavacuums. Then we can get the formula for
the polarization operator in a mixed representation for L; M = 0;
= 0
0R0(
= 0; r; r0) = 2e2 X Z d!
Im h!;l;m(r)h!2;l;m(r0)
2
l;m
2
with measure p
gg00
2
1 r r2 (compare to the derivation of the eq. (3.3))
To get the actual photon mass we need to integrate the above expression over the space
Where the last equation comes from the orthogonality conditions (4.12). While this
calculation shows that photon mass in the vacuum state of static coordinates is equal to zero,
BD vacuum may not correspond to the vaccuum state.
4.3
Contracting Poincare patch
For the case of the Contracting Poincare Patch we have the same modes and the same
expression for Debye mass of photon (4.3), but the integration over time goes from 0 (the
time is reserved in the case of CPP). So, we have
The equations of motion for modes are
The modes behave as
where h!;l;m(r) satis es the following equation
1
1
;
r2
fl;m;!(t; r; ; ) = 0
fl;m;! = p2!
1
e i!tYl;m( ; )h!;l;m(r);
these modes are complex conjugated to each other h!;l;m = h !;l;m and are orthogonal
1
Z
0
r2dr
1
r2 h!;l;m(r)h!0;l;m(r) / (!
!0):
m2Deb = lim e2
q!0
Z
0
h hi (p ) @!hi (jp + qj )
One can check that there are no additional divergences in the lower limit of integration over
0 ! 0, because all harmonics behave well as the argument goes to zero. As it was discussed
before eq. (3.9) in such a case we can set
= 0 and get immediately that m2Deb = 0.
Also there is a problem slightly mentioned in the previous subsection. Again before
taking the limit q ! 0 we see that the mass is a function of the product of momentum and time
m2Deb = m2(q ). And the limits of the distant future and the small momentum are di erent.
In order to prevent such an ambiguity we take the limit of small q and don't take the limit of
big . Immediately, we get that this limit is equivalent to the case when we keep q xed and
send
! 0. That gives us that m2Deb = 0. The limit when we take q to be equal in nity,
corresponds to the EPP and will give us the same answer as in the previous subsection.
In the previous section we showed that there is no photon mass for BunchDavies
vacuum, while for an alphavacuum there is a photon mass. One can ask what is a crucial
di erence between these two cases? The answer is the following. In the case of CPP, the
causal structure is quite simple. There is only a compact set of points, that are causally
connected to a given point. It leads to the absence of photon mass. Indeed, the coordinate
representation of the Debye mass is as usual
Also there is no singularity on the lower limit of the integration over 0, because in this
case the correlator goes to the zero. Now we have to integrate over spatial coordinates
only in a ball of radius , because only there lie points causally connected to the point of
observation ( ; ~x). Therefore we get
The integrand of (4.16) is a continuous function without singularities, because we integrate
over a compact space we can interchange integrals over time and space, and get zero
due to the charge conservation. However, if we calculate the higher loops, there are IR
divergences [3, 11]. They may break the above proof. Indeed, we can simply interchange
the integrals over space and time, because we do not have divergences on the lower limit
of the integration over 0.
The conclusions about di erent photon masses in the case of EPP and CPP cases may
be surprising, because as it was said above these patches are di erent only by the direction
of time arrow. Nevertheless, this small di erence brings a huge discrepancy between the
dynamics of the systems. E.g., in the CPP the initial Cauchy surface is spacelike, while
in the EPP it is lightlike, in the CPP there are IR divergences, while in the EPP there is
only large IR contributions [3, 11].
Also, one can state that any point in the EPP lays in some CPP, so we should conclude
that mEDPebP = mCDPebP = 0 even for alphavaccum. This contradiction is easy to resolve. When
we get the mEDPebP we indeed integrate the same propagators, but not over the whole space,
but only over the half of the space. So it can lead to the di erent results.
4.4
In the case of Global DeSitter we have a problem with de ning a photon mass. This can be
seen as follows. The spatial crosssection is a threedimensional sphere S3, that is compact.
Hence, the spatial momentum of an electric potential A0(x; t) is discrete and numerates
by M~ with the eigenfunctions YM~ (~x). Because there is a gauge transformation A0(x; t) !
A0(x; t) + f (t), where f (t) is arbitrary, it is easy to see that the case of M~ = ~0; Y~0 (~x) = 1
corresponds to the gauge degrees of freedom. It gives that physical photons must have a
nonzero spatial momentum and we can not see whether there is a gap or not. In such a
case formally we have to assign a zero Debye mass in order to keep gauge invariance. In
the case of Minkowski spacetime we had a similar problem, that the photons with ~k = 0
are really unphysical, but we can investigate the spectrum near ~k because ~k is nondiscrete.
The above reasoning also gives us that the polarization operator should be equal to zero
as we choose M~ = 0. More formally, the polarization operator is
where QM is de ned as
00(M; M 0; t1; t2) / h j[QM (t1); QM0 (t2)]j i
;
QM (t) =
Z p
d
d 1x YM (x) Q(x; t);
and
is a determinant of the spatial metric.
Because of M is discrete, we cannot take the limit M ! 0 and that leads to
because of the charge conservation. Let us express the
00(t; xjt0; x0) as a series
X
M~ ;M~ 0
00(0; 0; t1; t2) / h j[Q0(t1); Q0(t2)]j i = 0;
00(t; xjt0; x0) =
YM~ (x)YM~ 0 (x0) 00(M~ ; M~ 0; t; t0)
Where M~ and M~ 0 numerate modes on the spatial section. The case M~ = 0 corresponds
to the zero mode, that is just a constant. Again, it is easy to see that such a term in the
eq. (4.19) vanishes. In fact,
(4.17)
(4.18)
(4.19)
HJEP06(218)3
X YM~ 0 (x0) 00(0; M~ 0; t; t0) =
M~ 0
/
Z p d
Z p d
d 1
x 00(t; xjt0; x0)Y0(x)
d 1
x h j[J0(x; t); J0(x0; t0)]j i
= h j[Q; J0(x0; t0)]j i = 0:
(4.20)
Where we have used that the total charge of the system is equal to zero Q j i = 0. It can
happen if and only if
00(0; M~ 0; t; t0)
0. Therefore we get that in the eq. (4.19) M~ 6= 0 and
M~ 0 6= 0. As we discussed above there is no such a notion as a Debye mass in the compact
case, but we can formally check that the gauge degrees of freedom does not acquire a mass.
So we can follow the usual procedure and integrate over t0 to calculate the mass
Z
CPP
(4.23)
HJEP06(218)3
corresponds to the causal connected points with the given point A. It is easy to see that the whole
shaded region lies in a Contracted Poincare Patch.
where we have used that R dd 1x YM~ (x) = 0 as M~ 6= 0. The di erence with Minkowski
spacetime is that the index in this case is discrete, which allows us to interchange the
integrals over time and space.
The above statement that may indicate that the calculations in the Contracting
Poincare Patch and in the Global DeSitter are in the agreement with each other [3].
Indeed, one can prove that if one uses the KeldyshSchwinger diagrammatic technique and
calculates the correction to an onepoint function, then there is no di erence between these
two calculations. The proof is quite simple,the KeldyshSchwinger technique is causal in
contrast with the Feynman one, therefore all contributions only can come from the past
light cone, but for a point in Globalde Sitter its causal past lays within a CPP (see the
gure 3). It proves that the calculation of onepoint functions in the CPP and in Global
de Sitter must be in agreement with each other.
4.5
Breaking deSitter invariance
In this subsection we consider the case when the de Sitter symmetry is broken. Namely
we consider a situation with an in ation described by the Expanding Poincare Patch, but
the scale parameter a(t) is switched o in a while
a(t) = eT tanh( Tt ) T
After the expansion phase the system will be in an excited state. We can nd this state by
gluing modes at the moments when de Sitter was switched o and on (see the paper [34]).
We get the following level population in the limit T ! 1
where
=
q
m2
D 1 2
2
np =
( 0; jpj > ; jpj < e 2T
e2
1
1
; e 2T < jpj <
and m is a mass of the scalar eld.
In such a case we can use the above formula for Minkowski spacetime that gives us
The formula is di erent from the Debye mass for a thermal state with GibbonsHawking
temperature T = 21 , it happens because of the in nite blueshift during the expansion
phase.
One can expect then, that we should get the same answer for the case of the CPP
with the contraction switched o
after a while. However this means that we will get a
contradiction with the above calculation of the Debye mass in CPP, because the same
reasoning as in that case leads to the zero photon mass. This contradiction is easy to
resolve. Indeed, we know that the limits of small q and distant future do not commute.
But in the case when we switch o the contraction there is no such a problem and we can
take the limit of the distant future and get the nonzero photon mass.
Let us consider the case when the EPP was turned on at some moment 0. In this
case after gluing modes will be described as follows
As one can notice, there is no singularity from the switching on the EPP, e ectively if one
changes all momentum to the physical ones in eq. (4.3), he gets that the border between
two states goes down 0 =
0 ! 0;
! 0. So, we would expect that the state will be
described by BunchDavies and that will not give any photon mass.
5
Debye mass in strong electric eld
In this section we derive the photon mass for Schwinger process. We assume that the
electric eld was never turned on, i.e. it always exists. The situation is similar to the de
Sitter one, where we assumed the eternal in ation. And indeed, the IR properties in this
two situation are similar and were discussed in many papers [
3, 4, 9, 12
].
We choose the spatial gauge A1 =
Et; A
? = 0. In this case the modes obeying the
following equations
fp(t) = fp? (p1 + eEt):
(5.1)
Solutions to this are parabolic cylindric functions. In this case we cannot de ne modes that
are similar to the BunchDavies ones in EPP, because in the limit of large times modes are
not simple plane waves. Therefore we choose the following boundary conditions [3, 12]:
fp? (pphys) =
8
>>< p2pphys
>>: p2pphys
2
2
e i p2pehEys + p2pphys
e i p2pehEys + p2pphys
2
ei p2pehEys ;
2
ei p2pehEys ;
pphys . 0;
pphys & 0:
where
j j
2
j j2 = j j
2
j j2 = 1
(5.2)
The photon mass in such a case can be calculated in the similar way and the result is:
p?+q? (p1 + q1 + eEt0)@!t0 fp? (p1 + eEt0) i
:
It is easy to see that there are no divergences as q? ! 0, the possible divergence may appear
only when q1 ! 0. One can notice that the dependence on time t can be removed. Indeed
let rede ne in (5.3) all momentums into the physical ones pphys = p1 + eEt. Then we have
HJEP06(218)3
0
Z
1
?
fp (p1 + q1 + eEt0)@!t0 fp (p1 + eEt0) i
:
?
Now we can calculate the photon mass. The integral has a singularity
0
Z
1
That eventually gives us
?
dt0fp (p1 + q1 + eEt0)@!t0 fp (p1 + eEt0) =
?
+ O(q1); q1 ! 0:
2
q1
Where a prime corresponds to the di erentiation with respect to the p1. One can notice
that this integral actually diverges quadrically, when p1 ! 1. It shows again that the
eternal electric eld is unphysical. Namely, a perturbation of the electric eld may lead to
the decay to another state [
9, 12
].
To understand the meaning of this divergence we consider the case when the electric
eld was turned on at some moment. So we have the electric impulse: the
eld was
switched on, worked for a time T and turned o
after. Constant electric eld uniformly
changes the physical momentum for an arbitrary mode. It leads that for a long enough
pulse any highfrequency mode excites or crosses the horizon [4]. It will give the following
level population [
1
]:
np =
(
0; p1 & 0;
p1 .
e eE (m2+p2?); eET . p1 . 0;
and will allow us to use the eq. (3.11). This formula comes from the gluing of modes when
the eld was turned on and switched o . Then the mass is
2e2
Z
4e2 Z
0
dp1 Z
2
d2p
!
2e2
Z
4e2 Z
0
2
4 2
e emE2 (eET )2:
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)
Where we extracted the leading contribution in the limit T ! 1. As one can see the
Debye mass indeed diverges quadratically with the growth of the length of the impulse.
Eventually it gives the following expression for photon mass after the impulse
mDeb =
e
2
2
m2
e 2eE jA1(1)
A1(
1)j :
(5.9)
This mass may be large for a su ciently long impulse. This means that any electric eld
is screened, including the initial impulse. That means that a long electric impulse is an
unphysical situation and in order to understand the situation better we have to take into
account the backreaction of the scalar eld on the electromagnetic eld. Let us stress the
di erence with the de Sitter case (4.24): in this case we got a
nite result because the
harmonics gets excited only when it experiences a huge blueshift, but the blueshift also
shrinks the phase volume for a given harmonic, which eventually makes its contribution
negligible. While in the case of an electric pulse the phase volume is constant and leads
to a
nite contribution for an arbitrary mode, that got excited. That leads to the linearly
growing photon mass in the case of an electric pulse (5.9).
6
In this paper we discuss the Debye and magnetic photon masses in the framework of particle
creation by the external electromagnetic and gravitational elds. We argue that for the
alpha vacuums in de Sitter spacetime there is a nonzero Debye mass, while it is vanishing
for the BunchDavies vacuum, which is an analog of the usual Minkowski vacuum. The
last fact can be considered as consequence of the analytical properties of Green functions
for BunchDavies vacuum. These properties also lead to the de Sitter invariance of the
loop corrections in the Expanding Poincare Patch [3].
Also, the cases of Global DeSitter and Contracting Poincare Patch were considered. It
was shown that for any chosen vacuum there is no photon mass. These observation lead to
the question of the physical meaning and relevance of the GibbonsHawking temperature.
The same calculations were done for the eternal electric eld and impulse, that features the
appearance of a very large Debye mass, that can indicate that the considered situations are
unphysical. In this paper we did not calculate the higher corrections to the photon mass
and used only initial propagators. As it was shown [3, 11] these propagators receive large
IR contributions. It would be interesting to investigate the e ect of these large IR loop
contributions to the Debye mass. Also it would be interesting to consider gravitational
mass and its impact on quantum eld systems.
Acknowledgments
The work is devoted to the memory of V.N.Diesperov. I am grateful to E.T. Akhmedov, I.R. Klebanov, J.Maldacena and A.M.Polyakov for important discussions and remarks. The work was supported in part by the US NSF under Grant No. PHY1620059.
Open Access.
This article is distributed under the terms of the Creative Commons Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
[INSPIRE].
HJEP06(218)3
Mathematical Physics, Cambridge Univ. Press, Cambridge U.K., (1984) [INSPIRE].
[3] D. Krotov and A.M. Polyakov, Infrared sensitivity of unstable vacua, Nucl. Phys. B 849
(2011) 410 [arXiv:1012.2107] [INSPIRE].
[4] A.M. Polyakov, Infrared instability of the de Sitter space, arXiv:1209.4135 [INSPIRE].
Nauk 136 (1982) 538] [INSPIRE].
[arXiv:0709.2899] [INSPIRE].
corrections in strong electric elds, JHEP 09 (2015) 085 [arXiv:1412.1554] [INSPIRE].
[10] E.T. Akhmedov, F.K. Popov and V.M. Slepukhin, Infrared dynamics of the massive 4
theory on de Sitter space, Phys. Rev. D 88 (2013) 024021 [arXiv:1303.1068] [INSPIRE].
[11] E.T. Akhmedov, Lecture notes on interacting quantum
elds in de Sitter space, Int. J. Mod.
Phys. D 23 (2014) 1430001 [arXiv:1309.2557] [INSPIRE].
[12] E.T. Akhmedov, N. Astrakhantsev and F.K. Popov, Secularly growing loop corrections in
strong electric elds, JHEP 09 (2014) 071 [arXiv:1405.5285] [INSPIRE].
[13] E.T. Akhmedov and S.O. Alexeev, Dynamical Casimir e ect and loop corrections, Phys. Rev.
D 96 (2017) 065001 [arXiv:1707.02242] [INSPIRE].
[14] P.R. Anderson and E. Mottola, Quantum vacuum instability of \eternal" de Sitter space,
Phys. Rev. D 89 (2014) 104039 [arXiv:1310.1963] [INSPIRE].
[15] P.R. Anderson, E. Mottola and D.H. Sanders, Decay of the de Sitter vacuum, Phys. Rev. D
97 (2018) 065016 [arXiv:1712.04522] [INSPIRE].
[16] R.P. Woodard, Perturbative quantum gravity comes of age, Int. J. Mod. Phys. D 23 (2014)
1430020 [arXiv:1407.4748] [INSPIRE].
[17] J.H. Jeans, The stability of a spherical nebula, Phil. Trans. Roy. Soc. Lond. A 199 (1902) 1.
[18] D.S. Gorbunov and V.A. Rubakov, Introduction to the theory of the early universe: hot big
bang theory, World Scienti c, Singapore, (2011) [INSPIRE].
QED: photon production, magnetic and Debye masses and conductivity, Phys. Rev. D 61
(2000) 085007 [hepph/9909259] [INSPIRE].
applications to magnetic and conduction problems, J. Phys. Soc. Jpn. 12 (1957) 570.
(2013).
Cambridge U.K., (2005) [INSPIRE].
eld theory in condensed matter physics, Cambridge Univ. Press,
89 (2002) 101301 [astroph/0205331] [INSPIRE].
eld theory in de Sitter space: renormalization by
[arXiv:1701.07226] [INSPIRE].
elds on curved spacetimes and a new look at the
[1] J.S. Schwinger , On gauge invariance and vacuum polarization , Phys. Rev . 82 ( 1951 ) 664 [2] N.D. Birrell and P.C.W. Davies , Quantum elds in curved space, Cambridge Monographs on [5] A. m. Polyakov , Phase transitions and the universe , Sov. Phys. Usp . 25 ( 1982 ) 187 [Usp . Fiz.
[6] A.M. Polyakov , De Sitter space and eternity, Nucl. Phys. B 797 ( 2008 ) 199 [7] A.M. Polyakov , Decay of vacuum energy, Nucl. Phys. B 834 ( 2010 ) 316 [arXiv: 0912 .5503] [8] E.T. Akhmedov and P. Burda , Solution of the DysonSchwinger equation on de Sitter background in IR limit , Phys. Rev. D 86 ( 2012 ) 044031 [arXiv: 1202 .1202] [INSPIRE].
[9] E.T. Akhmedov and F.K. Popov , A few more comments on secularly growing loop Cambridge U .K., ( 2011 ). Netherlands, ( 2012 ).
[19] D.S. Gorbunov and V.A. Rubakov , Introduction to the theory of the early universe: cosmological perturbations and in ationary theory , World Scienti c, Singapore, ( 2011 ) [20] A. Kamenev , Field theory of nonequilibrium systems , Cambridge University Press, [21] L. Pitaevskii and E. Lifshitz , Physical kinetics, volume 10 , Elsevier Science , The [22] D. Boyanovsky , H.J. de Vega and M. Simionato , Nonequilibrium quantum plasmas in scalar [24] L. Landau and E. Lifshitz , Statistical physics, volume 5 , Elsevier Science , The Netherlands, [25] A.B. Migdal , Qualitative methods in quantum theory, Front . Phys. 48 ( 1977 ) 1 [INSPIRE].
[26] A.M. Tsvelik , Quantum