376
G. Barnich and M. Bonte
context and from [24], that the analysis is done in radiation gauge, A 0 = 0 =
∇ ·
A,
imposed together with the constraint equations π 0 = 0,
∇ · ·
π = 0. When translated
to momentum space with the above boundary conditions, it follows directly that
the k 3 = 0 modes of (
A,
π) give rise to the two transverse polarizations, while the
k 3 = 0 mode of (
A,
π) is also divergence-free.
The divergence-free vector fields in position space associated with the k 3 = 0
mode are given by
A
NPG
i
= δ
3
i A
C
3,0 (x
a ), π
i
NPG = δ
i
3 π
3C
0 (x
a ).
(24)
The argument why they are non-trivial from a position space viewpoint in equation
(4.10) of [16] is incorrect. Let us focus on
π NPG , which has a direct interpretation
in electrostatics, the argument for
A NPG being the same. The vector field
π NPG
has a non-trivial longitudinal piece. The associated 1 form is co-closed without
being co-exact. This follows from the Helmholtz decomposition in the presence of
boundaries. Indeed, under suitable fall-off assumptions at infinity, there is a unique
decomposition
π =
∇ϕ +
∇ × ×
α,
(25)
ϕ(x) = −
d
3 x
(
∇ · ·
π )(x )
4π || x − −
x |
+
S
( n · ·
π dσ )(x )
4π || x − −
x |
,
(26)
α(x) =
d
3 x
(
∇ × ×
π )(x )
4π || x − −
x |
−
S
( n × ×
π dσ )(x )
4π || x − −
x |
.
(27)
When this decomposition is applied to
π NPG for the capacitor, the potential for the
longitudinal part comes entirely from the boundary contribution and is explicitly
given by
ϕ NPG (x) =
1
4π
dx
dy
π
3C
0 (x
)
ρ
2
+ (z − d)
2
−
1
2 −
ρ
2
+ z
2
−
1
2
,
(28)
while the potential for the transverse part comes entirely from the bulk contribution
and is explicitly given by
α
i
NPG (x) =
δ i
a
4π
dx
dy
ab ∂
b π
3C
0 (x
) ln
ρ 2 + (d − z) 2 + d − z
ρ 2 + z 2 − z
,
(29)
where ρ 2 = (x − x ) 2 + (y − y ) 2 .
One then has to decide how to deal with the transverse space R 2 . As usual, we
will put the system in a finite two-dimensional box in an intermediate stage. In
this box, we can adopt either perfectly conducting conditions as in [9, 24], or use
periodic conditions, which is what was done in [16]. In the large area limit, where
sums go to integrals, both approaches yield the same results for finite temperature
G. Barnich and M. Bonte
context and from [24], that the analysis is done in radiation gauge, A 0 = 0 =
∇ ·
A,
imposed together with the constraint equations π 0 = 0,
∇ · ·
π = 0. When translated
to momentum space with the above boundary conditions, it follows directly that
the k 3 = 0 modes of (
A,
π) give rise to the two transverse polarizations, while the
k 3 = 0 mode of (
A,
π) is also divergence-free.
The divergence-free vector fields in position space associated with the k 3 = 0
mode are given by
A
NPG
i
= δ
3
i A
C
3,0 (x
a ), π
i
NPG = δ
i
3 π
3C
0 (x
a ).
(24)
The argument why they are non-trivial from a position space viewpoint in equation
(4.10) of [16] is incorrect. Let us focus on
π NPG , which has a direct interpretation
in electrostatics, the argument for
A NPG being the same. The vector field
π NPG
has a non-trivial longitudinal piece. The associated 1 form is co-closed without
being co-exact. This follows from the Helmholtz decomposition in the presence of
boundaries. Indeed, under suitable fall-off assumptions at infinity, there is a unique
decomposition
π =
∇ϕ +
∇ × ×
α,
(25)
ϕ(x) = −
d
3 x
(
∇ · ·
π )(x )
4π || x − −
x |
+
S
( n · ·
π dσ )(x )
4π || x − −
x |
,
(26)
α(x) =
d
3 x
(
∇ × ×
π )(x )
4π || x − −
x |
−
S
( n × ×
π dσ )(x )
4π || x − −
x |
.
(27)
When this decomposition is applied to
π NPG for the capacitor, the potential for the
longitudinal part comes entirely from the boundary contribution and is explicitly
given by
ϕ NPG (x) =
1
4π
dx
dy
π
3C
0 (x
)
ρ
2
+ (z − d)
2
−
1
2 −
ρ
2
+ z
2
−
1
2
,
(28)
while the potential for the transverse part comes entirely from the bulk contribution
and is explicitly given by
α
i
NPG (x) =
δ i
a
4π
dx
dy
ab ∂
b π
3C
0 (x
) ln
ρ 2 + (d − z) 2 + d − z
ρ 2 + z 2 − z
,
(29)
where ρ 2 = (x − x ) 2 + (y − y ) 2 .
One then has to decide how to deal with the transverse space R 2 . As usual, we
will put the system in a finite two-dimensional box in an intermediate stage. In
this box, we can adopt either perfectly conducting conditions as in [9, 24], or use
periodic conditions, which is what was done in [16]. In the large area limit, where
sums go to integrals, both approaches yield the same results for finite temperature