17 Axion Electrodynamics in Magnetoelectric Media
473
g
μ
Cν
z, z = ˜
θg C (a, a)
p μ p ν −
η
μ
ν + n μ n ν
p 2
A C
z, z + i
μ α3
ν p α A C
z, z ,
A C
z, z = − ˜
θ
g C (z, a) g C
a, z
1 − p 2 ˜
θ 2 g 2
C (a, a)
,
(17.42)
will be responsible for the effect we are looking for. The partition in (17.41) yields the
full GF as G
μ
ν
x, x
= η
μ
ν G
x, x
+ G
μ
Cν
x, x
, where each term arises from
the respective contribution in (17.41). Since the MEP modifies the behavior of the
fields only in the interface, the energy-momentum tensor (EMT) T
μν in the bulk
retains the original expression of the unmodified Maxwell equations, failing to be
conserved only in the interface in the absence of free external sources [45]. Next
we determine the vacuum expectation value (VEV) of T
μν , to be called the vacuum
stress (VE), according to the basic relation G
μν
(x, x) = −i0|T A
μ
(x)A
ν
(x
)|0,
where T denotes time ordering. Following the standard point-splitting method we
obtain
T
μν
=
i
4π
lim
x→x
− ∂
μ
∂
ν G
λ
λ + ∂
μ
∂
λ G
λν
+ ∂
λ
∂
ν G
μ
λ
− ∂
λ
∂ λ G
μν
+
1
2
η
μν
∂
α
∂
α G
λ
λ − ∂
α
∂
β G
β
α
.
(17.43)
Again, we can write T
μν
= t
μν
+
T
μν
C
. The first term is the contribution to
the VE in the absence of the TI, yielding the well-known result in [55], which is
independent of a. After some calculation, the second term
T
μν
C
, to be called the
θ -vacuum stress (θ -VS), is
T
μν
C
= i ˜
θ
d
2 p
(2π) 2
dω
2π
p
μ p
ν
+ n
μ n
ν p
2
g C (a, a) lim
z→z
p
2
+ ∂
z ∂ z
A C
z, z
.
(17.44)
Next we require the renormalized VS: T
μν
ren = T
μν
− T C
μν
vac , where the first
(second) term is the VS in the presence (absence) of conducting plates [55, 56]. We
obtain
T
μν
C
ren
= −
π 2
720L 4
η
μν + 4n
μ n
ν
[u(θ, χ )H (a − z) + u(θ, 1 − χ)H (z − a)] , (17.45)
u(θ, χ ) =
120
π 4
∞
0
˜
θ 2 ξ 3 sh [ξχ] sh
3 [ξ (1 − χ )] sh
−3 [ξ ]
1 + ˜
θ 2 sh
2 [ξχ] sh
2 [ξ (1 − χ )] sh
−2 [ξ ]
dξ.
(17.46)
The notation is sh(x) = sinh(x) and χ = a/L with 0 < χ < 1. Our expression
(17.45) has the same tensorial structure as that in [55], except for a z dependence
because the EMT is not conserved at the interface, implying that the renormalized θ -VS is constant in each bulk but has a discontinuity in z = a consistent with
∂ z
T
00
C
ren
∝ δ(().
473
g
μ
Cν
z, z = ˜
θg C (a, a)
p μ p ν −
η
μ
ν + n μ n ν
p 2
A C
z, z + i
μ α3
ν p α A C
z, z ,
A C
z, z = − ˜
θ
g C (z, a) g C
a, z
1 − p 2 ˜
θ 2 g 2
C (a, a)
,
(17.42)
will be responsible for the effect we are looking for. The partition in (17.41) yields the
full GF as G
μ
ν
x, x
= η
μ
ν G
x, x
+ G
μ
Cν
x, x
, where each term arises from
the respective contribution in (17.41). Since the MEP modifies the behavior of the
fields only in the interface, the energy-momentum tensor (EMT) T
μν in the bulk
retains the original expression of the unmodified Maxwell equations, failing to be
conserved only in the interface in the absence of free external sources [45]. Next
we determine the vacuum expectation value (VEV) of T
μν , to be called the vacuum
stress (VE), according to the basic relation G
μν
(x, x) = −i0|T A
μ
(x)A
ν
(x
)|0,
where T denotes time ordering. Following the standard point-splitting method we
obtain
T
μν
=
i
4π
lim
x→x
− ∂
μ
∂
ν G
λ
λ + ∂
μ
∂
λ G
λν
+ ∂
λ
∂
ν G
μ
λ
− ∂
λ
∂ λ G
μν
+
1
2
η
μν
∂
α
∂
α G
λ
λ − ∂
α
∂
β G
β
α
.
(17.43)
Again, we can write T
μν
= t
μν
+
T
μν
C
. The first term is the contribution to
the VE in the absence of the TI, yielding the well-known result in [55], which is
independent of a. After some calculation, the second term
T
μν
C
, to be called the
θ -vacuum stress (θ -VS), is
T
μν
C
= i ˜
θ
d
2 p
(2π) 2
dω
2π
p
μ p
ν
+ n
μ n
ν p
2
g C (a, a) lim
z→z
p
2
+ ∂
z ∂ z
A C
z, z
.
(17.44)
Next we require the renormalized VS: T
μν
ren = T
μν
− T C
μν
vac , where the first
(second) term is the VS in the presence (absence) of conducting plates [55, 56]. We
obtain
T
μν
C
ren
= −
π 2
720L 4
η
μν + 4n
μ n
ν
[u(θ, χ )H (a − z) + u(θ, 1 − χ)H (z − a)] , (17.45)
u(θ, χ ) =
120
π 4
∞
0
˜
θ 2 ξ 3 sh [ξχ] sh
3 [ξ (1 − χ )] sh
−3 [ξ ]
1 + ˜
θ 2 sh
2 [ξχ] sh
2 [ξ (1 − χ )] sh
−2 [ξ ]
dξ.
(17.46)
The notation is sh(x) = sinh(x) and χ = a/L with 0 < χ < 1. Our expression
(17.45) has the same tensorial structure as that in [55], except for a z dependence
because the EMT is not conserved at the interface, implying that the renormalized θ -VS is constant in each bulk but has a discontinuity in z = a consistent with
∂ z
T
00
C
ren
∝ δ(().