472
A. Martín-Ruiz et al.
take both media with ε = μ = 1, however, non trivial optical properties as well as
anisotropy effects can also be considered.
We follow a method similar to that of [42, 55] that begins with the calculation
of the corresponding GF, from which the renormalized energy-momentum tensor
in the region between the plates is obtained. A subsequent calculation yields the
Casimir stress in the interface of the TI. We also consider the limit when the
plate P 2 goes back to infinity (L → ∞) to obtain the Casimir stress produced by a
single conducting plate in front of a semi-infinite TI. The boundary conditions on the
plates P 1 and P 2 are the usual ones for a perfect conductor: n μ ˜
F
μν
| P 1,2 = 0, where
n = (0, 0, 0, 1).
This calculation highlights two extensions of our method: (i) the inclusion of
time-dependence and (ii) the use of a convenient well-known free reduced GF to
obtain the full result. In the absence of free sources at the interface, the GF
G
μ
ν
x, x
= 4π
d
2 p
(2π )
2
e
ip ·x
dω
2π
e
−iω(t−t
) g
μ
ν
z, z
,
(17.38)
satisfies the (17.19), together with the BC of (17.16). From here on we do not write
the dependence of the reduced GF g
μ
ν on ω and p . In the Lorenz gauge, the equation
for g
μ
ν is
η
μ
ν ∂
2
+ i ˜
θδ (z − a)
3µα
ν p α
g
ν
σ
z, z
= η
μ
σ δ
z − z
,
(17.39)
where now we have ∂
2
= p
2
− ω
2
− ∂
2
z and p
α
=
ω, p , 0
.
The boundary contribution in z = L, which is not present in (17.39), is identically
zero in the distributional sense due to the BCs on the conducting plate P 2 . The next
step, which illustrates the flexibility of the method developed in the Sect. 17.3, is to
choose a free GF to integrate (17.39). It is clear that we must use the free reduced
GF corresponding to two perfectly conducting parallel plates located at z = 0 and
z = L, which is given by [42]
g C
z, z
=
sin [ pz < ] sin [ p (L − z > )]
p sin [ pL]
,
(17.40)
where the suffix C is for Casimir, z > (z < ) is the greater (smaller) value between the
coordinates z, z
, and p =
ω 2 − p 2 . Writting the solution of (17.39) as
g
μ
ν
z, z
= η
μ
ν g C
z, z
+ g
μ
Cν
z, z
.
(17.41)
we realize that the first term corresponds to the absence of the TI between the plates,
while the second term, to be called the reduced θ -GF,
A. Martín-Ruiz et al.
take both media with ε = μ = 1, however, non trivial optical properties as well as
anisotropy effects can also be considered.
We follow a method similar to that of [42, 55] that begins with the calculation
of the corresponding GF, from which the renormalized energy-momentum tensor
in the region between the plates is obtained. A subsequent calculation yields the
Casimir stress in the interface of the TI. We also consider the limit when the
plate P 2 goes back to infinity (L → ∞) to obtain the Casimir stress produced by a
single conducting plate in front of a semi-infinite TI. The boundary conditions on the
plates P 1 and P 2 are the usual ones for a perfect conductor: n μ ˜
F
μν
| P 1,2 = 0, where
n = (0, 0, 0, 1).
This calculation highlights two extensions of our method: (i) the inclusion of
time-dependence and (ii) the use of a convenient well-known free reduced GF to
obtain the full result. In the absence of free sources at the interface, the GF
G
μ
ν
x, x
= 4π
d
2 p
(2π )
2
e
ip ·x
dω
2π
e
−iω(t−t
) g
μ
ν
z, z
,
(17.38)
satisfies the (17.19), together with the BC of (17.16). From here on we do not write
the dependence of the reduced GF g
μ
ν on ω and p . In the Lorenz gauge, the equation
for g
μ
ν is
η
μ
ν ∂
2
+ i ˜
θδ (z − a)
3µα
ν p α
g
ν
σ
z, z
= η
μ
σ δ
z − z
,
(17.39)
where now we have ∂
2
= p
2
− ω
2
− ∂
2
z and p
α
=
ω, p , 0
.
The boundary contribution in z = L, which is not present in (17.39), is identically
zero in the distributional sense due to the BCs on the conducting plate P 2 . The next
step, which illustrates the flexibility of the method developed in the Sect. 17.3, is to
choose a free GF to integrate (17.39). It is clear that we must use the free reduced
GF corresponding to two perfectly conducting parallel plates located at z = 0 and
z = L, which is given by [42]
g C
z, z
=
sin [ pz < ] sin [ p (L − z > )]
p sin [ pL]
,
(17.40)
where the suffix C is for Casimir, z > (z < ) is the greater (smaller) value between the
coordinates z, z
, and p =
ω 2 − p 2 . Writting the solution of (17.39) as
g
μ
ν
z, z
= η
μ
ν g C
z, z
+ g
μ
Cν
z, z
.
(17.41)
we realize that the first term corresponds to the absence of the TI between the plates,
while the second term, to be called the reduced θ -GF,