76
5 Appendix
The electromagnetic polarization tensor is given by
μν
(k) = 16πα
d
3 p i
(2π) 3
1
2 E
[n e (E) + n ¯
e (E)]
×
p · k ( p
μ k
ν
+ k
μ p
ν
) − k
2 p
μ p
ν
− ( p · k)
2
g
μν
( p · k) 2 − (k 2 ) 2 /4
,
(5.19)
where k
μ
= (ω, k) and p
μ
= (E, p). The transverse and longitudinal polarizations
are defined as
T (ω, k) =
1
2
δ
i j
− k
i k
j
i j
(ω, k)
(5.20)
and
L (ω, k) =
00
(ω, k) .
(5.21)
The effective propagator of the photon (in the Coulomb gauge) has components
D
00
(ω, k) =
1
k 2 − L (ω, k)
(5.22)
and
D
i j
(ω, k) =
1
k 2 − T (ω, k)
δ
i j
− k
i k
j
.
(5.23)
The dispersion relationships are defined by the solutions of the equations
ω
2
T = k
2
+ T (ω T , k) and ω
2
L =
ω
2
L
k 2 + T (ω L , k)
(5.24)
In the degenerate limit, the distribution functions in Eq. (5.17) reduce to step
functions at the Fermi momentum p F =
3π 2 n e and we have
T (ω, k) = ω
2
P
3ω
2
2v
2
F k 2
1 −
ω
2
− v
2
F k
2
2v F ωk
log
ω + v F k
ω − v F k
(5.25)
and
L (ω, k) = ω
2
P
3ω
2v
3
F k
ω
2v F k
log
ω + v F k
ω − v F k
− 1
(5.26)
where ω P = 4α p
2
F v F /3π is the plasma frequency.
The energy loss rate Q (energy per volume and unit time) is written in terms of the
imaginary part of the polarization of the photon in the medium of charged particles.
The contribution of the longitudinal and transverse modes is obtained (by the optical
theorem) as
5 Appendix
The electromagnetic polarization tensor is given by
μν
(k) = 16πα
d
3 p i
(2π) 3
1
2 E
[n e (E) + n ¯
e (E)]
×
p · k ( p
μ k
ν
+ k
μ p
ν
) − k
2 p
μ p
ν
− ( p · k)
2
g
μν
( p · k) 2 − (k 2 ) 2 /4
,
(5.19)
where k
μ
= (ω, k) and p
μ
= (E, p). The transverse and longitudinal polarizations
are defined as
T (ω, k) =
1
2
δ
i j
− k
i k
j
i j
(ω, k)
(5.20)
and
L (ω, k) =
00
(ω, k) .
(5.21)
The effective propagator of the photon (in the Coulomb gauge) has components
D
00
(ω, k) =
1
k 2 − L (ω, k)
(5.22)
and
D
i j
(ω, k) =
1
k 2 − T (ω, k)
δ
i j
− k
i k
j
.
(5.23)
The dispersion relationships are defined by the solutions of the equations
ω
2
T = k
2
+ T (ω T , k) and ω
2
L =
ω
2
L
k 2 + T (ω L , k)
(5.24)
In the degenerate limit, the distribution functions in Eq. (5.17) reduce to step
functions at the Fermi momentum p F =
3π 2 n e and we have
T (ω, k) = ω
2
P
3ω
2
2v
2
F k 2
1 −
ω
2
− v
2
F k
2
2v F ωk
log
ω + v F k
ω − v F k
(5.25)
and
L (ω, k) = ω
2
P
3ω
2v
3
F k
ω
2v F k
log
ω + v F k
ω − v F k
− 1
(5.26)
where ω P = 4α p
2
F v F /3π is the plasma frequency.
The energy loss rate Q (energy per volume and unit time) is written in terms of the
imaginary part of the polarization of the photon in the medium of charged particles.
The contribution of the longitudinal and transverse modes is obtained (by the optical
theorem) as