Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 209
The relationship to the transverse and longitudinal components of
the polarization tensor is (Weldon 1982a)
ϵ L = 1 − π L /(ω 2 − k 2 ) and ϵ T = 1 − π T /ω 2 . (6.33)
This yields the well-known dispersion relations (Sitenko 1967)
ϵ L (ω, k) = 0 and ω 2 ϵ T (ω, k) = k 2 (6.34)
for the longitudinal and transverse modes.
Calculations of the polarization tensor from the forward scattering
amplitudes on microscopic medium constituents are usually performed
in a cartesian basis and thus yield an expression for Π µν . The longitudinal
and transverse components are projected out by virtue of
π L = e ∗µ
L Π µν e ν L and π T = e ∗µ
± Π µν e ν ±, or explicitly in the medium frame
(Weldon 1982a)
π L = (1 − ω 2 /k 2 ) Π 00 and π T = 1 2 (Tr Π − π L), (6.35)
with Tr Π = g µν Π µν .
Recall that the dispersion relations are given by ω 2 −k 2 = π T,L (ω, k).
Thus the frequency ω(k) and the “effective mass” of a given mode are
generally complicated functions of k, notably in a medium involving
bound electrons where various resonances occur. It can be shown on
general grounds (Jackson 1975), however, that for frequencies far above
all resonances the transverse mode has a particle-like dispersion relation
ω 2 − k 2 = m 2 T where m T is the “transverse photon mass” which is a
constant independent of the wave number or frequency.
6.3.4 Lowest-Order QED Calculation of Π
On the level of quantum electrodynamics (QED) the potential V =
1
2 A µΠ µν A ν which modifies the free Lagrangian is interpreted as the
self-energy of the photons in the medium. As a Feynman graph, it corresponds
to an insertion of Π µν into a photon line of four-momentum K
and thus corresponds to forward scattering on the medium constituents
(Fig. 6.1), entirely analogous to the interpretation of the refractive index
in terms of a forward scattering amplitude in Sect. 6.2.1. One then
concludes that Π µν (K) is the truncated matrix element for the forward
scattering of a photon with momentum K, i.e. it is the matrix element
of the medium constituents alone, uncontracted with the photon
polarization vectors ϵ µ and ϵ ν .
In general, the calculation of Π requires the methods of field theory
at finite temperature and density. To lowest order, however, this