Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 207
part of K and which obeys K · e L = 0,
e L ≡ ωK − K2 U
k √ K 2
= (k2 , ωk)
k √ K 2 , (6.27)
where the second expression refers to the medium rest frame. There
remain two directions orthogonal to e g and e L , or equivalently, to K
and U. If k is taken to point in the z-direction two possible choices are
the unit vectors e x and e y , respectively. However, in order to retain the
azimuthal symmetry around the k direction the “circular polarization
vectors” e ± = (e x ± ie y )/ √ 2 are needed. Then
e ± ≡ (0, e ± ) (6.28)
in the rest frame of the medium; a suitable covariant formulation is
also possible. The basis vectors obey e ∗ ± = e ∓ while e g,L are real for
K 2 > 0 (time-like) and e ∗ g,L = −e g,L for K 2 < 0 (space-like). They are
normalized according to e ∗ ± · e ± = e ∓ · e ± = −1 and e ∗ L · e L = ∓1 and
e ∗ g · e g = ±1, depending on K 2 being time- or space-like. Evidently, e g
and e L switch properties between a time- and space-like K.
This choice of basis vectors is only possible if K 2 ≠ 0. If K is
light-like one must make some other choice, for example (1, 0) and
(0, ˆk) in the rest frame of the medium. Because the goal is to describe
electromagnetic excitations in a medium, and because usually K 2 ≠ 0
for such waves, this is no serious limitation. It will turn out that the
dispersion relation of (longitudinal) plasmons crosses the light-cone, i.e.
there is a wave number for which ω 2 − k 2 = 0. The degeneracy of e g
with e L at this single point will cause no trouble.
The most general polarization tensor compatible with the gauge
condition Eq. (6.24) must be constructed from e ± and e L alone. Azimuthal
symmetry about the k direction requires that they occur only
in the scalar combinations e µ a e ∗ν
a . Therefore, one defines the projection
operators on the basis vectors
P µν
a
≡ −e µ a e ∗ν
a , (a = ±, L). (6.29)
The most general polarization tensor is then given as
Π µν = ∑
π a P a µν , (6.30)
a=±,L
where the π a are functions of the Lorentz scalars K 2 and U · K or
equivalently of ω and k in the medium frame. They represent the
medium response to circularly and longitudinally polarized A’s.