Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 205
particles which constitute the currents move themselves under the influence
of electromagnetic fields. Therefore, the interaction between
fields and currents must be calculated self-consistently. If the fields are
sufficiently weak one may assume that the reaction of the currents to
the fields can be described as a linear response. (For a general review
of linear-response theory in electromagnetism see Kirzhnits 1987.)
In general this statement cannot be made locally in the sense that
the currents at space-time point (t, x) were only linear functions of
A(t, x). Within the restrictions imposed by causality the relationship
between fields and currents is nonlocal; for example, a solution
of Maxwell’s equations with prescribed currents requires integrations
over the sources in space and time. After a Fourier transformation,
however, the assumption of a linear response can be stated as
J µ ind = −Πµν A ν . (6.22)
The polarization tensor Π(K) with K = (ω, k) is a function of the
medium properties.
Besides the induced current there may be an externally prescribed
one J ext which is unrelated to the response of the microscopic medium
constituents to the fields; the total current is J = J ind +J ext . Maxwell’s
equations (6.19) are then in Fourier space
(−K 2 g µν + K µ K ν + Π µν )A ν = J µ ext. (6.23)
Invariance under a gauge transformation A ν → A ν + K ν α requires that
Π µν K ν = 0. Because the external and total currents are conserved
the induced current is conserved as well, leading to K · J ind = 0 or
K µ Π µν = 0. Altogether
K µ Π µν = Π µν K ν = 0 (6.24)
which is an important general property of the polarization tensor.
Considering the Maxwell equations in Coulomb gauge in the absence
of external currents, the transversality of A still implies that it
provides only two wave polarization states, albeit with modified dispersion
relations due to the presence of Π. With regard to the Φ equation
note that in an isotropic medium the induced charge density ρ ind must
be a spatial scalar and so can depend only on Φ and the combination
k · A = 0 which is the only available scalar linear in A. Therefore, the
homogeneous equation for Φ is
(k 2 + Π 00 )Φ = 0. (6.25)
Because Π 00 is a function of ω and k this is a wave equation with the
dispersion relation k 2 + Π 00 (ω, k) = 0. The electric field associated