Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 203
as Langmuir waves or plasmons. 31 Still, one should not think of photons
as becoming literally massive like a massive vector boson which also
carries three polarization states. A photon mass is prohibited by gauge
invariance which remains intact. However, the medium singles out an
inertial frame and thus breaks Lorentz invariance, an effect which is
ultimately responsible for the possibility of a third polarization state.
It is useful, then, to begin with some general aspects of photon propagation
in a medium which are unrelated to specific assumptions about
the medium constituents. Notably, begin with the classical Maxwell
equations for the electric and magnetic fields
∇ · E = ρ, ∇ × B − Ė = J,
∇ · B = 0, ∇ × E + Ḃ = 0. (6.15)
An additional condition is that the electric charge density ρ and current
density J obey the continuity equation
∂ · J = ˙ρ − ∇ · J = 0, (6.16)
where J = (ρ, J) and ∂ = (∂ t , ∇). Covariantly, Maxwell’s equations are
∂ µ F µν = J ν , ϵ µνρσ ∂ µ F ρσ = 0, (6.17)
where F µν is the antisymmetric field-strength tensor with the nonvanishing
components F 0i = −F i0 = −E i , and F ij = −F ji = −ϵ ijk B k .
Applying ∂ µ to the inhomogeneous equation and observing that F µν
is antisymmetric and ∂ µ ∂ ν symmetric under µ ↔ ν reveals that for
consistency J must obey the continuity equation.
An equivalent formulation arises from expressing the field strengths
in terms of a four-potential A = (Φ, A) by virtue of
F µν = ∂ µ A ν − ∂ ν A µ , (6.18)
which amounts to E = −∇Φ − Ȧ and B = ∇ × A. This representation
is enabled by the homogeneous set of Maxwell equations which are then
automatically satisfied. The inhomogeneous set now takes the form
A − ∂ (∂ · A) = J, (6.19)
where = ∂ · ∂ = ∂ µ ∂ µ = ∂ 2 t − ∇ 2 .
31 They are sometimes called “longitudinal plasmons” in contrast to “transverse
plasmons.” In this nomenclature the term “plasmon” refers to any excitation of the
electromagnetic field in a medium while “photon” refers to an excitation in vacuum.