Stars as Laboratories for Fundamental Physics - MPP Theory Group

206 Chapter 6
with this third polarization degree of freedom is proportional to kΦ,
along the direction of propagation—hence the term longitudinal excitation.
There is no magnetic field associated with it. Physically, it
corresponds to a density wave of the electrons much like a sound wave.
Obviously this mode requires being carried by a medium, as opposed
to the transverse waves which propagate in vacuum as well.
The wave equation Eq. (6.23) corresponds to a Langrangian density
in Fourier space which involves a new term −V with V = 1A 2 µΠ µν A ν
which plays the role of a medium-induced potential energy for the
field A. In vacuum Π µν can be constructed only from g µν and K µ K ν
which both violate the gauge condition ΠK = 0. Notably, this forbids a
photon mass term which would have to be of the form Π µν
mass = m 2 g µν .
In a medium an inertial frame is singled out, allowing one to construct
Π from the medium four-velocity U and to find a structure which obeys
the gauge constraint. Strictly speaking, however, the medium does not
induce an effective-mass term which would be of the form m 2 eff g µν and
which remains forbidden by gauge invariance.
6.3.3 Isotropic Polarization Tensor in the Lorentz Gauge
The Coulomb gauge is well suited to treat radiation in vacuum because
the propagating modes are neatly separated from the scalar potential,
and the gauge component is easily identified with the longitudinal part
of A. In a medium, however, the different appearance of Φ and A in
their respective wave equations is cumbersome. The Maxwell equations
in Lorentz gauge are symmetric between Φ and A which allows one to
treat all polarization states on the same footing.
In order to construct the most general Π(K) for an isotropic medium
it is useful to define four basis vectors for Minkowski space which are
adapted to the symmetry of the medium as well as to the Lorentz
condition (Weldon 1982a; Haft 1993). For that purpose one may use
the preferred directions in Minkowski space, namely K and the fourvelocity
of the medium U which is (1, 0) in its inertial frame. Moreover,
the notation ω and k is used for the frequency and wave vector of K
in the medium frame; they are covariantly given by ω = U · K and
k 2 = k 2 = (U · K) 2 − K 2 .
In Lorentz gauge the physical A fields obey K · A = 0. Therefore,
one defines a basis vector for the gauge degree of freedom by
e g ≡ K/ √ K 2 . (6.26)
Next, one chooses a vector which is longitudinal relative to the spatial