Stars as Laboratories for Fundamental Physics - MPP Theory Group

202 Chapter 6
Because Z is a constant for a fixed k the approximate equation of
motion EE405 corresponds to a Hamiltonian
H = H 0 + H int = 1 2 Z−1 ( ˙ϕ 2 k + ω 2 kϕ 2 k) + gϕ k ρ k . (6.14)
The free-field term is of the standard harmonic-oscillator form if one
substitutes ϕ k = √ Z ˜ϕ k , i.e. free particles are excitations of the field
˜ϕ k which has a renormalized amplitude relative to ϕ k .
In terms of the renormalized field the interaction Hamiltonian is now
of the form H int = √ Zg ˜ϕ k ρ k which means that particles in the medium
interact with an external source with a modified strength √ Zg. Therefore,
in Feynman graphs one must include one factor of √ Z for each
external line of the ϕ field. Equivalently, the squared matrix element
involves a factor Z for each external ϕ particle.
For relativistic modes where |ω 2 k − k 2 | ≪ ω 2 k the modification is
inevitably small, |Z−1| ≪ 1. For (longitudinal) plasmons, however, the
dispersion relation in a nonrelativistic plasma is approximately ω = ω P
with the plasma frequency ω P , i.e. they are far away from the light
cone, and then Z is a nonnegligible correction (Sect. 6.3). In the original
calculation of the plasma decay process γ → νν an incorrect Z was used
for the longitudinal excitations (Adams, Ruderman, and Woo 1963).
The correct factor was derived by Zaidi (1965).
6.3 Photon Dispersion
6.3.1 Maxwell’s Equations
For the astrophysical applications relevant to this book the photon refractive
index in a fully ionized plasma consisting of nuclei and electrons
will be needed. On the quantum level, this system is entirely described
by quantum electrodynamics (QED). It is sometimes referred to as a
QED plasma—in contrast with a quark-gluon plasma which is described
by quantum chromodynamics (QCD). The calculation of the refractive
index amounts to an evaluation of the forward scattering amplitude of
photons on electrons, a simple task except for the complications from
the statistical averaging over the electrons which are partially or fully
relativistic and exhibit any degree of degeneracy. Recently Braaten and
Segel (1993) have found an astonishing simplification of this daunting
problem (Sect. 6.3.4).
A more conceptual complication is the occurrence of a third photon
degree of freedom in a medium (Langmuir 1926), sometimes referred to