Stars as Laboratories for Fundamental Physics - MPP Theory Group

200 Chapter 6
The problem of the physical momentum flow associated with a wave
will be of no concern to the issues addressed in this book. In microscopic
reactions the quantity which appears in the law of “energy-momentum
conservation” is the wave vector. For example, in the plasma process
γ → νν the momenta of the outgoing neutrinos must balance against
the wave vector of the decaying electromagnetic excitation. In this
book dispersion effects will be important only for pulse propagation
from distant sources, for particle oscillation effects, and for energymomentum
conservation in microscopic reactions. In these cases the
naive interpretation of ¯hk as a particle’s momentum is safe. For the
remainder of this book the wave number (or pseudomomentum) and
the momentum of a field excitation will not be distinguished.
6.2.3 Wave-Function Renormalization
In particle reactions the main impact of medium-induced modifications
of the dispersion relations is on the kinematics, notably if a threshold
condition is involved. One is thus tempted to proceed with the usual
Feynman rules and take account of the dispersion relations only in
the phase-space integration, notably in the law of energy-momentum
conservation. In most practical cases this approach causes no problems,
although an exception are interactions involving longitudinal plasmons
(Sect. 6.3). Therefore, one should be aware that the matrix element
also must be modified because of the subtle issue of what one means
with a “particle” in a medium.
After a spatial Fourier transform the equations of motion for the
Fourier components ϕ k of a free field are those of a harmonic oscillator.
Interpreting the amplitude ϕ k and its velocity ˙ϕ k as conjugate variables,
the canonical quantization procedure leads to quantized energy levels
¯hω k , where ω k is the classical frequency of the mode k according to its
dispersion relation. Conversely, a quantized excitation with energy ¯hω k
has a certain field strength which determines its coupling strength to a
source, e.g. the coupling strength of a photon to an electron.
In a medium, the energy associated with a frequency ω k is still ¯hω k .
However, because of the presence of interaction energy between ϕ and
the medium, the field strength associated with a quantized excitation
is modified. For example, photons with a given frequency couple with
a different strength to electrons in a medium than they do in vacuum.
This modification can be lumped into a “renormalization factor” √ Z
of the coupling strength of external photon lines in a Feynman graph.