Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 201
In order to determine this factor from the dispersion relation consider
a scalar field ϕ in the presence of a medium which induces a refractive
index. This means that the Klein-Gordon equation in Fourier
space, including a source term ρ, is of the form
[−K 2 + Π(K)]ϕ(K) = gρ(K), (6.8)
where K = (ω, k) is a four-vector in Fourier space and g is a coupling
constant. Π(K) is the “self-energy” which includes a possible vacuum
mass m 2 and medium-induced contributions which are calculated from
the forward scattering amplitude. The homogeneous equation with
ρ = 0 has nonvanishing solutions only for K 2 = Π(K) which defines
the dispersion relation
ω 2 k − k 2 = Π(ω k , k). (6.9)
This equation determines implicitly the frequency ω k related to a wave
number k for a freely propagating mode.
A problem with Eq. (6.8) is the general dependence of Π on ω
and k which implies “dispersion,” i.e. in coordinate space it is not a
simple second-order differential equation. Otherwise the equation of
motion for a single field mode ϕ k would be (∂t 2 + k 2 + Π k )ϕ k = gρ k .
Apart from the source term this is a simple harmonic oscillator with
frequency ωk 2 = k 2 + Π k . The canonical quantization procedure then
leads to quantized excitations with energy ¯hω k —the usual “particles.”
In a medium one follows this procedure in an approximate sense by
expanding Π k (ω) ≡ Π(ω, k) to lowest order around ω k ,
Π k (ω) = Π k (ω k ) + Π ′ k(ω k )(ω − ω k ), (6.10)
where Π ′ k(ω) ≡ ∂ ω Π k (ω). To this order the Klein-Gordon equation is
[
−ω 2 + ω 2 k + Π ′ k(ω k )(ω − ω k ) ] ϕ k (ω) = gρ k (ω), (6.11)
where the dispersion relation Eq. (6.9) was used. To first order in ω−ω k
one may use 2ω = 2ω k = ω + ω k which allows one to write
where
Z −1 (ω 2 − ω 2 k) ϕ k (ω) = gρ k (ω), (6.12)
Z −1 ≡ 2ω k − Π ′ k(ω k )
2ω k
= 1 −
∣
∂Π(ω, k) ∣∣∣∣ω
. (6.13)
∂ω 2 2 −k 2 =Π(ω,k)