Stars as Laboratories for Fundamental Physics - MPP Theory Group

Particle Dispersion and Decays in Media 199
tempted to interpret p = ¯hk as the particle’s momentum. In vacuum a
particle’s velocity is p/E, a quantity which exceeds the speed of light
for space-like excitations. Occasionally one reads in the literature that
for this reason only those branches of a particle dispersion relation were
physical where |p| < E. Such statements are incorrect, however, and
the underlying concern about tachyonic propagation is unfounded.
The quantity p/E has no general physical relevance. Two significant
velocity definitions are the phase velocity and the group velocity of
a wave (Jackson 1975). The former is the speed with which the crest of
a plane wave propagates, i.e. it is given by the condition ωt − kz = 0 or
v phase = ω/k = n −1
refr . For a massive particle in vacuum v phase > 1. However,
the phase velocity can drop below the speed of light in a medium.
When this occurs for electromagnetic excitations in a plasma, electrons
can “surf” in the wave which thus transfers energy at a rate proportional
to the fine structure constant α (Landau 1946), an effect known
as Landau damping. As long as v phase > 1 the photon propagation is
damped only by Thomson scattering which is an effect of order α 2 .
The group velocity v group = dω/dk is the speed with which a wave
packet or pulse propagates. In terms of the refractive index it is
v −1
group = n refr (ω) + ω dn refr /dω (6.7)
(Jackson 1975). For a massive particle in vacuum with ω 2 = (k 2 +m 2 ) 1/2
it is v group = k/ω < 1, and also in a medium normally v group < 1. Near
a resonance it can happen that v group > 1, but there is still no reason
for alarm. The fast variation of n refr (ω) as well as the presence of a
large imaginary part near a resonance imply that the issue of signal
propagation is much more complicated than indicated by the simple
approximations which enter the definition of the group velocity. For a
detailed discussion of electromagnetic signal propagation in dispersive
media see Jackson (1975).
Evidently a naive interpretation of ¯hk as a particle momentum can
be quite misleading. Another example relates to the difficulty of separating
the momentum flow of a (light) beam in a medium into one part
carried by the wave and one carried by the medium. There was a longstanding
dispute in the literature with famous researchers on different
sides of an argument that was eventually resolved by Peierls (1976);
see also Gordon (1973). Experimentally, it was addressed by shining a
laser beam vertically through a water-air interface and measuring the
deformation of the surface due to the force which must occur because of
a photon’s change of momentum between the two media (Ashkin and
Dziedzic 1973).