  In a medium that contains a neutrino background in addition to the matter
particles, the neutrinos contribute to the photon self-energy as a result of
the effective electromagnetic vertex that they acquire in the presence of
matter. We calculate the contribution to the photon self-energy in a dense
plasma, due to the presence of a gas of charged particles, or neutrinos, that
moves as a whole relative to the plasma. General formulas for the transverse
and longitudinal components of the photon polarization tensor are obtained in
terms of the momentum distribution functions of the particles in the medium,
and explicit results are given for various limiting cases of practical
interest. The formulas are used to study the electromagnetic properties of a
plasma that contains a beam of neutrinos. The transverse and longitudinal
photon dispersion relations are studied in some detail. Our results do not
support the idea that neutrino streaming instabilities can develop in such a
system. We also indicate how the phenomenon of optical activity of the neutrino
gas is modified due to the velocity of the neutrino background relative to the
plasma. The general approach and results can be adapted to similar problems
involving relativistic plasmas and high-temperature gauge theories in other
environments.
