We derive semiclassical neutrino-electron transport equations in the collis
ionless (Vlasov) limit from the coupled Dirac equations, incorporating the
charged and neutral weak current-current as well as electromagnetic interac
tions. A corresponding linear response theory is derived. In particular, we
calculate the response functions for a variety of beam-plasma geometries,
which are of interest in a supernova scenario. We apply this to the study o
r plasmons and to a new class of collective pharon resonance modes, which a
re characterized by omega <q. We find that the growth rates of the unstable
modes correspond to a strongly temperature (<proportional to>(TnuTe3)-T-2)
and linearly momentum dependent e-folding length of about 10(10) km under
typical conditions for type II supernovas. This appears to rule out such lo
ng-wavelength collective modes as an efficient means of depositing neutrino
energy into the plasma sphere.