  We study neutrino-nucleon scattering and absorption in a dense, magnetized
nuclear medium. These are the most important sources of neutrino opacity
governing the cooling of a proto-neutron star in the first tens of seconds
after its formation. Because the weak interaction is parity violating, the
absorption and scattering cross-sections depend asymmetrically on the
directions of the neutrino momenta with respect to the magnetic field. We
develop the moment formalism of neutrino transport in the presence of such
asymmetric opacities and derive explicit expressions for the neutrino flux and
other angular moments of the Boltzmann transport equation. For a given neutrino
species, there is a drift flux of neutrinos along the magnetic field in
addition to the usual diffusive flux. This drift flux depends on the deviation
of the neutrino distribution function from thermal equilibrium. Hence, despite
the fact that the neutrino cross-sections are asymmetric throughout the star,
asymmetric neutrino flux can be generated only in the outer region of the
proto-neutron star where the neutrino distribution deviates significantly from
thermal equilibrium. In addition to the asymmetric absorption opacity arising
from nucleon polarization, we find the contribution of the electron (or
positron) ground state Landau level. For neutrinos of energy less than a few
times the temperature, this is the dominant source of asymmetric opacity.
Lastly, we discuss the implication of our result to the origin of pulsar kicks:
in order to generate kick velocity of a few hundred km/s from asymmetric
neutrino emission using the parity violation effect, the proto-neutron star
must have a dipole magnetic field of at least $10^{15}-10^{16}$ G.
