  We construct the chiral radiation transport equation for left-handed
neutrinos in the context of radiation hydrodynamics for core-collapse
supernovae. Based on the chiral kinetic theory incorporating quantum
corrections due to the chirality of fermions, we derive a general relativistic
form of the chiral transfer equation with collisions. We show that such quantum
corrections explicitly break the spherical symmetry and axisymmetry of the
system. In the inertial frame, in particular, we find that the so-called side
jump leads to quantum corrections in the collisions between neutrinos and
matter. We also derive analytic forms of such corrections in the emission and
absorption rates for the neutrino absorption process. These corrections result
in the generation of kinetic helicity and cross helicity of matter, which
should then modify the subsequent evolution of matter. This theoretical
framework can be applied to investigate the impacts of the chirality of
neutrinos on the evolution of core-collapse supernovae.
