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.
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