Generalized Boltzmann formalism for oscillating neutrinos

In the standard approaches to neutrino transport in the simulation of core-collapse supernovae, one will often start from the classical Boltzmann equation for the neutrinos spatial, temporal, and spectral evolution. For each neutrino species, and its anti-particle, the classical density in phase space, or the associated specific intensity, will be calculated as a function of time. The neutrino radiation is coupled to matter by source and sink terms on the right-hand-side of the transport equation and together with the equations of hydrodynamics this set of coupled partial differential equations for classical densities describes, in principle, the evolution of core collapse and explosion. However, with the possibility of neutrino oscillations between species, a purely quantum-physical effect, how to generalize this set of Boltzmann equations for classical quantities to reflect oscillation physics has not been clear. To date, the formalisms developed have retained the character of quantum operator physics involving complex quantities and have not been suitable for easy incorporation into standard supernova codes. In this paper, we derive generalized Boltzmann equations for quasi-classical, real-valued phase-space densities that retain all the standard oscillation phenomenology, including resonant flavor conversion (the MSW effect), neutrino self-interactions, and the interplay between decohering matter coupling and flavor oscillations. With this formalism, any code(s) that can now handle the solution of the classical Boltzmann or transport equation can easily be generalized to include neutrino oscillations in a quantum-physically consistent fashion.