Liouville equations for neutrino distribution matrices

The classical notion of a single-particle scalar distribution function or phase space density can be generalized to a matrix in order to accommodate superpositions of states of discrete quantum numbers, such as neutrino mass/flavor. Such a "neutrino distribution matrix" is thus an appropriate construct to describe a neutrino gas that may vary in space as well as time and in which flavor mixing competes with collisions. The Liouville equations obeyed by relativistic neutrino distribution matrices, including the spatial derivative and vacuum flavor mixing terms, can be explicitly but elegantly derived in two new ways: from a covariant version of the familiar simple model of flavor mixing, and from the Klein-Gordon equations satisfied by a quantum "density function" (mean value of paired quantum field operators). Associated with the latter derivation is a case study in how the joint position/momentum dependence of a classical gas (albeit with Fermi statistics) emerges from a formalism built on quantum fields.