Liouville equations for neutrino distribution matrices

TL;DR

This paper develops Liouville equations for neutrino distribution matrices, extending classical phase space concepts to quantum flavor states, with derivations from covariant models and Klein-Gordon equations, illuminating quantum-classical connections.

Contribution

It introduces explicit derivations of relativistic neutrino distribution matrix equations using covariant flavor mixing models and quantum field-based Klein-Gordon equations.

Findings

Derived Liouville equations for neutrino distribution matrices.

Connected quantum field formalism with classical gas behavior.

Provided a case study on quantum to classical transition in neutrino gases.

Abstract

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.