Some Comments of Vlasov-Liouville Equation and its Relation with Gravitational Field

TL;DR

This paper explores formulating the Vlasov-Liouville equation for spinor fields in curved space-time, emphasizing the role of covariant derivatives and spin connections in the presence of gravitational and gauge fields.

Contribution

It introduces a framework for expressing the Vlasov-Liouville equation for spinor fields coupled to gauge fields within curved space-time using covariant derivatives and spin connections.

Findings

Derived a covariant form of the Vlasov-Liouville equation in curved space-time.

Clarified the role of spin connections in coupling gauge fields to spinor fields.

Provided insights into the interplay between gravitational and gauge fields in kinetic equations.

Abstract

We discuss here the possibility to write the Liouville-Vlasov equation for the Wigner-function of a spinor field coupled to a gauge field with field strength tensor $F^{\mu\nu}$ in a curved space-time versus a local Lorentz manifold (introduction of local Lorentz coordinates) with an appropriate definition of a covariant derivative carried out using a spin connection $B_{\mu}^{ab}(x)$.