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

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 $B_{\mu}^{ab}(x)$