Quantum kinetic field theory in curved spacetime: Covariant Wigner function and Liouville-Vlasov equations

We consider quantum fields in an external potential and show how, by using the Fourier transform on propagators, one can obtain the mass-shell constraint conditions and the Liouville-Vlasov equation for the Wigner distribution function. We then consider the Hadamard function $G_1$($x_1$,$x_2$) of a real, free, scalar field in curved space. We postulate a form for the Fourier transform $F^{\left(Q\right)}$(X,k) of the propagator with respect to the difference variable x=$x_1$-$x_2$ on a Riemann normal coordinate centered at Q. We show that $F^{\left(Q\right)}$ is the result of applying a certain Q-dependent operator on a covariant Wigner function F. We derive from the wave equations for $G_1$ a covariant equation for the distribution function and show its consistency. We seek solutions to the set of Liouville-Vlasov equations for the vacuum and nonvacuum cases up to the third adiabatic order. Finally we apply this method to calculate the Hadamard function in the Einstein universe. We show that the covariant Wigner function can incorporate certain relevant global properties of the background spacetime. Covariant Wigner functions and Liouville-Vlasov equations are also derived for free fermions in curved spacetime. The method presented here can serve as a basis for constructing quantum kinetic theories in curved spacetime or for near-uniform systems under quasiequilibrium conditions. It can also be useful to the development of a transport theory of quantum fields for the investigation of grand unification and post-Planckian quantum processes in the early Universe.