Wigner distribution function and phase-space formulation of quantum cosmology

We present a phase-space formulation of quantum cosmology in terms of the Wigner distribution functions, which are useful for statistical descriptions of the quantum states of the Universe. Using a conformal formulation of field theory in curved space we derive a set of Wheeler-DeWitt-Vlasov equations and a generalized mass-shell condition satisfied by the Wigner functions up to the second adiabatic order. For minisuperspaces with diagonal metric and potentials of one variable, we find approximate solutions to these equations and exact solutions for the Bianchi type-II universe as an example. They all possess the correct classical limits.