On the Liouville formalism in quantum field theory

A formalism is investigated of quantum field theory based on the Liouville equation (the Liouville formalism). The Liouville equations are obtained. These are functional Wigner equation in the bosonic case, and functional Naiman equation in the fermionic case. the Green functions of a classical Liouville equation is obtained in the form of a path integral. These are necessary to construct the perturbation theory in the vivinity of the classical field. It is shown that the fermion path integral can be represented in δ-functional form if a fermionic field interacts with a classical field