Feynman and Wiener Path Integrals Representations of the Liouville Evolution

We show that the Liouville equation for any Hamiltonian system can be identified with the Schrödinger equation obtained by the Schrödinger-Dirac quantization of some “extended” Hamiltonian system whose phase space is the product of two copies of the initial phase space and hence whose configuration space coincides with the initial phase space. This identification allows us to represent some solutions of the Cauchy problem for the Liouville equation by Feynman path integrals [1] over trajectories in the phase space of the extended Hamiltonian systems.