  Starting from Feynman's Lagrangian description of quantum mechanics, we
propose a method to construct explicitly the propagator for the Wigner
distribution function of a single system. For general quadratic Lagrangians,
only the classical phase space trajectory is found to contribute to the
propagator. Inspired by Feynman's and Vernon's influence functional theory we
extend the method to calculate the propagator for the reduced Wigner function
of a system of interest coupled to an external system. Explicit expressions are
obtained when the external system consists of a set of independent harmonic
oscillators. As an example we calculate the propagator for the reduced Wigner
function associated with the Caldeira-Legett model.
