Feynman Path Integral and Landau Density Matrix in Probability Representation of Quantum States

Using construction of probability distributions describing density operators of quantum system states, the relation of Feynman path integral with the time evolution of the density operator (Landau density matrix) (as well as the state wave function) is found. The explicit expression of the probability in terms of the Green function of the Schrödinger equation is obtained. The equation for the Green function determined by arbitrary integral of motion is written. The examples of the probability distributions describing the evolution of the free particle states, as well as of the states of the systems with time dependent integrals of motion (like the oscillator) are discussed.