  A full consideration of classical and quantum systems with radiation
(electromagnetic/gravitational) requires the involvement of a mathematical
description in the generalized phase space of high kinematical values. Based on
the dispersion chain of equations of quantum mechanics, we construct a
generalization of the von Neumann equation for the density matrix in the phase
space of fourth-order kinematical values. The paper introduces a new extended
definition of the fourth rank Wigner function, which is constructed from the
wave functions of the second rank. A new extended Moyal equation (PSI-Moyal
equation) for the Wigner function of the fourth rank is obtained. Theorems on
the properties of the new PSI-Moyal equation and its solutions are proved. An
example of a model quantum system is considered in detail.
