  A non-separable wave-like integro-differential equation for the time
evolution of the Wigner distribution function in phase space is educed from the
corresponding separable kinetic equation. It is shown that it leads to
non-local dispersion effects that may involve complex group velocities and also
display non-unitary features as entropy production. By employing the quantum
hydrodynamical description a non-local evolution wave equation is also derived
by synthesizing the Hamilton-Jacobi equation with that of continuity, which
predicts the generation of quadrupole quantum effects in the propagation of the
spatial probability density. Extension of the formalism for a boson scalar
field is also presented along with a brief commentary on the issue of
irreversibility.
