  The concept of phase space distribution functions and their evolution is used
in the case of en enlarged phase space. In particular, we include the intrinsic
spin of particles and present a quantum kinetic evolution equation for a scalar
quasi-distribution function. In contrast to the proper Wigner transformation
technique, for which we expect the corresponding quasi-distribution function to
be a complex matrix, we introduce a spin projection operator for the density
matrix in order to obtain the aforementioned scalar quasi-distribution
function. There is a close correspondence between this projection operator and
the Husimi (or Q) function used extensively in quantum optics. Such a function
is based on a Gaussian smearing of a Wigner function, giving a positive
definite distribution function. Thus, our approach gives a Wigner-Husimi
quasi-distribution function in extended phase space, for which the reduced
distribution function on the Bloch sphere is strictly positive. We also discuss
the gauge issue and the fluid moment hierarchy based on such a quantum kinetic
theory.
