  The von Neumann trace form of quantum statistical mechanics is transformed to
an integral over classical phase space. Formally exact expressions for the
resultant position-momentum commutation function are given. A loop expansion
for wave function symmetrization is also given. The method is tested for
quantum harmonic oscillators. For both the boson and fermion cases, the grand
potential and the average energy obtained by numerical quadrature over
classical phase space are shown to agree with the known analytic results. A
mean field approximation is given which is suitable for condensed matter, and
which allows the quantum statistical mechanics of interacting particles to be
obtained in classical phase space.
