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

Quantum Statistical Mechanics in Classical Phase Space. Test Results for Quantum Harmonic Oscillators