Quantum Monte Carlo in Classical Phase Space with the Wigner-Kirkwood Commutation Function. Results for the Saturation Liquid Density of ${}^4$He

A Metropolis Monte Carlo algorithm is given for the case of a complex phase space weight, which applies generally in quantum statistical mechanics. Computer simulations using Lennard-Jones ${}^4$He near the $\lambda$-transition, including an expansion to third order of the Wigner-Kirkwood commutation function, give a saturation liquid density in agreement with measured values.