Phase-Space Methods for Fermions

Phase-space representations first arose from the attempt to describe quantum mechanics in terms of distributions over classical variables [1]. For example, Wigner introduced a function of phase-space variables W(x, p) that would classically correspond to a joint-probability distribution: an integration over x gives the marginal distribution for p and vice-versa. However in quantum mechanics, such a function is not guaranteed to be positive; Wigner interpreted this feature as a quantum correction to classical statistical mechanics [2]. Besides providing insight into the quantum-classical correspondence, phasespace distributions lead to powerful calculation tools. Where they can be interpreted as true probability distributions, the phase-space functions can be sampled with stochastic trajectories, leading to efficient calculations of quantum dynamics or equilibrium states (see also Chapters ??, ??, ??, ?? ??).