The Wigner function is recovered exactly as the expectation of signed weights times delta functions along classical phase-space trajectories, with the weights carrying all non-classical corrections from the Moyal bracket.
Polkovnikov, Phase space representation of quan- tum dynamics, Annals of Physics325, 1790 (2010), arXiv:0905.3384 [cond-mat.stat-mech]
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Weighted Phase-Space Paths for Exact Wigner Dynamics
The Wigner function is recovered exactly as the expectation of signed weights times delta functions along classical phase-space trajectories, with the weights carrying all non-classical corrections from the Moyal bracket.