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.
Nelson, Derivation of the schr¨ odinger equation from newtonian mechanics, Physical Review150, 1079 (1966)
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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.