Mathematica(@mathemetica):W(q, p) is allowed to be negative. That single liberty turns a classical phase-space density into a complete encoding of a quantum state. Wigner defined it in 1932 so that the ordinary integrals W(q, p) = (1/π) ∫ ψ*(q + y) ψ(q − y) exp(2 i p y) dy recover the correct position and momentum probabilities as marginals, while expectation values of Weyl-ordered operators become ordinary averages against W. Only Gaussian wave packets stay non-negative everywhere; every other pure state must oscillate through negative regions, the visible imprint of quantum interference. The same idea, discretized by parity, paints each point of the Bloch sphere with a characteristic pattern of positive and negative values.

2026.09.22 13:43

W(q, p) is allowed to be negative. That single liberty turns a classical phase-space density into a complete encoding of a quantum state. Wigner defined it in 1932 so that the ordinary integrals W(q, p) = (1/π) ∫ ψ*(q + y) ψ(q − y) exp(2 i p y) dy recover the correct position and momentum probabilities as marginals, while expectation values of Weyl-ordered operators become ordinary averages against W. Only Gaussian wave packets stay non-negative everywhere; every other pure state must oscillate through negative regions, the visible imprint of quantum interference. The same idea, discretized by parity, paints each point of the Bloch sphere with a characteristic pattern of positive and negative values.