Phase Space: Wigner Negativity and Decoherence

Wigner function W(x,p) = (1/πℏ)∫ψ*(x+y)ψ(x−y)e^{2ipy/ℏ}dy is a quasi-probability distribution on phase space. Classical states have W≥0; negative regions signal non-classical quantum correlations. A Schrödinger cat state |α⟩+|−α⟩ shows fringes of negativity between two displaced Gaussians. Fock states |n⟩ have W with (−1)ⁿ sign at origin. Decoherence (coupling to environment) exponentially suppresses the interference fringes at rate γ ∝ |2α|² (distance² between components), making W non-negative and the state classical.