The Moyal bracket {f,g}★ = (2/ℏ)f sin(ℏ/2 ←∂_q→∂_p - ←∂_p→∂_q) g is the quantum deformation of the Poisson bracket. Wigner functions can be negative — quantum signature.
Weyl Quantization: The Weyl map sends classical phase-space functions f(q,p) to operators f̂ via the Weyl transform. The inverse — the Wigner function W(q,p) — encodes the full quantum state as a quasi-probability distribution. Unlike classical probabilities, W can be negative (negativity = non-classicality). The Moyal product (★) deforms classical multiplication: f★g = fg + (iℏ/2){f,g}_Poisson + O(ℏ²), and the Moyal bracket replaces the commutator in the phase-space picture. This is the foundation of deformation quantization (Kontsevich 1997, Fields Medal).