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Wigner Function

The Wigner function provides a quasi-probability distribution in phase space, directly arising from the path integral formulation, offering insights into quantum behavior.
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The statement of the theorem

The Wigner function W(q,p,t)W(q, p, t) provides a quasi-probability distribution in phase space, defined via the Fourier transform of the density matrix ρ(q,q′,t)\rho(q, q', t): \nW(q,p,t)=12πℏ∫−∞∞eipτ/ℏ⟨q−τ/2∣ρ(t)∣q+τ/2⟩dτW(q, p, t) = \frac{1}{2\pi \hbar} \int_{-\infty}^{\infty} e^{i p \tau / \hbar} \langle q - \tau/2 | \rho(t) | q + \tau/2 \rangle d\tau
Source: Wikipedia