The Wigner transform [1], associates to every quantum operator $\hat A$ a real phase space function $A$, its symbol, via

$$\text {sym}(\hat A) := \int \!\!dy\; e^{\frac {i}{\hbar }py} \langle q-\frac {y}{2} | \hat A | q+ \frac {y}{2} \rangle = A(q,p).$$

(1)

Its inverse, the Weyl transformation [2], gives a systematic way of associating quantum operators to classical distribution functions and is used in the Weyl quantization.