6.2.3 Operator-Theoretic Structure

The Wigner function construction is a special case of a correspondence between quantum-mechanical operators and ordinary c-number phase space functions.

For operators which are functions of coordinates or momenta only, such as $x^n$ or $p^n$, the Weyl transform agrees with the classical function corresponding to $A$. This can not be the case in general, as is clear from the non-commutativity of the operator product. For example the Weyl transform of $xp$ is $xp + \imath \hbar / 2$.

The crucial question is: what does the operator product look like in phase space? This question was answered by Moyal in [Moy49] by introduction of the star product.

Let $A_w, B_w$ be the Weyl transforms of the operators $A,B$. Then the Weyl transform $C_w$ of the operator $AB$ is given by the Moyal star product $A_w \star B_w$:

$$C_w(x,p) = \frac{1}{\hbar^2 \pi^2}\int du dv dw dz A_w(x + u, p + v) B_w(x + w, p + z) \exp(\frac{2 \imath}{\hbar}(uz - vw)) .$$ (6.49)

The $\star$-multiplication of c-number phase space functions is in complete isomorphism to Hilbert-space operator multiplication.

The von Neumann equation

$$\imath \hbar \frac{\partial \rho(x,t)}{\partial t} = [H, \rho(x,t)]$$ (6.50)

becomes the quantum Liouville (or Wigner) equation:

$$\imath \hbar \frac{\partial f(x,t)}{\partial t} = H_w \star f - f \star H_w .$$ (6.51)