Let $\overleftarrow{a}$ and $\overrightarrow{a}$ represent the action of the operator $a$ in arguments to the left and to the right of it, respectively. Define, then,

$$\star := \exp \left \{ \frac{i \hbar}{2} \left( \overleftarrow{\partial_x} \overrightarrow{\partial_p} - \overleftarrow{\partial_p} \overrightarrow{\partial_x} \right) \right\} \, .$$

The Wigner function for an eigenstate $\langle x | \psi \rangle = \psi(x)$ of $\hat{H}$, defined by

$$ f(x,p) := \frac{1}{2 \pi} \int d\gamma \, \, \bar{\psi} \left(x - \frac{\gamma}{2} \right) \psi \left(x + \frac{\gamma}{2} \right) e^{-i \gamma p} \, ,$$

satisfies the $*$-genvalue equation

$$ H(x,p) \star f(x,p) = E f(x,p) \, ,$$

where $\hat{H}|\psi \rangle = E | \psi \rangle$ and $H(x,p)$ is the classical Hamiltonian. This shows that the Wigner's function finds itself in a central position within the deformation quantisation scheme. This is not at all surprising, since this formalism was also developed to contextualise such a function.

What, if it exists, is the place of Wigner's function within the geometric quantisation approach?


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