way to associate a quantum operator to a classical distribution function, which is now referred to as *Weyl quantization*. This complex procedure will not be discussed further here but its inverse, the *Wigner transform* will be useful for our formalism. This transformation associates to every quantum operator $\hat{A}$ a real phase-space function $A(x,v)$:

$$A(x,v) = {\mbox{sym}}(\hat{A}) := \int e^{\frac{i}{\hbar}vy}\left\langle x-\frac{y}{2}
\right|\hat{A}\left|x+\frac{y}{2}\right\rangle dy,
\label{eq:Wigner_transform}$$

where $\langle \cdot|\cdot\rangle$ is the usual Bra-ket notation for quantum states. The transformation of a product of operators is given by

$${\mbox{sym}}(\hat{A}\hat{B}) :={\mbox{sym}}(\hat{A}) \star {\mbox{sym}}(\hat{B}),$$

where the *Moyal star product* $\star$ contains the quantum mixing of the operators. This product of functions in phase space is defined as $$A(x,v)\star B(x,v) := A(x,v)~
e^{\frac{i\hbar}{2}\left(\overleftarrow{\partial_x}\overrightarrow{\partial_v} -
\overleftarrow{\partial_v}\overrightarrow{\partial_x} \right)}~B(x,v)$$ and is a central element in this formulation of quantum mechanics. Defining the *Moyal bracket* [@Moyal1949] by

$$\left\lbrace A, B\right\rbrace_M := A\star B - B \star A$$

the commutator of operators is associated to the Moyal brackets of two symbols in the following way:

$${\mbox{sym}}\left(\left[\hat{A},\hat{B}\right]\right)
=\left\lbrace{\mbox{sym}}\left(\hat{A}\right),{\mbox{sym}}\left(\hat{B}\right)\right\rbrace_M.$$

The dynamical equation in this formulation can be written in a simple way using these brackets and reads

$$i\hbar \partial_t f = \left\lbrace \hat{H}, f \right\rbrace_M,$$

where $\hat{H}$ is the Hamiltonian of the system. The 