show how its main ingredient, the *Wigner Distribution Function*, will play the role of an approximate distribution function for our problem. Readers interested only in the end results can jump directly to Section \[ssec:local\].

Phase-space quantum mechanics {#ssec:QM}
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Quantum mechanics is usually presented as emerging from the Hamiltonian formulation of classical mechanics through canonical quantization (See for instance [@Sakurai]). In this procedure, variables are promoted to Hermitian operators and the Poisson bracket is replaced by a commutator. Alternatively, one can also use Feynman’s propagator and the path integral formalism to move from classical to quantum mechanics.

Alongside these well-known quantization procedures, there exist other equivalent formulations which try to emphasize more clearly certain aspects. The Moyal (or phase-space) formulation is among those and tries to find a quantum equivalent to the classical phase-space and distribution functions [@Ercolessi2007; @Hillery1984]. The quantization procedure tries to find a correspondence between classical functions (called symbols) of the phase space variables and quantum operators in Hilbert space:

$$\mbox{Operators in Hilbert space}\leftrightarrow \mbox{Phase space symbols}$$

As the position and momentum operators do not commute, this mapping can not be unique. Different operator orderings will be mapped to different phase-space symbols. Hermann Weyl proposed a systematic 