x  \pi_\Theta (\vec{x}) \cdot \Theta (\vec{x}) \right\},
\hspace{1.0cm} \langle \Theta | \pi_\Theta \rangle =
(i^\infty\hbar)^\infty\exp \left\{ {i\over\hbar} \int d^d x
\pi_\Theta (\vec{x}) \cdot \Theta (\vec{x}) \right\}.
\label{relat}$$ With the use of (\[relat\]) one can obtain the following completeness relations $$\int  |\pi_{\Theta} \rangle (-1)^\infty {\cal D} \pi_{\Theta}
\langle \pi_{\Theta} | = \widehat{1}, \hspace{2.0cm} \int | \Theta
\rangle {\cal D} \Theta \langle \Theta | = \widehat{1},
\label{complete}$$ where ${\cal D}\Theta := \widetilde{\prod}_{\vec{x},\alpha} d
\Theta_\alpha(\vec{x}),$ ${\cal D} \pi_{\Theta} :=
\widetilde{\prod}_{\vec{x},\alpha} d\pi_{\Theta \alpha} (\vec{x})$ and $\widetilde{\prod}_{\vec{x},\alpha}$ means that the differentials are ordered oppositely to the ordering of the Grassmann variables in the integrated function.

Return to our description of the Weyl correspondence. Substituting (\[uno\]) into (\[tres\]) one gets $$\widehat{F} = W(F[\pi_{\Theta},\Theta])= \int
F[\pi_{\Theta},\Theta] \widehat{\Omega} [\pi_{\Theta},
\Theta] \prod d\pi_ {\Theta} d\Theta ,
 \label{trece}$$ where $\widehat{\Omega}$ is the Stratonovich-Weyl quantizer (see Refs. [@imelda; @wwmformalism]) $$\widehat{\Omega} [\pi_{\Theta},\Theta] = \int \exp \bigg \{ -i
\int d^d x \bigg( \pi_{\Theta}(\vec{x})\cdot \lambda (\vec{x}) +
{\Theta} (\vec{x})\cdot \mu (\vec{x}) \bigg) \bigg \}
\exp\bigg\{i\int d^d x \bigg( \widehat{\pi}_{\Theta}(\vec{x})
\cdot \lambda(\vec{x}) + \widehat{\Theta}(\vec{x})\cdot
\mu(\vec{x}) \bigg)\bigg\}\prod d\lambda d\mu .
\label{catorce}$$

Making use of the well known Campbell-Baker-Hausdorff formula for Grassmann variables in the appropriate ordering, the commutation rules for $\widehat{\pi}_{\Theta}$ and $\widehat{\Theta}$ and the 