relation $$F[\pi_{\Theta},\Theta] = {\rm tr} \bigg \{ \widehat{\Omega}
[\pi_{\Theta},\Theta] \widehat{F} \bigg\}. \label{docho}$$

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The Moyal $\star$-Product {#SP}
-------------------------

We are at the position to define the Moyal $\star$-product in field theory involving Grassmann scalar fields. Let $F=F[\pi_{\Theta},\Theta]$ and $G=G[\pi_{\Theta},\Theta]$ be some functionals on $\cal{Z}_{\mathbb{G}}$ that correspond to the field operators $\widehat{F}$ and $\widehat{G}$ respectively, i.e. $F[\pi_{\Theta},\Theta]=W^{-1}(\widehat{F})={\rm tr} \bigg \{
\widehat{\Omega} [\pi_{\Theta},\Theta] \widehat{F} \bigg \}$ and $G[\pi_{\Theta},\Theta]=W^{-1}(\widehat{G}) = {\rm tr} \bigg \{
\widehat{\Omega}[\pi_{\Theta},\Theta] \widehat{G} \bigg \}$. The functional which corresponds to the operator product $\widehat{F}
\widehat{G}$ will be denoted by $(F \star
G)[\pi_{\Theta},\Theta]$. So we have $$(F \star G)[\pi_{\Theta},\Theta]:= W^{-1}(\widehat{F}
\widehat{G})= {\rm tr} \bigg \{
\widehat{\Omega}[\pi_{\Theta},\Theta] \widehat{F} \widehat{G}
\bigg \}. \label{vsiete}$$ Using Eqs. (\[trece\]) and (\[vsiete\]) and performing some simple but rather lengthy calculations one gets $$(F \star G) [\pi_{\Theta},\Theta] = \bigg({ i\hbar \over 2}
\bigg)^{2\infty}\int \ F[\pi_{\Theta} ', \Theta '] G[\pi_{\Theta}
'', \Theta '']$$ $$\times \exp \bigg \{ -{2i \over \hbar} \int d^d x \bigg(
\pi_{\Theta} (\Theta' - \Theta'' ) + \pi_{\Theta}' (\Theta'' -
\Theta ) + \pi_{\Theta}''(\Theta - \Theta ') \bigg) \bigg \} \prod
d\pi_{\Theta} ' d\Theta ' \prod d\pi_{\Theta}'' d \Theta '' .
\label{vocho}$$

Introduce new variables: $\Psi ' = \Theta ' -
\Theta,$ $\Psi '' = \Theta 