'' - \Theta,$ $\Pi ' = \pi_{\Theta} '
- \pi_{\Theta}$, $\Pi ''= \pi_{\Theta}'' - \pi_{\Theta} .$ Using the expansion of $F[\pi_{\Theta} ', \Theta '] = F[\pi_{\Theta} +
\Pi ', \Theta + \Psi ']$ and $G[\pi_{\Theta} '', \Theta ''] =
G[\pi_{\Theta} + \Pi '', \Theta + \Psi ''] $ into the Taylor series and performing some manipulations we obtain $$\big(F \star  G\big)[\pi_{\Theta},\Theta] = F[\pi_{\Theta},\Theta]
\exp\bigg\{{i\hbar\over 2} \buildrel{\leftrightarrow}\over {\cal
P}_{\mathbb{G}}\bigg\} G[\pi_{\Theta},\Theta], \label{treinta}$$ with $$\buildrel{\leftrightarrow}\over {\cal P}_{\mathbb{G}} :=  \int d^d
x \bigg({{\buildrel{\leftarrow}\over {\delta}}\over \delta
\Theta(\vec{x})} {{\buildrel{\rightarrow}\over {\delta}}\over
\delta \pi_{\Theta}(\vec{x})} + {{\buildrel{\leftarrow}\over
{\delta}}\over \delta \pi_{\Theta}(\vec{x})}
{{\buildrel{\rightarrow}\over {\delta}}\over \delta
\Theta(\vec{x})}\bigg), \label{tuno}$$ where $\overleftarrow{\delta}$ and $\overrightarrow{\delta}$ are the right and left functional derivatives, respectively. The last formula corresponds exactly to the Poisson bracket for two functionals $F$ and $G$ which is given by $$F\buildrel{\leftrightarrow}\over {\cal P}_{\mathbb{G}} G :=\{
F[\pi_{\Theta},\Theta],G[\pi_{\Theta},\Theta] \}_{P}$$ $$= (-1)^{\epsilon_{F}} \int d^d x \bigg\{ {\delta
F[\pi_{\Theta},\Theta] \over \delta\Theta(\vec{x})}{\delta
G[\pi_{\Theta},\Theta] \over \delta \pi_{\Theta}(\vec{x})}
 + {\delta
F[\pi_{\Theta},\Theta] \over \delta \pi_{\Theta}(\vec{x})} {\delta
G[\pi_{\Theta},\Theta] \over \delta \Theta(\vec{x})}\bigg\},
\label{poissonsusy}$$ with $\epsilon_F=0,1$ depending on whether the corresponding functional $F$ is even or odd, respectively. This Poisson bracket has been reported in the literature for many years [@casa; @marinov; @sberezin; @dewittbook; @bordemann; @bordemannII; @duetsch; @zachosfermions; @hirshfeld; @clifford; @imelda].

The associated symplectic 