relations (\[transla\]) and (\[complete\]), it is possible to reexpress $\widehat{\Omega}[\pi_{\Theta},\Theta]$ in a very useful form $$\widehat{\Omega}[\pi_{\Theta},\Theta] = (-i)^\infty \int {\cal D}
\mu \exp \bigg \{ -i \int d^d x \ \Theta(\vec{x}) \cdot
\mu(\vec{x}) \bigg \} \bigg{|} \pi_{\Theta} - {\hbar\mu \over 2}
\bigg\rangle \bigg\langle \pi_{\Theta} + {\hbar\mu \over
2}\bigg{|}$$ $$= (i)^\infty  \int {\cal D} \lambda \exp \bigg \{ - i \int d^d x
\pi_{\Theta}(\vec{x}) \cdot \lambda(\vec{x}) \bigg\} \bigg{|}
\Theta - {\hbar \lambda \over 2} \bigg\rangle \bigg\langle \Theta
+ {\hbar \lambda \over 2} \bigg{|}. \label{dnueve}$$ Now let us define the “trace” as follows $$\begin{aligned}
{\rm tr}\{ \widehat{\cal O}\} &:=& (i\hbar)^{-\infty}\int{\cal D}
\Theta
\big\langle \Theta \big| \widehat{\cal O} \big| \Theta \big\rangle\nonumber\\
&=&(i\hbar)^{-\infty}\int{\cal D} \pi_{\Theta} \big\langle
\pi_{\Theta} \big| \widehat{\cal O} \big| \pi_{\Theta}
\big\rangle, \nonumber\end{aligned}$$ for any operator $\widehat{\cal O}$ (compare with [@imelda]).

With this definition one can check also that the Stratonovich-Weyl operator satisfies the following properties

$${\rm tr} \big \{ \widehat{\Omega}[\pi_{\Theta},\Theta] \big \} =
1, \label{dseis}$$

$${\rm tr} \bigg \{ \widehat{\Omega}[\pi_{\Theta},\Theta]
\widehat{\Omega}[\pi_{\Theta} ', \Theta '] \bigg \} =
\prod_{\vec{x},\alpha} \big(\Theta_\alpha(\vec{x}) -
\Theta'_\alpha(\vec{x})\big) \big(\pi_{{\Theta}_\alpha}(\vec{x}) -
\pi'_{{\Theta}_\alpha}(\vec{x})\big) =: \delta (\Theta - \Theta' ,
\pi_{\Theta} - \pi_{\Theta} '). \label{dsiete}$$

Finally, multiplying (\[trece\]) by $\widehat{\Omega}[\pi_{\Theta},\Theta]$ and taking into account the property (\[dsiete\]) one easily gets the fundamental 