[*i.e.*]{}, $\Theta$ is the map $\Theta: M^{d+1} \to \mathbb{G}$. Canonical variables of this classical Grassmann field will be denoted by $\Theta(\vec{x},t)$ and $\pi_{\Theta} (\vec{x},t)$ with $(\vec{x},t) \in M^{d+1}=
\mathbb{R}^d \times \mathbb{R}.$

We deal with fields at the instant $t=0$ and we denote $\Theta(\vec{x},0) \equiv \Theta(\vec{x})$ and $\pi_{\Theta}
(\vec{x},0) \equiv \pi_{\Theta}(\vec{x})$. It is worth while to mention that some of the functional formulas and their manipulations are formal. It is also important to notice that since we will deal with Grassmann variables all the computations and results obtained in the present paper are valid under the specified conventions and ordering of factors given in this section. In the present section we study the deformation quantization of the Grassmann fields, including: the Stratonovich-Weyl quantizer, the Moyal $\star$-product, the Wigner functional [@wwmformalism; @tata; @hillery], and the normal ordering. In this paper we follow the conventions and notation used in Ref. [@imelda].

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The Stratonovich-Weyl Quantizer
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Let $F[\pi_{\Theta},\Theta]$ be a functional on the phase space ${\cal
Z}_{\mathbb{G}}\equiv \{ (\pi_{\Theta}, \Theta)\}$ and let $\widetilde{F}[\lambda, \mu]$ be its Fourier transform $$\widetilde{F}[\lambda,\mu] = \int F[\pi_{\Theta}, \Theta ]
\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\} \prod d \pi_ {\Theta }  