is the relativistic field generalization of the respective operator given in [@imelda] by the formula (64), and in the present case it reads $$\widehat{\cal N} := \exp \left\{ {1\over 2} \sum_{i=1}^N \int d^d
p f(E_{\vec{p}}) \bigg( {\overrightarrow{\delta}^2 \over \delta
b_i(\vec{p}) \delta b_i^*(\vec{p})} - {\overrightarrow{\delta}^2
\over \delta d_i^*(\vec{p}) \delta d_i(\vec{p})} \bigg) \right\},$$ where $f(E_{\vec{p}})$ is a function to be determined, $b_i(\vec{p})$, $b_i^*(\vec{p})$ stand for the $W^{-1}-images$ of the annihilation or creation operators (respectively) for particles, and $ d_i(\vec{p})$, $d_i^*(\vec{p})$ denote the $W^{-1}-images $ of annihilation or creation operators for antiparticles.

Then the Berezin-Wick or normal quantization is given by $$\widehat{F}_{\widehat{\cal N}} = \int F_{\widehat{\cal N}}
\widehat\Omega[\pi_\Theta,\Theta] \prod d\pi_{\Theta} d\Theta,$$ and the Weyl mapping of $F_{\widehat{\cal N}}$ gives the normal ordering of the Weyl image of $F_{\widehat{\cal N}}$ i.e. $$: \widehat{F} :=: W(F):\ \buildrel{df}\over {=} W(F_{\widehat
{\cal N}}) \ \buildrel{df}\over {=} W_{\widehat {\cal N}} (F).$$ In the next section the normal ordering in the case of the free Dirac field will be considered in more detail.

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Deformation Quantization of the Dirac Free Field {#dirac}
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The aim of this section is to provide an example of the application of the deformation quantization of Grassmann fields to the Dirac free field. In 