generic Fermi field. First, we construct the Stratonovich-Weyl (SW) quantizer for this Fermi field. Using a modified notion of the [*trace*]{} of an operator one finds that the properties of the fermionic SW quantizer have a similar form as the corresponding properties in the bosonic case. Then the Moyal $\star$-product and the Wigner functional are found. Finally, the normal ordering for the generic Fermi field is briefly discussed. In Sec. \[dirac\] the deformation quantization formalism developed in Sec. \[scalar\_field\] is applied to the case of the Dirac free field. Using the oscillator variables we were able to simplify the considerations. In particular, with the use of these variables we have found the Wigner functional corresponding to the ground state and also the Wigner functional of an arbitrary excited state. Finally, in this section the Dirac propagator within the deformation quantization is computed. Some conclusions and final remarks in Sec. \[remarks\] close the paper.

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Deformation Quantization of Grassmann Scalar Field {#scalar_field}
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Consider a scalar Grassmann field on the Minkowski spacetime $M^{d+1}$ of signature $(-,-,\dots,-,+).$ By a Grassmann scalar field we will understand a smooth function $\Theta$ over $M^{d+1}$ which takes values in the [*field*]{} of (anti-commuting) Grassmann numbers $\mathbb{G}$, 