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We develop Wigner - Weyl formalism for the lattice models. For thedefiniteness we consider Wilson fermions in the presence of $U(1)$ gauge field.The given technique reduces calculation of the two point fermionic Greenfunction to solution of the Groenewold equation. It relates Wignertransformation of the Green function with the Weyl symbol $Q_W$ of Wilson Diracoperator. We derive the simple expression for $Q_W$ in the presence of varyingexternal $U(1)$ gauge field. Next, we solve the Groenewold equation to allorders in powers of the derivatives of $Q_W$. Thus the given technique allowsto calculate the fermion propagator in the lattice model with Wilson fermionsin the presence of arbitrary background electromagnetic field. Thegeneralization of this method to the other lattice models is straightforward.

Wigner - Weyl formalism and the propagator of Wilson fermions in the presence of varying external electromagnetic field