Wigner-Weyl formalism for the lattice models

We discuss application of Wigner-Weyl formalism to the lattice models of condensed matter physics and relativistic quantum field theory. For the noninteracting models our technique relates Wigner transformation of the fermionic Green's function with the Weyl symbol QW of lattice Dirac operator. We take as an example of the model defined on rectangular lattice the model of Wilson fermions. It represents the regularization of relativistic quantum field theory and, in addition, describes qualitatively certain topological materials in condensed matter physics. In this model we derive expression for QW in the presence of the most general case of external U(1) gauge field. Next, we solve the Groenewold equation to all orders in powers of the derivatives of QW.