Discrete Wigner-Weyl calculus for the finite lattice

We develop the approach of Felix Buot to construction of Wigner-Weyl calculus for the lattice models. We apply this approach to the tight-binding models with finite number of lattice cells. For simplicity we restrict ourselves to the case of rectangular lattice. We start from the original Buot definition of the symbol of operator. This definition is corrected in order to maintain self-consistency of the algebraic constructions. It appears, however, that the Buot symbol for simple operators does not have a regular limit when the lattice size tends to infinity. Therefore, using a more dense auxiliary lattice we modify the Buot symbol of operator in order to build our new discrete Weyl symbol. The latter obeys several useful identities inherited from the continuum theory. Besides, the limit of infinitely large lattice becomes regular. We formulate Keldysh technique for the lattice models using the proposed Weyl symbols of operators. Within this technique the simple expression for the electric conductivity of a two dimensional non-equilibrium and non-homogeneous system is derived. This expression smoothly approaches the topological one in the limit of thermal equilibrium at small temperature and large system area.