Abstract
We develop Wigner–Weyl formalism for the lattice models. For the definiteness we consider Wilson fermions in the presence of U(1) gauge field. The given technique reduces calculation of the two point fermionic Green function to solution of the Groenewold equation. It relates Wigner transformation of the Green function with the Weyl symbol QW of Wilson Dirac operator. We derive the simple expression for QW in the presence of varying external U(1) gauge field. Next, we solve the Groenewold equation to all orders in powers of the derivatives of QW. Thus the given technique allows to calculate the fermion propagator in the lattice model with Wilson fermions in the presence of arbitrary background electromagnetic field. The generalization of this method to the other lattice models is straightforward.
Highlights
We develop Wigner - Weyl formalism for the lattice models
V we present our results on the Weyl symbol of Wilson Dirac operator
VI we propose the procedure of the iterative solution of the Groenewold equation, which allows to calculate completely the propagator of Wilson fermions in the presence of external electromagnetic field
Summary
We start from the definition of the average of operator Awith respect to quantum state Ψ ∫. For simplicity we consider the case of one - dimensional space R1. The generalization of our expressions to the case of D - dimensional space RD is straightforward. In terms of the matrix elements in momentum space: AW (x, p) ≡ dqeiqx ⟨p + q/2| A |p − q/2⟩. Wigner function for the state Ψ is defined as the Weyl symbol of the corresponding density operator:.
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