Feynman Rules in terms of the Wigner transformed Green functions

TL;DR

This paper introduces a new diagrammatic method using Wigner transformed propagators and Moyal products to simplify calculations in inhomogeneous quantum field models, reducing integrations needed.

Contribution

It formulates Feynman rules in terms of Wigner transformed Green functions, extending diagrammatic techniques to inhomogeneous systems with fewer integrations.

Findings

Reduces the number of integrations in propagator calculations.

Provides a straightforward extension to various models.

Potentially simplifies calculations of non-dissipative currents.

Abstract

In the models defined on the inhomogeneous background the propagators depend on the two space - time momenta rather than on one momentum as in the homogeneous systems. Therefore, the conventional Feynman diagrams contain extra integrations over momenta, which complicate calculations. We propose to express all amplitudes through the Wigner transformed propagators. This approach allows us to reduce the number of integrations. As a price for this the ordinary products of functions are replaced by the Moyal products. The corresponding rules of the diagram technique are formulated using an example of the model with the fermions interacting via an exchange by scalar bosons. The extension of these rules to the other models is straightforward. This approach may simplify calculations in certain particular cases. The most evident one is the calculation of various non - dissipative currents.