Feynman rules in terms of the Wigner transformed Green functions

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. (C) 2020 The Author(s). Published by Elsevier B.V.