Out-of-Equilibrium Two-Dimensional Yukawa Theory in a Strong Scalar Wave Background

We consider 2D Yukawa theory in a strong scalar wave background. We use operator and functional formalisms. In the latter the Schwinger–Keldysh diagram technique is used to calculate retarded, advanced and Keldysh propagators. We use simplest states in the two formalisms in question, which appear to be different from each other. As a result, the two Keldysh propagators found in different formalisms do not coincide, while the retarded and advanced ones do coincide. We use these propagators to calculate physical quantities such as the fermion stress–energy flux and the scalar current. One needs to know the latter to address the backreaction problem. It happens that while in the functional formalism (for the corresponding simplest state) we find zero fermion flux, in the operator formalism (for the corresponding simplest state) the flux is not zero and is proportional to a Schwarzian derivative. Meanwhile the scalar current is the same in both formalisms if the background field is large and slowly changing.