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We consider 2D Yukawa theory in the strong scalar wave background. We useoperator and functional formalisms. In the latter the Schwinger--Keldyshdiagrammatic technique is used to calculate retarded, advanced and Keldyshpropagators. We use simplest states in the two formalisms in question, whichappear to be different from each other. As the result two Keldysh propagatorsfound in different formalisms do not coincide, while the retarded and advancedones do coincide. We use these propagators to calculate physical quantities.Such as the fermion stress energy flux and the scalar current. The latter oneis necessary to know to address the backreaction problem. It happens that whilein the functional formalism (for the corresponding simplest state) we find zerofermion flux, in the operator formalism (for the corresponding simplest state)the flux is not zero and is proportional to a Schwarzian derivative. Meanwhilethe scalar current is the same in both formalisms, if the background field islarge and slowly changing.

Out of equilibrium two-dimensional Yukawa theory in a strong scalar wave background