  We consider 2D Yukawa theory in the strong scalar wave background. We use
operator and functional formalisms. In the latter the Schwinger--Keldysh
diagrammatic 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 the result 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. The latter one
is necessary to know 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.
