  QFT approaches elaborated for treating quantum effects in time-dependent
external electric fields are not directly applicable to time-independent
nonuniform electric fields that are given by a step potential and their
generalization for the such potentials was not sufficiently developed. Such
fields can also create particles from the vacuum, the Klein paradox being
closely related to this process. We believe that the present work presents a
consistent solution of the latter problem. Quantizing the Dirac and scalar
fields with time independent backgrounds, we have found in- and out-creation
and annihilation operators that allow one to have particle interpretation of
the physical system under consideration. To justify the proposed
identification, we have performed a detailed mathematical and physical analysis
of solutions of the corresponding relativistic wave equations with a subsequent
QFT analysis. We elaborated a nonperturbative technique that allows one to
calculate all characteristics of zero-order processes such scattering,
reflection, and electron-positron pair creation, and to calculate Feynman
diagrams that describe all characteristics of processes with interaction
between the in-, out-particles and photons. These diagrams have formally the
usual form, but contain special propagators. Expressions for these propagators
in terms of in- and out-solutions are presented. We apply the elaborated
approach to two popular exactly solvable cases: to the Sauter potential and to
the Klein step.
