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.