Quantization of charged fields in the presence of critical potential steps

QED with strong external backgrounds that can create particles from the vacuum is well developed for the so-called t-electric potential steps, which are time-dependent external electric fields that are switched on and off at some time instants. However, there exist many physically interesting situations where external backgrounds do not switch off at the time infinity. E.g., these are time-independent nonuniform electric fields that are concentrated in restricted space areas. The latter backgrounds represent a kind of spatial x-electric potential steps for charged particles. They can also create particles from the vacuum, the Klein paradox being closely related to this process. Approaches elaborated for treating quantum effects in the t-electric potential steps are not directly applicable to the x-electric potential steps and their generalization for x-electric potential steps was not sufficiently developed. We believe that the present work represents a consistent solution of the latter problem. We have considered a canonical quantization of the Dirac and scalar fields with x-electric potential step and have found in-and out-creation and annihilation operators that allow one to have particle interpretation of the physical system under consideration. To identify in-and out-operators we have performed a detailed mathematical and physical analysis of solutions of the relativistic wave equations with an x-electric potential step with subsequent QFT analysis of correctness of such an identification. We elaborated a nonperturbative (in the external field) technique that allows one to calculate all characteristics of zero-order processes, such, for example, scattering, reflection, and electron-positron pair creation, without radiation corrections, and also 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 of x-electric potential steps, namely, to the Sauter potential and to the Klein step.