# Spinor field singular functions in QED with strong external backgrounds

## TL;DR

This paper develops singular functions in strong-field QED for two types of external backgrounds, enabling nonperturbative calculations of transition amplitudes and physical quantities, and demonstrates their equivalence in certain limits.

## Contribution

It introduces a unified approach to construct singular functions in strong-field QED for electric step backgrounds, facilitating nonperturbative analysis and showing their equivalence in large-time or large-distance limits.

## Key findings

- Constructed in- and out-solutions of the Dirac equation for two electric backgrounds.
- Derived Fock-Schwinger proper-time integral representations for singular functions.
- Established the equivalence of different regularizations in large-time or large-distance limits.

## Abstract

We construct and study singular functions in strong-field $QED$ with two external electromagnetic fields that represent principally different types of external backgrounds, the first one belongs to the class of so-called $t$-potential electric steps (electric-like fields that are switched on and off at initial and final time instants), and the second one belongs to the class of so-called $x$-potential electric steps (time-independent electric-like fields of constant direction that are concentrated in a restricted spatial area). As the first background ($T$-constant electric field) is chosen an uniform electric field which acts during a finite time interval $T$ , whereas as the second background ($L$-constant electric field) is chosen a constant electric field confined between two capacitor plates separated by a large distance $L$. For the both cases we find \textrm{in}- and \textrm{out}-solutions of the Dirac equation in terms of light cone variables. With the help of these solutions, we construct Fock-Schwinger proper-time integral representations for all the singular functions that provide nonperturbative (with respect to the external backgrounds) calculations of any transition amplitudes and mean values of any physical quantities. Considering calculations in the $T$-constant field and in the $L$-constant field as different regularizations of the corresponding calculations in the constant uniform electric field, we have demonstrated their equivalence for sufficiently large $T$ and $L$.