# Regularization, renormalization and consistency conditions in QED with x-electric potential steps

## TL;DR

This paper advances the nonperturbative formulation of QED with x-electric potential steps by introducing new renormalization and regularization methods, enabling better analysis of vacuum instability, current, and energy-momentum tensor matrix elements, and addressing backreaction effects.

## Contribution

It proposes novel renormalization and volume regularization procedures in QED with x-steps, linking physical matrix elements with vacuum instability characteristics and considering backreaction effects.

## Key findings

- New regularization parameter {} fixed by physical considerations
- Ability to distinguish physical parts of current and energy-momentum tensor matrix elements
- Restriction on the product EL for the consistency of backreaction analysis

## Abstract

The present article is an important addition to the nonperturbative formulation of QED with x-steps presented by Gavrilov and Gitman in Phys. Rev. D. 93, 045002 (2016). Here we propose a new renormalization and volume regularization procedures which allow one to calculate and distinguish physical parts of different matrix elements of operators of the current and of the energy-momentum tensor, at the same time relating the latter quantities with characteristics of the vacuum instability. For this purpose, a modified inner product and a parameter {\tau} of the regularization are introduced. The latter parameter can be fixed using physical considerations. In the Klein zone this parameter can be interpreted as the time of the observation of the pair production effect. In the refined formulation of QED with x-steps, we succeeded to consider the backreaction problem. In the case of an uniform electric field E confined between two capacitor plates separated by a finite distance L, we see that the smallness of the backreaction implies a restriction (the consistency condition) on the product EL from above.