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

S P Gavrilov,D M Gitman

doi:10.1140/epjc/s10052-020-8337-4

Abstract

The present article is an important addition to the nonperturbative formulation of QED with x-steps presented by Gavrilov and Gitman (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 back-reaction 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 back-reaction implies a restriction (the consistency condition) on the product EL from above.

Highlights

Of such a model, particle creation is closely related to a violation of the vacuum stability with time
Scattering, particle creation from the vacuum and particle annihilation by the t-steps were considered in the framework of the relativistic quantum mechanics; see Refs. [1,2,3,4,5]; a more complete list of relevant publications can be found in [4,5]
In the case when we deal with a one-particle problem and relativistic quantum mechanics is applicable such a timeindependent approximation of external fields is quite common

Summary

Introduction

Our first task is to demonstrate that in the leading term approximation these quantities can be obtained using the vacuum matrix elements of the operators of the current J μ(x) and of the EMT Tμν(x) with respect to the initial and final vacua defined, according to the general formulation of strong-field QED [6,7,8] 2, we recall briefly some basic facts of QED with x-steps focusing on details important for treating the above-mentioned problem We consider such local quantities as vacuum means of operators of the electric current and of the energy–momentum tensor (EMT).

Regularization

Renormalization

Back-reaction and consistency conditions

Brief summary