Regularization, renormalization and consistency conditions in QED with x -electric potential steps
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Regular Article - Theoretical Physics
Regularization, renormalization and consistency conditions in QED with x-electric potential steps S. P. Gavrilov1,3,a , D. M. Gitman1,2,4,b 1
Department of Physics, Tomsk State University, 634050 Tomsk, Russia P.N. Lebedev Physical Institute, 53 Leninskiy Prospect, 119991 Moscow, Russia 3 Department of General and Experimental Physics, Herzen State Pedagogical University of Russia, Moyka Embankment 48, 191186 St. Petersburg, Russia 4 Institute of Physics, University of São Paulo, CP 66318, São Paulo, SP CEP 05315-970, Brazil
2
Received: 23 May 2020 / Accepted: 10 August 2020 © The Author(s) 2020
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 τ 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 E L from above.
1 Introduction The effect of particle creation by strong electromagnetic and gravitational fields has been attracting attention for a long time. The effect has a pure quantum nature and was first considered in the framework of the relativistic quantum mechanics with the understanding that all the questions can be answered only in the framework of quantum field theory (QFT). QFT with external background is to a certain extent an appropriate model for such calculations. In the framework a e-mails: b e-mail:
[email protected]; [email protected]
of such a model, particle creation is closely related to a violation of the vacuum stability with time. Backgrounds (external fields) that violate the vacuum stability are electric-like fields that are able to produce nonzero work when interacting with charged particles. The following statement of the problem is usually considered. Assuming that the initial state of the quantized field of matter is the vacuum (often called the initial vacuum, or in-vacuum), its evolution with time under the influence of some electric like field of finite length and duration is studied. After the external field is switched off one can define a new vacuum (final vacuum, or out-vacuum) and calculate pairs of particles (out-particles) appearing above
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