  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.
