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

The present article is an important additionas to the consistent nonperturbative formulation of QED with x-steps elaborated by us in Phys. Rev. D. 93, 045002 (2016). There we succeeded to calculate global effects (global quantities) such as the vacuum to vacuum transition amplitude, mean differential and total numbers of created particles, and so on. Here we formulate regularization and renormalization procedures for calculating quasi-local quantities such as the vector of electric current and the energy-momentum tensor of created particles. With the help of properly calculated mean values of the latter quantities, we get an opportunity to study a backreaction problem. Considering the case of an uniform electric field confined between two capacitor plates separated by a finite distance L, we have discovered that the smallness of the backreaction implies restrictions from above on the dimensionless parameter eELT. We call these restritions consistency conditions by analogy with the already known consistency conditions derived by us in Phys. Rev. Lett. 101, 130403 (2008) for QED with t-steps.


I. INTRODUCTION
The effect of particle creation by strong electromagnetic and gravitational fields has been attracting attention already for a long time. The effect has a pure quantum nature and was first considered in the framework of the relativistic quantum mechanics with 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 extend an appropriate model for such calculations. In the framework of such a model, the particle creation is closely related to a violation of the vacuum stability with time. Backgrounds (external fields) that violate the vacuum stability are electriclike fields that are able to produce nonzero work when interacting with charged particles. Depending on the structure of such backgrounds, different approaches for calculating the effect were proposed and realized. Initially, the effect of particle creation was considered for time-dependent external electric fields that are switched on and off at the initial and the final time instants, respectively. In what follows, we call such kind of external fields the t-electric potential steps (t-step or t-steps in what follows). 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]. A general nonperturbative with respect to the external background formulation of quantum electrodynamics (QED) with t-steps was developed in Refs. [6]. However, there exist many physically interesting situations where external backgrounds formally are not switched off at the time infinity, the corresponding backgrounds formally being not t-steps. As an example, we may point out time-independent non-uniform electric fields that are concentrated in restricted space areas. The latter fields represent a kind of spatial or, as we call them conditionally, x-electric potential steps (xstep or x-steps in what follows) for charged particles. The x-step can also create particles from the vacuum, the Klein paradox being closely related to this process [7][8][9]. Approaches elaborated for nonperturbative calculations in QED with t-steps are not directly applicable to QED with x-steps. Some heuristic calculations of the particle creation by x-steps in the framework of the relativistic quantum mechanics were presented by Nikishov in Refs. [2,10] and later developed by Hansen and Ravndal in Refs. [11]. One should also mention the Damour work [12], that contributed significantly in applying semiclassical methods for treating strong field problems in astrophysics. Using the Damour's approach, mean num-bers of pairs created by a strong uniform electric field confined between two capacitor plates separated by a finite distance was calculated in Ref. [13]. A detailed historical review can be found in Refs. [11,14]. Nikishov had tested his way of calculation using the special case of a constant and uniform electric field, which is possible both for t-steps and x-steps, see [2,10,15]). At that time, however, no justification for such calculations from the QFT point of view was known. In the recent article [16], quantizing the Dirac and the Klein-Gordon (scalar) fields in the presence of x-steps, we have formulated a consistent nonperturbative formulation of QED with such steps. In that work we have considered a canonical quantization of the Dirac and scalar (Klein-Gordon) fields in the presence of x-steps as backgrounds in terms of adequate in-and out-particles focusing on nonperturbative calculation techniques of different quantum processes, such as scattering, reflection, and electron-positron pair creation. New problem was related to the time-independence of the background under consideration. Whereas, when considering t-step that vanish at the time infinity, one can naturally introduce certain in-and out-creation and annihilation operators starting first with the Schrödinger representation and then passing to the Heisenberg picture, this way of action is not possible in time-independent backgrounds. Here one must quantize in the Heisenberg picture from the very beginning. Then a new problem appears: how to identify in-and out-operators. When doing this, we believe that a time independent x-step, as well as any constant electromagnetic field, is an idealization. In fact, the field of an x-step is a part of a time-dependent field that was switched on at a remote time instant before a time instant t in , then after t in it was acting as a constant field during a very large period of time T , and finally it was switched off at another remote time instant t out = t in + T .
It was believed that if T is large enough one can ignore effects of switching the external field on and off. To identify in-and out-operators it was important to perform a detailed mathematical and physical analysis of solutions of the Dirac equation with x-steps. Using the developed approach we succeeded to calculate global effects (global quantities) of zero order with respect to the radiative interaction, such that the vacuum to vacuum transition amplitude, mean differential and total numbers of created particles, and so on. However, many important technical and principal questions still need to be answered. For example, despite the fact that the task of calculating local mean values was formally formulated, it was already clear (by the analogy with the t-step case) that its full solution would require appropriate regularization and renormalization. In the present article we consider namely the latter problem and some important applications of its solution. The article is organized as follows: In Sec. II, we recall briefly basic fact of QED with x-steps focusing on details important for treating the above mentioned problem. In Sec. III, we consider such local quantities as vacuum means of operators of the electric current and of the energy-momentum tensor (EMT). These means should be properly regularized and normalized. The corresponding procedures are considering in this section in detail. In Sec. IV, using properly calculated vacuum mean values of electric current and EMT, we discuss the backreaction problem.
Creating pairs from the vacuum the external electric field is loosing its energy and should be depleted over time. Thus, the applicability of the constant field approximation is limited by a smallness of the backreaction. We study the backreaction problem considering the case of an uniform electric field confined between two capacitor plates separated by a finite distance L. Here the smallness of the backreaction implies restriction from above on the dimensionless parameter eELT . We call the corresponding equations as the consistency conditions. We compare the obtained restrictions with the consistency conditions derived by us in Ref. [22] for QED with t-steps.

II. QED WITH x-ELECTRIC CRITICAL POTENTIAL STEPS, BRIEF REVIEW
Here we briefly recall some basic points of QED with x-steps [16].
First, we clarify details of the definition of x-step important for the problem of regularization considered in this paper. Potentials of an external electromagnetic field 1 A µ (X) in d = D + 1 dimensional Minkowski space-time parametrized by coordinates X = (X µ , µ = 0, 1, . . . , D) = (t, r) , X 0 = t, r = X 1 , . . . , X D . Without loss of generality we suppose potentials that correspond to an x-step have the form so that the magnetic field is zero and the electric field E is constant and directed along the axis X 1 (could be inhomogeneous in this direction), such that E (X) = E (x) = 1 We use the relativistic units ℏ = c = 1 in which the fine structure constant is α = e 2 /cℏ = e 2 .
(E (x) , 0, ..., 0) , E (x) = −A ′ 0 (x) . In addition, we suppose 2 that the first derivative of the scalar potential A 0 (x) does not change its sign for any x ∈ R. The basic characteristics of potentials and fields corresponding to any x-step read A 0 (x) are some constant quantities, which means that the electric field under consideration is switched off at spatial infinity. For definiteness sake, we suppose that A ′ 0 (x) ≤ 0. We assume that the basic Dirac particle is an electron with the mass m and the charge −e, e > 0, and the positron is its antiparticle. Further, for convenience, we use the following notation: U(x) = −eA 0 (x). The electric field under consideration accelerates electrons along the x axis in the negative direction and positrons along the x axis in the positive direction.
An important clarification is that and the points x FL and x FR are separated from the origin by macroscopic but finite distances.
The magnitude of the corresponding x-potential step is Without loss of generality, we choose a gauge U R = −U L which emphasizes the existing symmetry between electrons and positrons. Mainly, we are interested in the pair creation from the vacuum that exists for critical x-steps, ∆U > 2m.
Let us consider the Dirac equation with a x-step, and its solutions of the form n (x) satisfy the following secondorder differential equation: These solutions are parametrized by the set of quantum numbers n = (p 0 , p ⊥ , σ) where p 0 is the total energy, p ⊥ is the transversal momentum (the index ⊥ stands for 2, ..., D components of d-dimensional quantities), and σ is the spin polarization. To construct complete sets necessary for our purposes it is sufficient to chose only one value of χ. Further, for convenience, we choose a specific fixation, and then do not use this index at all, ϕ (χ) n (x) = ϕ n (x). We construct two types of complete sets of solution in the form (2.5). The first one ζ ψ n (X) is defined by functions ϕ n (x) denoted as ζ ϕ n (x) and the second one ζ ψ n (X) is defined by functions ϕ n (x) denoted as ζ ϕ n (x). Asymptotically, these functions have the forms: Thus, asymptotically the solutions ζ ψ n (X) and ζ ψ n (X) describe particles with given real momenta p L/R along the x axis. Note that if U R = −U L then there is a symmetry |π 0 (R)| = |π 0 (L)| p 0 →−p 0 with respect of change p 0 → −p 0 . The solutions ζ ψ n (X) and ζ ψ n (X) are orthonormalized with respect to the inner product on the x = const hyperplane, where V ⊥ is the spatial volume of the (d − 1) dimensional hypersurface orthogonal to xdirection and T is the time duration of the electric field action. Both V ⊥ and T are macroscopically large.
It is assumed that each pair of solutions ζ ψ n (X) and ζ ψ n (X) with any n ∈ Ω 1 ∪ Ω 3 ∪ Ω 5 is complete in the space of solutions with the corresponding n. A mutual decompositions of such solutions have the form where η L = η R = 1 for n ∈ Ω 1 , η L = η R = −1 for n ∈ Ω 5 , and η L = −η R = 1 for n ∈ Ω 3 .
Decomposition coefficients g have the following origin: They satisfy the unitary relations In the case of the critical x-steps if 2π ⊥ ≤ ∆U there exist five ranges Ω k , k = 1, ..., 5, of quantum numbers n, where solutions ζ ψ n (X) and ζ ψ n (X) have similar forms and properties for a given π ⊥ , see Ref. [16]. A detailed consideration of various physical processes shows that: In the range only positrons that are also subjected to the total reflection but have an unbounded motion in x → +∞ direction. No particle creation in these four ranges is possible.
In the range Ω 3 the quantum field description of processes is essential. In this range there exist in-and out-electrons that can be situated only to the left of the step, and in-and out-positrons that can be situated only to the right of the step. In this range, all the partial vacua with given n are unstable, and particle creation from vacuum is possible. These pairs consist of out-electrons and out-positrons that appear on the left and on the right of the step and move there to the left and to the right, respectively. At the same time, the in-electrons that move to the step from the left are subjected to the total reflection. After being reflected they move to the left of the step already as out-electrons. Similarly, the in-positrons that move to the step from the right are subjected to the total reflection. After being reflected they move to the right of the step already as out-positrons. Note that the range Ω 3 often referred to as the Klein zone.
Sometime, we denote by n k quantum numbers from a corresponding range, n k ∈ Ω k . It was shown that pairs of solutions ζ ψ n (X) and ζ ψ n (X) from the ranges Ω 1 , Ω 3 , and Ω 5 can be interpreted as in-and out-solutions, out − solutions : − ψ n 1 , + ψ n 1 ; + ψ n 5 , − ψ n 5 ; + ψ n 3 , + ψ n 3 . (2.14) Sets (2.14) together with solutions ψ n 2 and ψ n 4 , whose interpretation is not related to asymptotic states and which are not essential for the problems discussed in this article, form a complete systems in the Hilbert space of Dirac spinors at any fixed time instant t, see [16].
Then we consider Dirac Heisenberg operatorΨ (X), which is assigned to the Dirac field ψ (X) . It satisfies the anticommutation relations 15) and the Dirac equation (2.4. We decompose it in the complete system mentioned above. Operator-valued coefficients in such decompositions do not depend on coordinates. The introduced division of the quantum numbers n in the five ranges, implies a representation Ψ (X) = 5 i=1Ψ i (X). For each of three operatorsΨ i (X), i = 1, 3, 5, there exist two possible decompositions, given by Eq. (2.14), in creation and annihilation operators according to the existence of two different complete sets of solutions with the same quantum numbers n in the ranges Ω 1 , Ω 3 , and Ω 5 , There exist only one complete set of solutions with the same quantum numbers n 2 and n 4 . Therefore, there exists only one possible decomposition for each of the two operatorŝ We interpret all a and b as annihilation and all a † and b † as creation operators. All a and a † are interpreted as describing electrons and all b and b † as describing positrons. All the operators labeled by the argument "in" are interpreted as in-operators, whereas all the operators labeled by the argument "out" are interpreted as out-operators. This identification is confirmed by a detailed mathematical and physical analysis in Ref. [16].
Taking into account the orthogonality and orthonormalization relations, we find that the standard anticommutation relations for the Heisenberg operatorΨ (X) yield the standard anticommutation rules for the introduced creation and annihilation in-or out-operators.
Note that commutation relations between sets of in and out-operators follow from the linear canonical transformation that relates in and out-operators.
Differential mean numbers of electrons N a n (out) and positrons N b n (out), n ∈ Ω 3 , created from the vacuum are equal, N b n (out) = N a n (out) = N cr n , and have the forms The total number of pairs created from the vacuum is the sum of the differential mean numbers N cr n , (2.20)

A. Regularization
Formally, the nonperturbative technics elaborated in QED with x-steps, see II and Ref. [16], allows one to derive appropriate formulas for both amplitudes of transition processes and mean values of physical quantities. However, calculating mean values, we often meet divergences that indicate the need to use a certain regularization. Below we consider the later problem studying vacuum mean values of operators of the current J µ (x) and of the energy-momentum tensor (EMT) T µν (x), We recall that the operators J µ and T µν have the form where P µ = i∂ µ +eA µ (X). In the general case the introduced means depend on the structure of the electric field, and, in particular, they depend on the coordinate x due to x-dependence of the external field. As will be shown later, properly calculated means (3.1) determine the vector of electric current and EMT of created particles.
We assume that x-steps of the form ( Since the time interval T and the distances |x FL − x L | and |x R − x FR | are macroscopic, one may believe that measuring characteristics of particles in the regions S L and S R , we are able to evaluate the effect of pair creation in the area S int for the time interval T . To this end, in order for particles created in the area S int not to leave the regions S L and S R in time are not causally related to processes in the area S int . Physically sensible to believe that the field of an x-step, given by Eq. (2.2), should be considered as a part of a time-dependent inhomogeneous electric field E pristine (X) directed along the x-direction, which was switched on very fast before the time instant t in , by this time it had time to spread to the whole area S int and disappear in the regions S L and S R .
Then it was switching off very fast just after the time instant t out .
We assume that the leading contribution to the number density of electron-positron pairs created by such a field is independent of fast switching-on and -off if the time duration T is sufficiently large and an electric field is strong enough. In this case the large T corresponds to a large density of states that are occupied by created pairs. Such an universal behavior of the leading contribution is associated with the big state density that is a large parameter.
Note that exact results beyond the slowly varying regime obtained for the case x L → −∞ and x R → ∞ [17] support this assumption.
Results of calculating the means are often presented by some volume integrals. These integrals appear as a result of use time-independent inner product of solutions of the Dirac equation, ψ n (X) and ψ ′ n ′ (X) on t = const hyperplane, see [16], where V ⊥ is the spatial volume of the (D − 1) dimensional hypersurface orthogonal to the electric field direction. The improper integral over x in the right-hand side of Eq. (3.3) can be reduced to a principal value, where the macroscopic quantities |x FL | > K (L) > 0 and x FR > K (R) > 0 satisfy the following relation We note that only the contribution from the area S int in the right-hand side of Eq. (3.3) depends on the external field E. The smoothness of the scalar potential A 0 (x) allows us to believe that this contribution is finite.
Without introducing additional assumptions about the nature of the external field, it is possible to calculate integrals of the form (3.3) in the case when the areas S L and S R are much wider than the area S int , Conditionally, we call this case a space constrained field. In fact, namely, such a case was initially considered in our work [16]. In this case the principal value of integral (3.3) is determined by integrals over the areas x ∈ −K (L) , x L and x ∈ x R , K (R) where the electric field is zero. Under condition (3.4) it was shown that the following orthonormality relations hold on the t = const hyperplane ( ζ ψ n , ζ ψ n ′ ) = ζ ψ n , ζ ψ n ′ = δ n,n ′ M n , n ∈ Ω 1 ∪ Ω 3 ∪ Ω 5 ; (ψ n , ψ n ′ ) = δ n,n ′ M n , n ∈ Ω 2 ∪ Ω 4 , where coefficients g ′ are defined by Eq. (2.11). Wave functions with different quantum numbers n are orthogonal, and besides ζ ψ n , −ζ ψ n = 0, n ∈ Ω 1 ∪ Ω 5 , ζ ψ n and −ζ ψ n are independent; ζ ψ n , ζ ψ n = 0, n ∈ Ω 3 , ζ ψ n and ζ ψ n are independent. (3.7) We recall that identification (2.14) was proposed and justified in [16].
In the latter work, the above constructions were sufficient for an unequivocal calculation Similar to QED with t-step, see Refs. [6], mean values and probability amplitudes can be described by Feynman type diagrams with two kinds of particle propagators in the external field under consideration. In particular, the probability amplitudes are calculated using a generalized causal in-out propagator S c (X, X ′ ), while mean values are calculated by some matrix propagators involving the so-called in-in propagator S in (X, X ′ ) and out-out whereT stands for the chronological ordering operation. These propagators can be expressed via singular functions S ± (x, x ′ ): An explicit form of these singular functions for QED with x-step was found in Ref. [16], their representations for the case under consideration, are given in what follows by Eqs. (3.13).

B. Renormalization
One can show that in-in and out-out means (3.1) can be expressed via the propagators S c in and S c out as follows: Sometimes it is also useful to consider in-out matrix elements, which are expressed via the causal in-out propagator S c (X, X ′ ), Since the vacuum is stable in the ranges Ω k , k = 1, 2, 4, 5 the corresponding contributions of the propagators S c , S c in , and S c out to vacuum mean values coincide, see Ref. [16] for details. Using asymptotic behavior of the solutions ζ ψ n (X) and ζ ψ n (X) one can verify that contributions due to these ranges to vacuum currents and energy fluxes in the areas S L and S R absent. Similar contributions to the diagonal elements T µµ (x) c do not depend on the electric field and have to be neglected according to the standard renormalization procedure. Therefore, contributions due to the propagator S c can affect only the vacuum polarization in the field region S int .
We are interested in vacuum instability effects, which are formed exclusively in the Klein zone Ω 3 . As was mentioned earlier, under condition (3.5) the principal value of integral (3.3) is determined by integrals over the areas x ∈ −K (L) , x L and x ∈ x R , K (R) , where the electric field is zero. This implies that the only areas S L and S R determine contributions of the vacuum instability effects to means (3.1). For this reason, in what follows, we consider these means only in the areas S L and S R , Thus, we can be limited only to contributions to singular functions that are formed in the Klein zone (we denote such contributions further with a tilde at the top), whereψ n (X) = ψ † n (X) γ 0 and the amplitudes w n are: see Ref. [16]. Then means (3.1) can be expressed via singular functions (3.13) as follows: It follows from Eqs. (3.15) that due to the cylindrical symmetry of a problem the transversal components of the mean currents and non-diagonal components of mean values of the EMT vanish, Relations between in-in and out-out means(3.12) and in-out matrix elements (3.11) can be written in terms of a singular function S p (X, X ′ ), which, for QED with x-steps, was found in Ref. [16], These relations have the form where Note that electric current densities (3.21) (as well as all componentsj 1 n ) are conserved along the axis x, while the energy flux densitiesT 10 (L) in the left area andT 10 (R) in the right area of the axis x are different due to the potential energy difference ∆U. These quantities correspond to the areas where the electric field is absent and they have no contributions that do not depend on the electric field. So if the electric field vanishes, E → 0, in the area S int then at the same time vanish the pair production, N cr n → 0. That is why the above densities characterize in a sense real particles and cannot change after the electric field is switched off.
Normal forms of the operators J 1 and T 10 with respect to the out-vacuum are N out J 1 = J 1 − 0, out J 1 0, out , N out T 10 = T 10 − 0, out T 10 0, out .
We suppose that all the measurements are performed during a macroscopic time T when the external field can be considered as constant. In this case the time τ , defined by Eq.
(3.8), coincides with the time T , i.e., τ = T . Thus, we have Here j 1 n is the flux density of particles created with given n and n∈Ω 3 j 1 n = N cr (T V ⊥ ) −1 is the total flux density of created particles, where N cr (see Eq. (2.20)) is the total number of pairs created from the vacuum. The density j 1 n and longitudinal current density J 1 cr are conserved in the x-direction.
The density j 1 n is formed by electrons on the left of the area S int , whereas it is formed by positrons on the right of the area S int . Differential mean numbers of electrons and positrons from pairs with a given n are equal. The differential numbers N cr n characterize electrons created on the left of the area S int and at the same time characterize positrons created on the right of the area S int . Thus, on the left of the area S int and on the right of the area S int there are fluxes consisting of all electrons and all positrons created from vacuum by the external electric field, which, on the one hand, is natural from physical considerations, and on the other hand confirms the correctness of the applied approach.
Using Eqs. (2.7), (3.13), and (3.15), we find that averaging over the in-and out-vacuum of the charge and diagonal components of EMT operators give the same results, One can see how quantities (3.27) are related to direct characteristics of the vacuum instability. To this end, in these quantities, we separate contributions of matrix elements (3.12) on the left of the area S int and on the right of the area S int . Using representations (3.13), one can see thatS − (X, X ′ ) on the area x ∈ S R andS + (X, X ′ ) on the area x ∈ S L represent x-dependent interference terms, containing factors ∼ exp i2 p R (x − x R ) and exp −i2 p L (x − x L ) respectively. Such terms do not contribute to the fluxes J 1 (x) c and T 10 (x) c . They do not contribute to any integrals over x and, therefore, they can be neglected in the densities J 0 (x) c and T µµ (x) c . Now we analyze the opposite case, namely, let us consider representations of the functionS − (X, X ′ ) in the area x ∈ S L and of the functionS + (X, X ′ ) on the area x ∈ S R . Using relations (2.10) and (3.14), and denying the x-dependent interference terms, we obtaiñ  Thus, we believe that the means J 0 (x) in and T µµ (x) in at x ∈ S L/R have to be renormalized to the ones J 0 (x) ren in and T µµ (x) ren in as follows: Let us calculate the charge density and the longitudinal current density of created particles. To this end we turn to Eq. (3.25). We see that created electrons with a given n are moving to the left with the velocity v L , whereas the current density ej 1 n is moving to the right. During the time T these electrons carry the charge ej 1 n T over a unit area V ⊥ of the surface x = x L . Taking into account that this charge is evenly distributed over the cylindrical volume of the length v L T , we obtain that charge density of created electrons with given n is ej 1 n / −v L . Note that this quantity coincides with density −ej 0 n (L), where j 0 n (L) is given by Eq. (3.31), −ej 0 n (L) = ej 1 n / −v L . Created positrons with a given n are moving to the right with the velocity v R , the current density ej 1 n moves in the same direction. During the time T these positrons carry the same charge ej 1 n T over a unit area V ⊥ of the surface x = x R . This charge is evenly distributed over the cylindrical volume of the length v R T . Thus, the charge density of created positrons with a given n is ej 1 n /v R . This quantity coincides with density ej 0 n (R), where j 0 n (R) is given by Eq. (3.31), ej 0 n (R) = ej 1 n /v R . This means that the charge density of created pairs is Comparing expressions (3.31) and (3.33) we can relate the quantity J 0 cr (x) with the one J 0 (x) p . One can easily see that J 0 cr (x) and J 1 cr are two nonzero components of d dimensional Lorentz vector J µ cr (x) (note that its transversal components are zero) that presents current density of created pairs, In the same manner, one can derive the following relations for nonzero components of EMT: We see, for example, that J 0 (x) ren in and J 1 (x) in are two nonzero components of d dimensional Lorentz vector J µ (x) ren in = J µ (x) p , whereas J 0 (x) ren out and − J 1 (x) out are two nonzero components of the same Lorentz vector. Similar relation holds for the vacuum means of EMT. We stress that calculated physical quantities (3.37) are determined by precisely the final particles. Thus, the proposed procedure allows one to relate characteristics of created particles with quasi local quantities.

IV. BACKREACTION AND CONSISTENCY CONDITIONS
Let us consider a volume V = V ⊥ (x R − x L ) , that contains the area S int = (x L , x R ) .
The total energy of created particles in the volume V is given by the corresponding volume integral of the energy density T 00 cr (t, x) . The corresponding energy conservation low reads: where Σ is a d − 1 dimensional surface surrounding the volume V and df k , k = 1, ..., d − 1, are components of the surface element df. Taking into account that T 00 cr (t, x) in Eq. (4.1) does not depend on transversal coordinates, T k0 cr (x) = 0 for k = 1 and using Eqs. (3.25) and (3.26) we obtain that the rate of the energy density change of created particles in the area S int per unit of the spatial volume of the hypersurface orthogonal to the x-direction is: where N cr , given by Eq. (2.20), is the total number of pairs created from the vacuum. It characterizes the loss of the energy that created particles carry away from the region S int .
At the same time, constant rate (4.2) determines the power of the constant electric field spent on the pair creation. Integrating this rate over the time duration of an electric field from t in to t out , and using the notation we find the total energy density of created pairs per unit of the orthogonal hypersurface as We assume that the constant external electric field which represents the x-step exists during a macroscopically large time period T . The time period T is macroscopically large if Condition (4.4) ensures that switching on and off effects are negligible under some stabilization conditions; see [18].
Now we are able to discuss the backreaction problem. It is clear that creating pairs from the vacuum the external electric field is loosing its energy and should change (weaken) with time. Thus, the applicability of the constant field approximation, which is used in the formulation of QED with x-step, is limited by the smallness of the backreaction. The obtained above results allow us to find conditions that provide this smallness, we call these consistency conditions in what follows. These conditions can be be obtained from the requirement that the energy density (4.3) is essentially smaller than the energy density of the external electric field (per unit of the orthogonal hypersurface).
To do this, we consider the case of an uniform electric field confined between two capacitor plates separated by a finite distance L. We call such a configuration the L-constant electric field. In this case ∆U = eEL. The particle creation in such a case was studied in Ref. [19].
If L ≫ (eE) −1/2 max 1, m 2 /eE , (4.5) the L-constant electric field can reasonably imitate a small-gradient electric field. Besides, in such case boundary effects are negligible and the density ∆T 00 cr (x) is uniform in the leading term approximation, ∆T 00 cr (x) = ∆T 00 cr . In this case the number density in the RHS of Eq. (4.3) has the form N cr V ⊥ = Ln cr , n cr = T r cr , r cr = where n cr is the number density of pairs created per unit volume of the field area and J = 2 [d/2]−1 is the number of the spin degrees of freedom (J = 1 for scalar particles) [19].
Taking into account that x R − x L = L and integrating over x in Eq. (4.3) we find the total energy density of created pairs per the volume unit is ∆T 00 cr = eELT r cr . (4.7) Note that the density r cr , given by Eq. (4.6), can be identified with the pair-production rate in the case of a constant uniform external electric field; see Ref. [18]. The number density T -constant field was derived in Refs. [20,21] and has the form T 00 (t out ) = eET 2 r cr . (4.8) The densities (4.7) and (4.8) are related as follows which means that ∆T 00 cr ≪ T 00 (t out ) . Thus, taking into account that the area occupied by the external field is finite significantly reduces energy densities of created pairs per unit volume for given field strength E and its duration T . In turn, this also affects the degree of the backreaction.
In problems of high-energy physics it is usually assumed that just from the beginning there exists a classical electric field having a given energy density. The system of fermions interacting with this field is supposed to be closed, that is, its total energy is conserved. Such an assumption is consistent only if the backreaction due to the pair creation is relatively small with respect to the background. For the same reasons QED with either strong T -constant or L-constant fields can be considered as a consistent model only if the backreaction is small.
Below we derive conditions which provide such a smallness for QED with x-step. Similar to the case of QED with t-step (see Ref. [22]), we call them the consistency conditions. The consistency conditions follow from the supposition that the energy density ∆T 00 cr , arising precisely due to the action of a L-constant electric field, should be essentially smaller than the energy density of the external electric field. As such a density in d = 4 dimensions we take the classical energy density E (0) of the electric field, E (0) = E 2 /8π. Thus, the condition of the smallness of back-reaction can written as ∆T 00 cr ≪ E 2 /8π. Taking into account Eq. (4.7), we obtain the sought-for consistency conditions as a restriction from above on the dimensionless parameter eELT : eELT ≪ π 2 Jα exp π m 2 eE . (4.10) Here α is the fine structure constant, α = e 2 /cℏ = e 2 , and J = 2.
Note that using the appropriate number of the spin degrees of freedom inequality (4. (4.11) One can easily extend these results to d dimensions, using expressions for r cr , given by Eq. (4.6).
Since π 2 /Jα ≫ 1, there exists a range of values of parameters E, L and T that satisfies both the inequalities. We recall once again that consistency conditions the case of QED with t-step (specifically for the case of the T -constant field) were obtained in Ref. [20], they have the following form max 1, m 2 /eE 2 ≪ eET 2 ≪ π 2 Jα exp π m 2 eE . (4.12) Taking into account that in the case under consideration the dimensionless parameter satisfies the inequality eELT ≪ eET 2 , one can see that consistency condition (4.10) is much less restrictive from above than the one (4.12).

V. CONCLUDING REMARKS
We consider the present article as a natural and important continuation of the consistent formulation of QED with x-steps elaborated in main in Ref. [16]. Using the developed approach we succeeded to calculate there global effects (global quantities) of zero order with respect to the radiative interaction, such that the vacuum to vacuum transition amplitude, mean differential and total numbers of created particles, and so on. Here we complement the previous constructions, formulating regularization and renormalization procedures for calculating such quasi-local quantities as the vector of electric current and the energymomentum tensor of created particles.
With the help of properly calculated mean values of the latter quantities, we succeeded to solve the backreaction problem for the case under consideration. Considering the case of an uniform electric field confined between two capacitor plates separated by a finite distance L, we see that the smallness of the backreaction implies restriction from above on the dimensionless parameter eELT . We call them consistency conditions by analogy with already known consistency conditions derived by us in Ref. [22] for QED with t-steps.
It should be noted that, recently, it was proposed a new formulation of a locally constant field approximations (LCFAs) in QED with x-steps which does not rely on the Heisenberg-Euler action [23] (similar approximation was formulated in QED with t-steps slowly varying with time [24]). As a part of this formulation, we have constructed universal approximate representations for the total number and current density of created particles in arbitrary weakly inhomogeneous x-steps. We hope that the regularization and renormalization procedures proposed in the present article will allow to formulate a LCFA adequate for calculating the vacuum means of the current density and EMT in QED with arbitrary weakly inhomogeneous x-steps.