Exactly solvable cases in QED with t-electric potential steps
aa r X i v : . [ h e p - t h ] J un Exactly solvable cases in QED with t -electric potential steps T. C. Adorno a , ∗ S. P. Gavrilov a,c , † and D. M. Gitman a,b,d ‡ a Tomsk State University, Russia b P.N. Lebedev Physical Institute, Russia c Herzen State Pedagogical University of Russia,St. Petersburg, Russia d Institute of Physics,University of São Paulo, Brazil (Dated: July 30, 2018)
Abstract
In this paper, we present in detail consistent QED (and scalar QED) calculations of particlecreation effects in external electromagnetic field that correspond to three most important exactlysolvable cases of t -electric potential steps: Sauter-like electric field, T -constant electric field, andexponentially growing and decaying electric fields. In all these cases, we succeeded to obtain newresults, such as calculations in modified configurations of the above mentioned steps and detailedconsiderations of new limiting cases in already studied before steps. As was recently discovered byus, the information derived from considerations of exactly solvable cases allows one to make somegeneral conclusions about quantum effects in fields for which no closed form solutions of the Dirac(or Klein-Gordon) equation are known. In the present article we briefly represent such conclusionsabout an universal behavior of vacuum mean values in slowly varying strong electric fields.PACS numbers: 12.20.Ds,11.15.Tk,11.10.Kk Keywords: Particle creation, Schwinger effect, time-dependent external field, Dirac and Klein-Gordon equa-tions. ∗ [email protected] / [email protected] † [email protected], [email protected] ‡ [email protected] ONTENTS
I. Introduction 2II. Vacuum instability description based on exact solutions 5III. Sauter-like electric field 13IV. T -constant electric field 19V. Peak electric field 25A. General form of peak electric field 25B. Slowly varying field 28C. Short pulse field 34D. Exponentially decaying field 35VI. Universal behavior of the vacuum mean values in slowly varying electric fields 39A. Total density of created pairs 39B. Time evolution of vacuum instability 44C. Vacuum polarization 50VII. Concluding remarks 54A. Some asymptotic expansions 55References 57 I. INTRODUCTION
Quantum field theories (QFTs) with external backgrounds (external fields) are to a cer-tain extent the most appropriate models for calculating quantum effects in strong fields ofelectromagnetic, gravitational, or other nature. These calculations must be nonperturba-tive with respect to the interaction with strong backgrounds. One of the most interestingeffect of such kind that attracts attention already for a long time is the particle creationfrom the vacuum by strong external backgrounds [1–6]. In the framework of the QFT, theparticle creation is closely related to a violation of the vacuum stability with time. Not all2ackgrounds violate the vacuum stability. For example, electromagnetic backgrounds thatviolate the vacuum stability have to be electriclike fields that are able to produce nonzerowork when interacting with charged particles. In such backgrounds any process is accompa-nied by new created particles and, thus, turns out to a many-particle process. That is whyany consistent consideration of quantum processes in the vacuum violating backgrounds hasto be done in the framework of a QFT; a consideration in the framework of the relativisticquantum mechanics is restricted and may lead to paradoxes and even incorrect results. Itshould be noted that at present, methods for the above mentioned nonperturbative calcu-lations are consistently formulated only in QED (and scalar QED) with some specific typesof external backgrounds. Namely, these types are time-dependent external electric fieldsthat are switched on and off at the initial and the final time instants respectively, see Refs.[6–9], we refer to such kind of external fields as the t -electric potential steps, and some time-independent inhomogeneous external electric fields, which are called conditionally, x -electricpotential steps, see Refs. [10]. It should be stressed that the possibility of nonperturbativecalculations in the both above cases is based on the existence of exact solutions of the Dirac(or Klein-Gordon) equation in the corresponding background fields. At present, there areknown only few such cases, which we call exactly solvable cases in QED with t -electric or x -electric potential steps. Note that solutions of the Dirac equation with x -electric potentialsteps are quite similar to solutions obtained for t -electric steps where time t is replaced bythe spatial coordinate x . Some of quantum effects in each of these cases were consideredin the literature using different approaches (relativistic quantum mechanics, QED) with acertain levels of consistency and details.The case of homogeneous and constant electric field was examined by Schwinger [1], whoobtained the probability for a vacuum to remain a vacuum. Such field admit analyticalsolutions of the Dirac and Klein-Gordon equations and was frequently used in various QFTcalculations; see Ref. [11] for a review. Different semiclassical and numerical methods wereapplied to study Schwinger’s effective action, see Refs. [2–5] for a review. Nikishov found themean number of pairs created by the homogeneous and constant electric field and establishedrelation between the Schwinger’s and Feynman’ representations of a causal propagator inthis external field [12, 13]. Subsequently he expanded and deepened his approach to thisproblem in Refs. [14–18].In QED (and scalar QED) with t -electric potential steps, there exist a few exactly solvable3ases that have real physical importance. Those are Sauter-like (or adiabatic or pulse)electric field, T -constant electric field (a uniform electric field which effectively acts during asufficiently large but finite time interval T ), an exponentially decaying electric field, and theircertain combinations. The particle creation effect in the cases of the Sauter-like electric fieldwas studied in Refs. [19–21], in the case of T -constant electric field in Refs. [20, 22–30], andin the case of exponential electric fields in Refs. [31, 32]. One can see that quantum effectswhich can be studied using the exactly solvable cases are important in astrophysics, neutrinophysics, cosmology, condense matter physics, and so on. In particular, the particle creationeffect due to the Sauter-like and T -constant electric fields is crucial for understanding theconductivity of graphene or Weyl semimetals in the nonlinear regime as was reported, e.g.,in Refs. [27, 33–40]. Note that the cases of a constant and exponentially decaying electricfields have many similarities with the case of the de Sitter background, e.g., see Refs. [41–43]and references therein. One can also notice that the case of harmonically alternating electricfield is also exactly solvable [44, 45]. In this case the alternating direction of the electricfield can also contribute to the particle creation effect, but it is not an example of t -electricpotential step. Using exactly solvable cases one can develop new approximation methodsof calculating quantum effects in QFT with unstable vacuum, see reviews [2–6]. However,many already known results are scattered over different publications and many new resultwere not published at all.In this article, we present in detail consistent QED (and scalar QED) calculations ofzero order quantum effects in external electromagnetic field that correspond to three mostimportant exactly solvable cases of t -electric potential steps: Sauter-like electric field, T -constant electric field, and exponentially growing and decaying electric fields. In all thesecases we succeeded to obtain new results, such as calculations in modified configurations ofthe above mentioned t -electric potential steps and a detailed consideration of new limitingcases (asymptotics) in already studied before t -electric potential steps. Considering all thecases, we tried to cite properly all previous relevant works. In Sec. II, we briefly recall basicformulas for treatment of zero-order processes in the framework of QED (and scalar QED)with t -electric potential steps. In Sets. III, IV and V, we study quantum effects in thethree most important exactly solvable cases of t -electric potential steps, in their modifiedconfigurations, and calculate carefully important limiting cases. As was recently discover in Processes that do not involve photons. x -electric potential steps. II. VACUUM INSTABILITY DESCRIPTION BASED ON EXACT SOLUTIONS
Potentials A µ ( x ) , x = ( x µ ) = ( x = t, r ) , r = ( x i ) of external electromagnetic fields corresponding to t -electric potential steps are defined as A = 0 , A ( t ) = ( A = A x ( t ) , A l = 0 , l = 2 , ..., D ) , A x ( t ) t →±∞ −→ A x ( ±∞ ) , (2.1)where A x ( ±∞ ) are some constant quantities, and the time derivative of the potential A x ( t ) does not change its sign for any t ∈ R . For definiteness, we suppose that ˙ A x ( t ) ≤ ⇒ A x ( −∞ ) > A x (+ ∞ ) . (2.2)The magnetic field is always zero and electric fields are homogeneous and have the form E ( t ) = ( E x ( t ) , , ..., , E x ( t ) = − ˙ A x ( t ) = E ( t ) ≥ , E ( t ) | t |→∞ −→ . (2.3)We stress that electric fields under consideration are switched off as | t | → ∞ and do nothave local minima.As an example of a t -electric potential step, we refer to the so-called Sauter-like (oradiabatic or pulse) electric field. This field and its vector potential have the form E ( t ) = E cosh − ( t/T S ) , A x ( t ) = − T S E tanh ( t/T S ) . (2.4)where the parameter T S > sets time scale. The Greek indexes span the Minkowisky space-time, µ = 0 , , . . . , D , and the Latin indexes span theEuclidean space, i = 1 , . . . , D . In what follows, we use the system of units where ℏ = c = 1 . ( d = D + 1) -dimensional Minkowskispace-time and with an external electromagnetic field of the form (2.1) reads i∂ t ψ ( x ) = H ( t ) ψ ( x ) , H ( t ) = γ ( γ P + m ) ,P x = − i∂ x − U ( t ) , P ⊥ = − i ∇ ⊥ , U ( t ) = qA x ( t ) , (2.5)where H ( t ) is the one-particle Dirac Hamiltonian; ψ ( x ) is a [ d/ -component spinor; [ d/ stands for the integer part of d/ ; m = 0 is the electron mass; the index ⊥ stands forcomponents of the momentum operator that are perpendicular to the electric field. Here, γ µ are the γ -matrices in d dimensions [46], [ γ µ , γ ν ] + = 2 η µν , η µν = diag(1 , − , − , . . . | {z } d ) . The number of spin degree of freedom is J ( d ) = 2 [ d/ − .We choose the electron as the main particle with the charge q = − e , where e > is theabsolute value of the electron charge, and we refer to U ( t ) as the potential energy of anelectron in the electric field.Let us consider solutions of Dirac equation (2.5) of the following the form ψ n ( x ) = exp ( i pr ) ψ n ( t ) , n = ( p , σ ) ,ψ n ( t ) = (cid:8) γ i∂ t − γ [ p x − U ( t )] − γ p ⊥ + m (cid:9) φ n ( t ) , (2.6)where ψ n ( t ) and φ n ( t ) are time-dependent spinors. In fact, these are states with a definitemomentum p = ( p x , p ⊥ ) . We can separate spin variables by the substitution φ n ( t ) = ϕ n ( t ) v χ,σ , χ = ± , σ = ( σ , σ , . . . , σ [ d/ − ) , σ s = ± , (2.7)where v χ,σ is a set of constant orthonormalized spinors, satisfying the following equations: γ γ v χ,σ = χv χ,σ , v † χ,σ v χ ′ ,σ ′ = δ χ,χ ′ δ σ,σ ′ . (2.8)In the dimensions d > , one can subject the spinors v χ to some supplementary conditions,which, for example, may be chosen as iγ s γ s +1 v χ,σ = σ s v χ,σ , for even d,iγ s +1 γ s +2 v χ,σ = σ s v χ,σ , for odd d. (2.9)6uantum numbers σ s describe the spin polarization [in the dimensions d = 2 , there areno spin degrees of freedom that are described by the quantum numbers σ ], and, togetherwith the additional index χ , provide a convenient parametrization of the solutions. Thenthe scalar functions ϕ n ( t ) have to obey the second order differential equation (cid:26) d dt + [ p x − U ( t )] + π ⊥ − iχ ˙ U ( t ) (cid:27) ϕ n ( t ) = 0 , π ⊥ = q p ⊥ + m . (2.10)The quantization of the Dirac field in the background under consideration is based on theexistence of solutions to the Dirac equation with special asymptotics as t → ±∞ , see [6, 23]for details. For instance, we let the electric field be switched on at t in and switched off at t out , so that the interaction between the Dirac field and the electric field vanishes at all timeinstants outside the interval t ∈ ( t in , t out ) , and the Dirac equation in the Hamiltonian formis given by [ i∂ t − H ( t )] ζ ψ n ( x ) = 0 , t ∈ ( −∞ , t in ][ i∂ t − H ( t )] ζ ψ n ( x ) = 0 , t ∈ [ t out , + ∞ ) . (2.11)where the additional quantum number ζ = ± labels asymptotic states, respectively. Theseasymptotic states are solutions of eigenvalue problems, H ( t ) ζ ψ n ( x ) = ζ ε n ζ ψ n ( x ) , t ∈ ( −∞ , t in ] , ζ ε n = ζ p ( t in ) ,H ( t ) ζ ψ n ( x ) = ζ ε n ζ ψ n ( x ) , t ∈ [ t out , + ∞ ) , ζ ε n = ζ p ( t out ) ,p ( t ) = q [ p x − U ( t )] + π ⊥ , . (2.12)In these asymptotic states, ζ = + correspond to free electrons and ζ = − corresponds tofree positrons. We call the time interval t ∈ ( −∞ , t in ] as the in-region, where the in-set { ζ ψ n ( x ) } is defined. The time interval t ∈ [ t out , + ∞ ) is called the out-region, where theout-set (cid:8) ζ ψ n ( x ) (cid:9) is defined. In these regions: ζ ϕ n ( t ) = ζ N e − i ζ ε n t , t ∈ ( −∞ , t in ] , ζ ϕ n ( t ) = ζ N e − i ζ ε n t , t ∈ [ t out , + ∞ ) , (2.13)where ζ N , ζ N , are normalization constants, and there exists an energy gap between theelectron and positron states.Then we construct two complete set of solutions to the Dirac equation, ζ ψ n ( x ) = exp ( i pr ) (cid:8) γ i∂ t − γ [ p x − U ( t )] − γ p ⊥ + m (cid:9) ζ ϕ n ( t ) v χ,σ , ζ ψ n ( x ) = exp ( i pr ) (cid:8) γ i∂ t − γ [ p x − U ( t )] − γ p ⊥ + m (cid:9) ζ ϕ n ( t ) v χ,σ . (2.14)7e suppose additionally that the in- { ζ ψ n ( x ) } and out-sets (cid:8) ζ ψ n ( x ) (cid:9) are complete andorthonormal with respect to the standard definition of the inner product [47], ( ψ, ψ ′ ) = Z ψ † ( x ) ψ ′ ( x ) d r , d r = dx ...dx D . (2.15)In calculating (2.15), we use the standard volume regularization, in which the scatteringproblem is confined by a large spatial box of volume V ( d − = L × · · · × L D , with the Diracspinors subject to periodic boundary conditions at the spatial walls. This inner product istime-independent. Since χ is not a physical quantum number if d > (the spin operator γ γ does not commute with the Dirac Hamiltonian (2.5) in case m = 0 ), one can select aparticular χ to calculate (2.15). For simplicity, we select the same χ in the cases of ψ ( x ) and ψ ′ ( x ) , so that the inner product is simplified, ( ψ n , ψ n ′ ) = V ( d − δ n,n ′ I , I = ψ † n ( x ) ψ n ′ ( x ) , I = ϕ ∗ n ( t ) (cid:16) i −→ ∂ t − i ←− ∂ t (cid:17) n i −→ ∂ t − χ [ p x − U ( t )] o ϕ n ( t ) . (2.16)Then solutions (2.14) can be subject to the orthonormality conditions ( ζ ψ n , ζ ′ ψ ′ n ′ ) = δ n,n ′ δ ζ,ζ ′ , (cid:16) ζ ψ n , ζ ′ ψ ′ n ′ (cid:17) = δ n,n ′ δ ζ,ζ ′ , (2.17)which, in particular, leads to the following expressions for the normalization constants: ζ N = ζ CY, ζ N = ζ CY, Y = V − / d − , ζ C = h p ( t in ) q ζ in i − / , ζ C = h p ( t out ) q ζ out i − / ,q ζ in / out = p (cid:0) t in / out (cid:1) − χζ (cid:2) p x − U (cid:0) t in / out (cid:1)(cid:3) . (2.18)Completeness relations for the in- and out-sets are given by X ζ,n ζ ψ n ( t, r ) ζ ψ † n ( t, r ′ ) = δ ( r − r ′ ) I = X ζ,n ζ ψ n ( t, r ) ζ ψ † n ( t, r ′ ) . (2.19)Due to property (2.16) inner products (cid:0) ζ ′ ψ l , ζ ψ n (cid:1) are diagonal in quantum numbers n and l, (cid:0) ζ ′ ψ l , ζ ψ n (cid:1) = δ l,n g (cid:0) ζ ′ | ζ (cid:1) , g (cid:16) ζ ′ | ζ (cid:17) = g (cid:0) ζ ′ | ζ (cid:1) ∗ . (2.20)The corresponding diagonal matrix elements g relate in- and out-solutions { ζ ψ n ( x ) } and (cid:8) ζ ψ n ( x ) (cid:9) for each n , ζ ψ n ( x ) = g (cid:0) + | ζ (cid:1) + ψ n ( x ) + g (cid:0) − | ζ (cid:1) − ψ n ( x ) , ζ ψ n ( x ) = g (cid:0) + | ζ (cid:1) + ψ n ( x ) + g (cid:0) − | ζ (cid:1) − ψ n ( x ) . (2.21)8ubstituting Eqs. (2.21), into the orthonormality conditions, we derive the unitarity re-lations X κ g (cid:0) ζ | κ (cid:1) g (cid:16) κ | ζ ′ (cid:17) = X κ g ( ζ | κ ) g ( κ | ζ ′ ) = δ ζ,ζ ′ . (2.22)Similar consideration is possible for a scalar fields that satisfies the Klein–Gordon (KG)equation with t -electric potential step. A formal transition to the case of scalar fields canbe done by setting χ = 0 in Eq. (2.10). The corresponding complete set of solutions to theKG equation reads φ n ( x ) = exp ( i pr ) ϕ n ( t ) , n = p . (2.23)One can also define complete in- { ζ φ n } and out-sets (cid:8) ζ φ n (cid:9) of solutions orthonormalwith respect to the adequate inner product [47], ( φ, φ ′ ) KG = i Z φ ∗ ( x ) (cid:16) −→ ∂ t − ←− ∂ t (cid:17) φ ′ ( x ) d r . (2.24)Namely, ( ζ φ n , ζ ′ φ ′ n ′ ) KG = ζ δ n,n ′ δ ζ,ζ ′ , (cid:16) ζ φ n , ζ ′ φ ′ n ′ (cid:17) KG = ζ δ n,n ′ δ ζ,ζ ′ , (2.25)with the normalization constants ζ C = [2 p ( t in )] − / , ζ C = [2 p ( t out )] − / . (2.26)Inner products (cid:0) ζ ′ φ l , ζ φ n (cid:1) KG are diagonal in quantum numbers n and l, (cid:0) ζ ′ φ l , ζ φ n (cid:1) KG = δ l,n g (cid:0) ζ ′ | ζ (cid:1) , g (cid:16) ζ ′ | ζ (cid:17) = g (cid:0) ζ ′ | ζ (cid:1) ∗ . (2.27)The corresponding diagonal matrix elements g relate in- and out-solutions, ζ φ n ( x ) = g (cid:0) + | ζ (cid:1) + φ n ( x ) − g (cid:0) − | ζ (cid:1) − φ n ( x ) , ζ φ n ( x ) = g (cid:0) + | ζ (cid:1) + φ n ( x ) − g (cid:0) − | ζ (cid:1) − φ n ( x ) , (2.28)and satisfy the unitarity relations X κ g (cid:0) ζ | κ (cid:1) κ g (cid:16) κ | ζ ′ (cid:17) = X κ g ( ζ | κ ) κ g ( κ | ζ ′ ) = ζ δ ζ,ζ ′ . (2.29)Decomposing the Dirac operator ˆΨ( x ) in the complete sets of in- and out-solutions [6, 23], ˆΨ ( x ) = X n (cid:2) a n (in) + ψ n ( x ) + b † n (in) _ ψ n ( x ) (cid:3) = X n (cid:2) a n (out) + ψ n ( x ) + b † n (out) − ψ n ( x ) (cid:3) , (2.30)9e introduce in- and out-creation and annihilation Fermi operators. Their nonzero anticom-mutation relations are, [ a n (in) , a † m (in)] + = [ a n (out) , a † m (out)] + = [ b n (in) , b † m (in)] + = [ b n (out) , b † m (out)] + = δ nm . (2.31)In these terms, the Heisenberg Hamiltonian is diagonalized at t ≤ t in and t ≥ t out , b H ( t ) = X n (cid:8) + ε n a + n (in) a n (in) + | − ε n | b + n (in) b n (in) (cid:9) , t ≤ t in , b H ( t ) = X n (cid:8) + ε n a + n (out) a n (out) + (cid:12)(cid:12) − ε n (cid:12)(cid:12) b + n (out) b n (out) (cid:9) , t ≥ t out , (2.32)where the diverging c-number parts have been omitted, as usual. The initial | ,in i and final | ,out i vacuum vectors, as well as many-particle in - and out-states, are defined by a n (in) | , in i = b n (in) | , in i = 0 , a n (out) | , out i = b n (out) | , out i = 0 , | in i = b + n (in) ...a + n (in) ... | , in i , | out i = b + n (out) ...a + n (out) ... | , out i . (2.33)Using the charge operator one can see that a † n , a n are the creation and annihilation oper-ators of electrons, whereas b † n , b n are the creation and annihilation operators of positrons,respectively.Transition amplitudes in the Heisenberg representation have the form M in → out = h out | in i . In particular, the vacuum-to-vacuum transition amplitude reads c v = h , out | , in i . Relativeprobability amplitudes of particle scattering, pair creation and annihilation are: w (+ | +) n ′ n = c − v h , out (cid:12)(cid:12) a n ′ (out) a † n (in) (cid:12)(cid:12) , in i = δ n,n ′ w n (+ | +) ,w ( −|− ) n ′ n = c − v h , out (cid:12)(cid:12) b n ′ (out) b † n (in) (cid:12)(cid:12) , in i = δ n,n ′ w n ( −|− ) ,w (+ − | n ′ n = c − v h , out | a n ′ (out) b n (out) | , in i = δ n,n ′ w n (+ − | ,w (0 | − +) nn ′ = c − v h , out (cid:12)(cid:12)(cid:12) b † n (in) a † n ′ (in) (cid:12)(cid:12)(cid:12) , in i = δ n,n ′ w n (0 | − +) . (2.34)The in - and out-operators are related by linear canonical transformations, a n (out) = g ( + | + ) a n (in) + g ( + | − ) b † n (in) , b † n (out) = g ( − | + ) a n (in) + g ( − | − ) b † n (in) . These relations allows one to calculate the differential mean numbers of electrons N an (out) and positrons N bn (out) created from the vacuum state as N an (out) = (cid:10) , in (cid:12)(cid:12) a † n (out) a n (out) (cid:12)(cid:12) , in (cid:11) = (cid:12)(cid:12) g (cid:0) − (cid:12)(cid:12) + (cid:1)(cid:12)(cid:12) ,N bn (out) = (cid:10) , in (cid:12)(cid:12) b † n (out) b n (out) (cid:12)(cid:12) , in (cid:11) = (cid:12)(cid:12) g (cid:0) + (cid:12)(cid:12) − (cid:1)(cid:12)(cid:12) , N cr n = N bn (out) = N an (out) . (2.35)10y N cr n we denote the differential numbers of created pairs. The total number of pairscreated from vacuum is given by the sum N = X n N cr n = X n (cid:12)(cid:12) g (cid:0) − (cid:12)(cid:12) + (cid:1)(cid:12)(cid:12) . (2.36)Similar consideration is possible for a quantum scalar fields ˆΦ ( x ) that satisfies the KGequation with t -electric potential steps. Decomposing the quantum fields ˆΦ ( x ) in the com-plete set of exact solutions { ± φ n ( x ) } and { ± φ n ( x ) } one introduces in- and out-creationand annihilation Bose operators and obtain quite similar representations of relative probabil-ity amplitudes and the differential numbers of created pairs via the corresponding diagonalmatrix elements g defined by Eq. (2.27) [6, 23].Both for fermions and bosons, relative probabilities (2.34), the vacuum-to-vacuum tran-sition amplitude c v , the probability for a vacuum to remain a vacuum P v as well as the totalnumber N of pairs created from vacuum can be expressed via the distribution N cr n , | w n (+ − | | = N cr n (1 − κN cr n ) − , | w n ( −|− ) | = (1 − κN cr n ) − ,P v = | c v | = Y n (1 − κN cr n ) κ , κ = +1 for fermions − . (2.37)The vacuum mean electric current, energy, and momentum are defined as integrals overthe spatial volume. Due to the translational invariance in the uniform external field, allthese mean values are proportional to the space volume. Therefore, it is enough to calculatethe vacuum mean values of the current density vector h j µ ( t ) i and of the energy-momentumtensor (EMT) h T µν ( t ) i , defined as h j µ ( t ) i = h , in | j µ | , in i , h T µν ( t ) i = h , in | T µν | , in i . (2.38)Here we stress the time dependence of mean values (2.38), which does exist due to a timedependence of the external field. We recall for further convenience the form of the operatorsof the current density and the EMT of the quantum Dirac field, j µ = q h ˆΨ( x ) , γ µ ˆΨ ( x ) i , T µν = 12 ( T canµν + T canνµ ) ,T canµν = 14 n [ ˆΨ( x ) , γ µ P ν ˆΨ ( x )] + [ P ∗ ν ˆΨ( x ) , γ µ ˆΨ ( x )] o ,P µ = i∂ µ − qA µ ( x ) , ˆΨ( x ) = ˆΨ † ( x ) γ . (2.39)11ote that the mean values (2.38) depend on the definition of the initial vacuum, | , in i and on the evolution of the electric field from the time t in of switching it on up to thecurrent time instant t , but they do not depend on the further history of the system. Therenormalized vacuum mean values h j µ ( t ) i and h T µν ( t ) i , t in < t < t out are sources in equationsof motion for mean electromagnetic and metric fields, respectively. In particular, completedescription of the back reaction is related to the calculation of these mean values for any t .Mean values and probability amplitudes are calculated by the help of different kind ofpropagators. The probability amplitudes are calculated using Feynman diagrams with thecausal (Feynman) propagator S c ( x, x ′ ) = i h , out | ˆ T ˆΨ ( x ) ˆΨ † ( x ′ ) γ | , in i c − v , (2.40)where ˆ T denotes the chronological ordering operation. A perturbation theory (with respectto radiative processes) uses the so-called in-in propagator S c in ( x, x ′ ) and S p ( x, x ′ ) propagator, S c in ( x, x ′ ) = i h , in | ˆ T ˆΨ ( x ) ˆΨ † ( x ′ ) γ | , in i , S p ( x, x ′ ) = S c in ( x, x ′ ) − S c ( x, x ′ ) . (2.41)All the above propagators can be expressed via the in- and out-solution as follows: S c ( x, x ′ ) = i P n + ψ n ( x ) ω n (+ | +) + ¯ ψ n ( x ′ ) , t > t ′ − P n − ψ n ( x ) ω n ( −|− ) − ¯ ψ n ( x ′ ) , t < t ′ , (2.42) S c in ( x, x ′ ) = i P n + ψ n ( x ) + ¯ ψ n ( x ′ ) , t > t ′ − P n − ψ n ( x ) − ¯ ψ n ( x ′ ) , t < t ′ , S p ( x, x ′ ) = − i X n − ψ n ( x ) w n (0 | − +) + ¯ ψ n ( x ′ ) . (2.43)The mean values of the operator (2.39) are expressed via the latter propagators as h j µ ( t ) i = Re h j µ ( t ) i c + Re h j µ ( t ) i p , h j µ ( t ) i c,p = iq tr [ γ µ S c,p ( x, x ′ )] | x = x ′ , h T µν ( t ) i = Re h T µν ( t ) i c + Re h T µν ( t ) i p , h T µν ( t ) i c,p = i tr [ A µν S c,p ( x, x ′ )] | x = x ′ ,A µν = 1 / (cid:2) γ µ ( P ν + P ′∗ ν ) + γ ν (cid:0) P µ + P ′∗ µ (cid:1)(cid:3) . (2.44)Here tr stands for the trace in the γ -matrices indices and the limit x → x ′ is understood asfollows: tr[ R ( x, x ′ )] x = x ′ = 12 (cid:20) lim t → t ′ − tr[ R ( x, x ′ )] + lim t → t ′ +0 tr [ R ( x, x ′ )] (cid:21) x = x ′ , where R ( x, x ′ ) is any two point matrix function.The function S p ( x, y ) vanishes in the case of a stable vacuum. In this case and only inthis case h j µ ( t ) i = Re h j µ ( t ) i c , h T µν ( t ) i = Re h T µν ( t ) i c . II. SAUTER-LIKE ELECTRIC FIELD
Here we consider quantum effects in a t -electric potential step which is formed by theso-called Sauter-like electric field given by Eq. (2.4). The origin of the name of the field isthe following: In his pioneer work [48] Sauter studied the Klein paradox considering the caseof an inhomogeneous field given by the x -electric potential step − LE tanh ( x/L ) , which iscalled at present the Sauter potential. The homogeneous t -electric step which we are goingto considered here has similar form in t coordinate. In a sense, solutions of the Dirac andKG equations with the Sauter-like potential are formally similar to solutions for the Sauterpotential.For the Sauter-like external field, the scalar functions ϕ n ( t ) (see the previous section)satisfy equation (2.10) with U ( t ) = T S eE tanh ( t/T S ) . This field switches-on and -off adia-batically at t in → −∞ and t out → + ∞ . In in- and out-regions, Dirac spinors ζ ψ n ( x ) and ζ ψ n ( x ) are solutions of the eigenvalue problem (2.12) and the plane-wave frequencies are ω ± = p ( ±∞ ) = q ( p x ∓ T S eE ) + π ⊥ . (3.1)In the case under consideration, Eq. (2.10) is an equation for a hypergeometric function[49]. Its solutions can be written as ϕ n ( t ) = y l (1 − y ) m f ( y ) , y = 12 [1 + tanh ( t/T S )] , where l and m are some constants, and the function f ( y ) is a solution of the Gauss hy-pergeometric differential equation [49]. We will use complete sets of solutions ζ ϕ n ( t ) and ζ ϕ n ( t ) , ζ ϕ n ( t ) = ζ N exp ( − iζ ω − t ) (cid:2) e t/T S (cid:3) iT S2 ( ζω − − ω + ) ζ u ( t ) , + u ( t ) = F ( a, b ; c ; y ) , − u ( t ) = F ( a + 1 − c, b + 1 − c ; 2 − c ; y ) ; ζ ϕ n ( t ) = ζ N exp ( − iζ ω + t ) (cid:2) e − t/T S (cid:3) iT S2 ( ω − − ζω + ) ζ u ( t ) , + u ( t ) = F ( a, b ; a + b + 1 − c ; 1 − y ) , − u ( t ) = F ( c − a, c − b ; c + 1 − a − b ; 1 − y ) ,a = iT S ω + − ω − ) + 12 + (cid:18) − (cid:0) eET (cid:1) − iχeET (cid:19) / , c = 1 − iT S ω − ,b = iT S ω + − ω − ) + 12 − (cid:18) − (cid:0) eET (cid:1) − iχeET (cid:19) / , (3.2)13here F ( a, b ; c ; y ) is the hypergeometric series in the variable y with the normalization F ( a, b ; c ; 0) = 1 [49]. As was already mentioned in Sec. II, the quantity χ can be chosen tobe either χ = +1 or χ = − , and ζ N and ζ N are normalization factors given by Eq. (2.18).A formal transition to the Bose case can be done by setting χ = 0 in Eqs. (3.2). In thiscase n = p and ζ N and ζ N are normalization factors given by Eq. (2.26).In Fermi case, using Kummer’s relations and Eqs. (2.20), one can find coefficients g ( + | − ) ∗ to be g (cid:0) + (cid:12)(cid:12) − (cid:1) ∗ = + C Γ ( c ) Γ ( a + b − c ) − C Γ ( a ) Γ ( b ) , (3.3)where + C and − C are constants given by Eq. (2.18) and Γ( a ) is the Euler gamma function.Then, using Eq. (2.35), we obtain the mean number of created pairs, N cr n = sinh (cid:8) πT S (cid:2) eET S + ( ω + − ω − ) (cid:3)(cid:9) sinh (cid:8) πT S (cid:2) eET S − ( ω + − ω − ) (cid:3)(cid:9) sinh ( πT S ω + ) sinh ( πT S ω − ) . (3.4)In 3+1 QED the corresponding formula was found first in [19].In the similar manner, in the Bose case, we obtain coefficients g ( + | − ) ∗ , where + C and − C are given by Eqs. (2.26) and parameters a, b, and c are given by Eq. (3.2) at χ = 0 .Here, the mean number for created pairs is N cr n = cosh (cid:20) π q ( T eE ) − (cid:21) + sinh (cid:2) πT S ( ω + − ω − ) (cid:3) sinh ( πT S ω + ) sinh ( πT S ω − ) . (3.5)It should be noted that mean numbers (3.4) and (3.5) are even functions of all themomentum p . In particular it can be seen that p x → − p x leads to ω + → ω − . The mostimportant parameter in the case under consideration is T S . Its value determines the effectivetime of electric field action.We begin our analysis by considering T S small and constant values for the asymptoticpotentials, U (+ ∞ ) = − U ( −∞ ) = U / eET S . In this case we deal with a very shortpulse field. The corresponding potential imitates sufficiently well a t -electric rectangularpotential step, and coincides with the latter as T S → . Thus, the Sauter-like potential canbe considered as a regularization of the rectangular step. We assume that sufficiently small T S for given ω ± satisfies the inequalities U T S ≪ , max { T S ω + , T S ω − } ≪ . (3.6)14n such a case mean numbers of created pairs are N cr n = U − ( ω + − ω − ) ω + ω − in Fermi case , (3.7) N cr n = T U / ω + − ω − ) ω + ω − in Bose case . (3.8)The number of created fermions in Eq. (3.7) does not depend on T S . However, in contrastwith the Fermi case, the limit T S → in Eq. (3.8) is possible only when the difference ( ω + − ω − ) is not very small, namely, when T U / ≪ ( ω + − ω − ) . (3.9)Only under the latter condition one can neglect an T S -depending term in Eq. (3.8) to obtain N cr n = ( ω + − ω − ) ω + ω − . (3.10)Unlike the Fermi case, where N cr n ≤ , in the Bose case, the mean number of createdparticles is unlimited in two ranges of the longitudinal kinetic momenta, namely when either ω + /ω − → ∞ or ω − /ω + → ∞ , N cr n ≈
14 max { ω + /ω − , ω − /ω + } . (3.11)One can see that in the Fermi and Bose cases N cr n → as π ⊥ → ∞ . On a sufficientlyhigh step and small transversal momentum, π ⊥ / U ≪ , one finds that the maximum meannumber of bosons is only limited by the potential difference U , max N cr n ≈ U π ⊥ . The maximum mean numbers of fermions N cr n → are in the range of small π ⊥ and | p x | ,when longitudinal kinetic momenta are large, ( p x ∓ U / ∼ U / .The Sauter-like potential is suitable for imitating a slowly alternating electric field. Tothis end the parameter T S is taken to be sufficiently large. Let us consider just this case,supposing that T S ≫ max (cid:16) / √ eE, m/eE (cid:17) . (3.12)For both the Fermi and Bose cases, one can check that the mean numbers (3.4) and (3.5)are negligibly small, N cr n ≪ e − πm /eE , (3.13)15or any given p ⊥ and for small kinetic momenta | p x ± eET S | = √ eEK S ≪ eET S , K S ≫ max (cid:16) , m/ √ eE (cid:17) , where K S is any given number.For the range of large longitudinal kinetic momenta, | p x ± eET S | > √ eEK S ⇐⇒ | p x | < eET S − √ eEK S , (3.14)and any given p ⊥ , mean numbers (3.4) and (3.5) have approximately the following form N cr n ≈ N as n = e − πτ , τ = T S ( ω + + ω − − eET S ) . (3.15)The function τ has a minimum at p x = 0 , τ = τ | p x =0 = T S (cid:20) q π ⊥ + ( eET S ) − eET S (cid:21) , (3.16)and is growing monotonically as | p x | and p ⊥ grow. One can see that mean numbers N as n areexponentially small in the range of large transversal momenta , π ⊥ & √ eEK S . Thereforethe following range of π ⊥ is of interest, π ⊥ ≪ √ eEK S . (3.17)In this range, the following approximation holds true τ ≈ eET π ⊥ ( eET S ) − p x , τ ≈ λ = π ⊥ eE . (3.18)The function τ takes its maximum value τ max = τ | | p x | = eET S −√ eEK S ≈ √ eET S λ K S as | p x | tends to its maximum. For m = 0 , we see that τ max → ∞ as √ eET S → ∞ . In thewide range of transversal momenta, π ⊥ ≪ eET S , the mean numbers N as n do not dependpractically on the parameter T S and coincide with differential numbers of created particlesin a constant electric field [12, 13] N as n ≈ N n = e − πλ . (3.19)The total number of pairs created from a vacuum (defined by Eq. (2.36)) by an uni-form electric field , is proportional to the space volume V ( d − as N cr = V ( d − n cr and thecorresponding number density n cr has the form n cr = 1(2 π ) d − X σ Z d p N cr n . (3.20)16n deriving Eq. (3.20) the sum over all momenta p was transformed into an integral. Thenthe integral in the right hand side of Eq. (3.20) can be approximated by an integral over asubrange Ω that gives the dominant contribution with respect to the total increment to thenumber density of created particles, Ω : n cr ≈ ˜ n cr = 1(2 π ) d − X σ Z p ∈ Ω d p N cr n . (3.21)Let us consider the number density of pairs created from the vacuum by the Sauter-likepotential with a large parameter T S . This quantity can be calculated using Eq. (3.21) withdifferential numbers N cr n approximated by Eqs. (3.15) and (3.18). In this case, the leadingterm, ˜ n cr , is formed over the range given by Eqs. (3.14) and (3.17), that is, this range ischosen as a realization of the subrange Ω in Eq. (3.21). In this approximation, the numbers N as n are the same for fermions and bosons and do not depend on the spin polarizationparameters σ s . Thus, in the Fermi case, probabilities and mean numbers summed over all σ s obtain the factor J ( d ) = 2 [ d/ − . We obtain that ˜ n cr = J ( d ) (2 π ) d − Z p ∈ Ω d p N cr n . (3.22)In the case of scalar bosons, J ( d ) = 1 .Taking into account Eqs. (3.15) and (3.18), we approximate integral (3.22) as ˜ n cr ≈ J ( d ) (2 π ) d − Z d p ⊥ I p ⊥ , I p ⊥ = 2 Z eET S −√ eEK S dp x e − πτ . (3.23)It is convenient to introduce a variable t , defined as τ = λt + τ . Taken into accountEq. (3.18) we can find a relation between t and p x and see that dp x = 12 eET S t − / ( t + 1) − / dt. (3.24)Neglecting the contribution from τ > τ max and using the variable t, one can represent thequantity I p ⊥ as follows I p ⊥ ≈ eET S Z ∞ dtt − / ( t + 1) − / e − πλ ( t +1) . (3.25)In particular, using Eq. (3.25), one can find the number density of created pairs with a given p ⊥ for large and small λ in the following form I p ⊥ ≈ eET S √ λ e − πλ if λ ≫ , I p ⊥ ≈ eET S if λ ≪ . (3.26)17inally, substituting Eq. (3.25) into integral (3.23) and performing the integration over p ⊥ ,we obtain ˜ n cr = J ( d ) T S δ (2 π ) d − ( eE ) d/ exp (cid:18) − π m eE (cid:19) , (3.27)where δ = Z ∞ dtt − / ( t + 1) − ( d +1) / exp (cid:18) − tπ m eE (cid:19) = √ π Ψ (cid:18) , − d π m eE (cid:19) . (3.28)Here Ψ ( a, b ; x ) is the confluent hypergeometric function [49]. This result was first obtainedin Ref. [20] . We see that the number density ˜ n cr , given by Eq. (3.27), is proportional to thetotal increment of the longitudinal kinetic momentum, ∆ U S = e | A x (+ ∞ ) − A x ( −∞ ) | =2 eET S .From this result one can find the vacuum-to-vacuum probability P v , defined by Eq. (2.37).Using the identity ln (1 ± x ) = ± x + . . . , and performing an integration following the con-siderations above, one gets the following approximation P v ≈ exp (cid:0) − µ S V ( d − ˜ n cr (cid:1) , µ S = ∞ X l =0 ( − (1 − κ ) l/ ǫ S l +1 ( l + 1) d/ exp (cid:18) − lπ m eE (cid:19) ,ǫ S l = δ − √ π Ψ (cid:18) , − d lπ m eE (cid:19) . (3.29)In 3+1 QED the same result was found in a different way [21].If the Sauter-like field is weak, m /eE ≫ , one can use asymptotic expression for the Ψ -function [49], Ψ (cid:0) / , (2 − d ) / lπm /eE (cid:1) = (cid:0) eE/lπm (cid:1) / + O (cid:16)(cid:2) eE/m (cid:3) / (cid:17) . (3.30)Then δ ≈ √ eE/m, ǫ S l ≈ l − and µ S ≈ . In the case of a very strong field, m /eE ≪ ,one obtains from Ref. [49] that the leading term for the Ψ -function does not depend on theparameter m /eE , Ψ (cid:0) / , (2 − d ) / πm /eE (cid:1) ≈ Γ ( d/ / Γ ( d/ / . (3.31)Then, for example, δ ≈ π/ if d = 3 and δ ≈ / if d = 4 . For the very strong field, lπm /eE ≪ , the leading contribution of ǫ S l has a quite simple form and does not dependon the dimension, ǫ S l ≈ . In this case µ S ≈ ∞ X l =0 ( − (1 − κ ) l/ ( l + 1) d/ . (3.32) Unlike Eq. (3.24) the relation between t and p x in Refs. [20] is given for small t only. This approximationis good enough for λ > . However, the final form of δ = √ π Ψ (cid:16) , − d ; π m eE (cid:17) is given correctly forarbitrary m /eE . V. T -CONSTANT ELECTRIC FIELD In this section we present a detailed study of the particle creation problem from the vac-uum by a T -constant field. This field corresponds to a regularized version of the constantfield E ( t ) = E , in which the electric field remains switched on for all the time t ∈ ( −∞ , + ∞ ) .This regularization was first considered in Ref. [29] and then developed in Ref. [20]. In thepresent section we explore additional peculiarities concerning particle creation, supplement-ing the previous considerations with new details. The T -constant electric field is constantwithin the time interval T and is zero outside of it, E ( t ) = , t ∈ I E , t ∈ II0 , t ∈ III = ⇒ A x ( t ) = − Et in , t ∈ I − Et , t ∈ II − Et out , t ∈ III , (4.1)where I denotes the in-region t ∈ ( −∞ , t in ] , II is the intermediate region where the electricfield is non zero t ∈ ( t in , t out ) and III is the out-region t ∈ [ t out , + ∞ ) and t out , t in areconstants, t out − t in = T . We choose t out = − t in = T / . It the in-region I and in out-regionIII, Dirac spinors are solutions of the eigenvalue problem (2.12).For t ∈ II , U ( t ) = eEt , equation (2.10) can be written in the form (cid:20) d d ξ + ξ − i χ + λ (cid:21) ϕ n ( t ) = 0 , (4.2)where ξ = eEt − p x √ eE , λ = π ⊥ eE . (4.3)The general solution of Eq. (4.2) is completely determined by an appropriate pair of thelinearly independent Weber parabolic cylinder functions (WPCFs) [49]: either D ρ [(1 − i) ξ ] and D − − ρ [(1 + i) ξ ] , or D ρ [ − (1 − i) ξ ] and D − − ρ [ − (1 + i) ξ ] , where ρ = i λ/ − (1 − χ ) / .Then taking into account Eq. (2.21), the functions − ϕ n ( t ) and + ϕ n ( t ) can be presented in19he form − ϕ n ( t ) = Y − C exp [ ip ( t in ) ( t − t in )] , t ∈ I − C { a D ρ [ − (1 − i ) ξ ] + a D − − ρ [ − (1 + i ) ξ ] } , t ∈ II g ( + | − ) + C exp [ − ip ( t out ) ( t − t out )] + κg ( − | − ) − C exp [ ip ( t out ) ( t − t out )] , t ∈ III ; + ϕ n ( t ) = Y g ( + | + ) + C exp [ − ip ( t in ) ( t − t in )] + κg ( − | + ) − C exp [ ip ( t in ) ( t − t in )] , t ∈ I + C { a ′ D ρ [(1 − i ) ξ ] + a ′ D − − ρ [(1 + i ) ξ ] } , t ∈ II + C exp [ − ip ( t out ) ( t − t out )] , t ∈ III (4.4)on the whole axis t . Here κ = 1 and the normalization constants are given by Eqs. (2.18).The functions − ϕ n ( t ) and + ϕ n ( t ) and their derivatives satisfy the following gluing condi-tions: + − ϕ n ( t in , out −
0) = + − ϕ n ( t in , out + 0) , ∂ t + − ϕ n ( t ) (cid:12)(cid:12) t = t in , out − = ∂ t + − ϕ n ( t ) (cid:12)(cid:12) t = t in , out +0 . (4.5)Using Eq. (4.5) and the Wronskian determinant of WPCFs [49], D ρ ( z ) ddz D − ρ − ( iz ) − D − ρ − ( iz ) ddz D ρ ( z ) = exp (cid:20) − iπ ρ + 1) (cid:21) , we find the coefficients a j and a ′ j ,a j = ( − j √ (cid:20) iπ (cid:18) ρ + 12 (cid:19)(cid:21) q ξ + λf (+) j ( ξ ) ,a ′ j = ( − j √ (cid:20) iπ (cid:18) ρ + 12 (cid:19)(cid:21) q ξ + λf ( − ) j ( ξ ) , j = 1 , , (4.6)where ξ , = ξ | t = t in , out = ∓ eET / − p x √ eE ; (4.7) f ( ± )1 ( ξ ) = ± i p ξ + λ ddξ ! D − ρ − [ ∓ (1 + i ) ξ ] ,f ( ± )2 ( ξ ) = ± i p ξ + λ ddξ ! D ρ [ ∓ (1 − i ) ξ ] . (4.8)Note that the following relations hold: p ( t in ) / √ eE = p ξ + λ and p ( t out ) / √ eE = p ξ + λ . Using Eqs. (4.6) one can determine the coefficients g ( ± | + ) and g ( ± | − ) . It should20e noted that we need to know explicitly only the coefficients g ( − | + ) and g ( + | − ) , which are g (cid:0) + | − (cid:1) = AB exp [( ρ + 1 / iπ/ , g (cid:0) − | + (cid:1) = A ′ B ′ exp [( ρ + 1 / iπ/ ,A = p ξ + λ p ξ + λ (cid:16)p ξ + λ + χξ (cid:17) (cid:16)p ξ + λ − χξ (cid:17) / , B = f (+)2 ( ξ ) f (+)1 ( ξ ) − f (+)1 ( ξ ) f (+)2 ( ξ ) ,A ′ = p ξ + λ p ξ + λ (cid:16)p ξ + λ − χξ (cid:17) (cid:16)p ξ + λ + χξ (cid:17) / , B ′ = f ( − )1 ( ξ ) f ( − )2 ( ξ ) − f ( − )2 ( ξ ) f ( − )1 ( ξ ) . (4.9)One can see that ξ | p x →− p x = − ξ then coefficients (4.9) obey the relations g (cid:0) + | − (cid:1)(cid:12)(cid:12) p x →− p x = g (cid:0) − | + (cid:1) . (4.10)From these relations, we see that | g ( − | + ) | is an even function of momenta p and does notdepend on a spin polarization.Taking into account Eq. (2.28), a formal transition to the Klein-Gordon case can be doneby setting χ = 0 and κ = − in Eqs. (4.4). In this case n = p and the normalization factorsare given by Eq. (2.26). In the Klein-Gordon case, the coefficients g are g (cid:0) + | − (cid:1) = exp ( − λπ/ A sc B | χ =0 , g (cid:0) − | + (cid:1) = − exp ( − λπ/ A sc B ′ | χ =0 ,A sc = (cid:18) q ξ + λ q ξ + λ (cid:19) / , (4.11)where B and B ′ are given by Eqs. (4.9). We stress that this results are new.The differential mean numbers of created pairs have the form N cr n = | g ( − | + ) | , seeEq. (2.35), where g ( − | + ) is given by Eqs. (4.9) for Dirac particles and by Eqs. (4.11) forKlein-Gordon particles. They depend only on the values ξ , for a given λ . The T -constantfield is a regularization for a constant uniform electric field and it is suitable for imitating aslowly varying field. That is why the T -constant field with a sufficiently large time interval T , √ eET ≫ max (cid:0) , m /eE (cid:1) , (4.12)is of interest. In what follows, we suppose that these conditions hold true and additionallyassume that √ λ < K ⊥ , (4.13)where K ⊥ is any given number satisfying the condition √ eET / ≫ K ⊥ ≫ max { , m /eE } . N cr n depend on the parameters ξ , and λ . Let forfermions χ = 1 and ρ = i λ/ ν . Since N cr n are even functions of p x , we can consider onlythe case of p x ≤ . In this case ξ ≥ √ eET / is large, ξ ≫ max { , λ } , and the asymptoticexpansions of WPCFs with respect to ξ are valid. As to the parameter ξ , the whole interval −√ eEL/ ≤ ξ ≤ + ∞ can be divided in three ranges: ( a ) − √ eET / ≤ ξ ≤ − K, ( b ) − K < ξ < K, ( c ) ξ ≥ K, (4.14)where K is any given number satisfying the condition √ eET / ≫ K ≫ K ⊥ . Using theasymptotic expansions of WPCFs with respect to ξ , [49], we get the following expansionsof the coefficients f ( − )1 , ( ξ ) , given by Eqs. (4.8), f ( − )1 ( ξ ) = e − i ξ / (cid:16) √ e i π/ ξ (cid:17) − ν − (cid:20) iξ + O (cid:0) ξ − (cid:1)(cid:21) ,f ( − )2 ( ξ ) = e i ξ / (cid:16) √ e − i π/ ξ (cid:17) ν (cid:20) i ν (1 − ν )4 ξ + O (cid:0) ξ − (cid:1)(cid:21) if ξ ≥ K ; f ( − )1 ( ξ ) = − e − i ξ / (cid:16) √ e i π/ | ξ | (cid:17) − ν − e iπν (cid:20) i (cid:18) ν + ν (cid:19) ξ − + O (cid:0) ξ − (cid:1)(cid:21) + e i ξ / (cid:16) √ e − i π/ | ξ | (cid:17) ν e iπν/ √ π
2Γ ( ν ) ξ ,f ( − )2 ( ξ ) = ie − i ξ / (cid:16) √ e i π/ | ξ | (cid:17) − ν − e iπν/ √ π Γ ( − ν ) (cid:2) O (cid:0) ξ − (cid:1)(cid:3) if ξ < , | ξ | ≥ K. (4.15)One can use Eq. (4.15) with respect to ξ and ξ for the cases (a) and (c). In the case(c), we find that the quantity N cr n is very small, N cr n ∼ max (cid:8) | ξ | − , | ξ | − (cid:9) if min {| ξ | , | ξ |} ≥ K. (4.16)In the case (a), we obtain N cr n = e − πλ " (cid:0) − e − πλ (cid:1) / √ λ (cid:18) sin φ | ξ | + sin φ | ξ | (cid:19) + O (cid:0) | ξ | − (cid:1) + O (cid:0) | ξ | − (cid:1) ,φ , = ( ξ , ) + λ ln (cid:16) √ | ξ , | (cid:17) − arg Γ ( iλ/ − π/ . (4.17)Consequently, the quantity (4.17) is almost constant over the wide range of longitudinalmomentum p x for any given λ satisfying Eq. (4.13). Note that the next-to-leading oscillatingterms in Eq. (4.17) presented here improve an approximation obtained before in Ref. [20].When √ eET → ∞ , one obtains the result in a constant uniform electric field, given byEq. (3.19), setting | ξ , | → ∞ in Eq. (4.17).22n the intermediate range (b), using the only asymptotics with respect to ξ given byEq. (4.15) and the exact form of f ( − )1 ( ξ ) given by Eq. (4.8), we find that N cr n = 14 e − πλ/ q ξ + λ (cid:18)q ξ + λ − ξ (cid:19) (cid:12)(cid:12)(cid:12) f ( − )1 ( ξ ) (cid:12)(cid:12)(cid:12) . (4.18)One can make some conclusions about the contribution of this region to the integral overthe longitudinal momentum in Eq. (3.20). Taking into account that N cr n is always less thanunity for fermions, one can get a rough estimation of the integral Z | ξ |
In this section we present a complete discussion concerning particle creation from thevacuum by a third set of exactly solvable t -electric potential steps, namely, the peak electricfield and the exponentially decaying electric field, both considered previously by us in Refs.[31, 32]. Here we supplement the former studies with new details and discussions particularto these fields. A. General form of peak electric field
The peak electric field E ( t ) is composed of two parts, one of them is increasing expo-nentially on the time-interval I = ( −∞ , , and reaches its maximal magnitude E > atthe end of the interval t = 0 , the second part decreases exponentially on the time-interval II = (0 , + ∞ ) having at t = 0 the same magnitude E. The vector potential A x ( t ) and thefield E x ( t ) are A x ( t ) = E k − (cid:0) − e k t + 1 (cid:1) , t ∈ I k − (cid:0) e − k t − (cid:1) , t ∈ II , E ( t ) = E e k t , t ∈ I e − k t , t ∈ II , (5.1)and A y = A z = E y = E z = 0 . Here k and k are positive constants. The field E ( t ) iscontinuous at t = 0 , but its time-derivative is not in the general case, lim t →− E ( t ) = lim t → +0 E ( t ) = E, ∀ t : E ( t ) ≤ E, lim t →− ˙ E ( t ) = k E = lim t → +0 ˙ E ( t ) = − k E . (5.2)Note that the so-called exponentially decreasing electric field with the potential A ed x ( t ) = E , t ∈ I k − (cid:0) e − k t − (cid:1) , t ∈ II , (5.3)25an be considered as a particular case of the peak field, when the latter switches on abruptlyat t = 0 , i.e., when k is sufficiently large, k → ∞ . Similarly can be treated the exponentiallyincreasing electric field.Exact solutions of the Dirac equation with the exponentially decreasing and the peakelectric fields have been obtained by us previously in Refs. [31, 32]. Following the same way,we introduce new variables η j , η = ih e k t , η = ih e − k t , h j = 2 eEk − j , j = 1 , , (5.4)in place of t and represent the scalar functions ϕ n ( t ) as ϕ jn ( t ) = e − η j / η ν j j ˜ ϕ j ( η j ) , ν j = iω j k j , ω j = q π j + π ⊥ , π j = p x − ( − j eEk j , (5.5)where the subscript j distinguishes quantities associated to the time-intervals I and II . Thefunctions ˜ ϕ j ( η j ) satisfy confluent hypergeometric equations [49], (cid:20) η j d dη j + ( c j − η j ) ddη j − a j (cid:21) ˜ ϕ j ( η j ) = 0 ,c j = 1 + 2 ν j , a j = 12 (1 + χ ) + ( − j iπ j k j + ν j . (5.6)In accordance with Eq. (2.8), the quantity χ can be chosen to be either χ = +1 or χ = − .A fundamental set of solutions for the latter equation consists of two linearly independentconfluent hypergeometric functions Φ ( a j , c j ; η j ) and η − c j j e η j Φ (1 − a j , − c j ; − η j ) , where Φ ( a, c ; η ) = 1 + ac η
1! + a ( a + 1) c ( c + 1) η
2! + . . . . (5.7)Thus, the general solution of Eq. (2.10) in the intervals I and II can be written as thefollowing linear superposition: ϕ jn ( t ) = b j y j ( η j ) + b j y j ( η j ) ,y j ( η j ) = e − η j / η ν j j Φ ( a j , c j ; η j ) ,y j ( η j ) = e η j / η − ν j j Φ (1 − a j , − c j ; − η j ) , (5.8)with arbitrary constants b j and b j . The Wronskian of the functions y is y j ( η j ) ddη j y j ( η j ) − y j ( η j ) ddη j y j ( η j ) = 1 − c j η j . (5.9)As can be seen from Eq. (5.1), the peak electric field is switched on at the infinitelyremote past t → −∞ and switched off at the infinitely remote future t → + ∞ . At these26egions, the exact solutions represent free particles and the appropriate superpositions fromEq. (5.8) obey the asymptotic conditions (2.13), where p ( −∞ ) = ω denotes energy ofinitial particles at t → −∞ , p (+ ∞ ) = ω denotes energy of final particles at t → + ∞ and ζ N and ζ N are given by Eq. (2.18).Using the initial conditions (2.13), we fix the constants b j and b j , and then we find thein- and out-electron and positron states in the intervals I and II : + ϕ n ( t ) = + N exp ( iπν / y ( η ) , − ϕ n ( t ) = − N exp ( − iπν / y ( η ) , t ∈ I ; + ϕ n ( t ) = + N exp ( − iπν / y ( η ) , − ϕ n ( t ) = − N exp ( iπν / y ( η ) , t ∈ II . (5.10)Taking into account the structure of exact solutions given by Eqs. (5.8) and (5.10), werepresent the functions − ϕ n ( t ) and + ϕ n ( t ) in the form + ϕ n ( t ) = g ( + | + ) + ϕ n ( t ) + κg ( − | + ) − ϕ n ( t ) , t ∈ I + N exp ( − iπν / y ( η ) , t ∈ II , (5.11) − ϕ n ( t ) = − N exp ( − iπν / y ( η ) , t ∈ I g ( + | − ) + ϕ n ( t ) + κg ( − | − ) − ϕ n ( t ) , t ∈ II , (5.12)already for any t . Here coefficients g are defined by Eq. (2.20). The constant κ is defined byEq. (2.37). Its introduction allows us to describe by one equation the case of scalar particlesas well, which is discussed in detail below. The functions − ϕ n ( t ) and + ϕ n ( t ) and theirderivatives satisfy the following continuity conditions: + − ϕ n ( t ) (cid:12)(cid:12) t = − = + − ϕ n ( t ) (cid:12)(cid:12) t =+0 , ∂ t + − ϕ n ( t ) (cid:12)(cid:12) t = − = ∂ t + − ϕ n ( t ) (cid:12)(cid:12) t =+0 . (5.13)Using Eqs. (5.13) and (5.9), one can find coefficients g (cid:0) ζ | ζ ′ (cid:1) and g (cid:0) ζ | ζ ′ (cid:1) from Eqs. (5.11)and (5.12), g (cid:0) − | + (cid:1) = C ∆ , C = − s q − ω q +2 ω exp (cid:20) iπ ν − ν ) (cid:21) , ∆ = (cid:20) k h y ( η ) ddη y ( η ) + k h y ( η ) ddη y ( η ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) t =0 ; (5.14) g (cid:0) + | − (cid:1) = C ′ ∆ ′ , C ′ = − s q +2 ω q − ω exp (cid:20) iπ ν − ν ) (cid:21) , ∆ ′ = (cid:26) k h y ( η ) ddη y ( η ) + k h y ( η ) ddη y ( η ) (cid:27) t =0 , (5.15)27espectively [32]. Comparing Eqs. (5.14) and (5.15) one can verify that the symmetry undera simultaneous change k ⇆ k and π ⇆ − π holds true, g (cid:0) + | − (cid:1) ⇆ g (cid:0) − | + (cid:1) . (5.16)A transition to solutions of the Klein-Gordon equation can be performed by setting χ = 0 and κ = − in Eqs. (5.11) and (5.12), and by using the normalization constants (2.26).Thus, in the case of scalar particles, the coefficient g ( − | + ) reads g (cid:0) − | + (cid:1) = C sc ∆ | χ =0 , C sc = (4 ω ω ) − / exp [ iπ ( ν − ν ) / , (5.17)where ∆ is given by Eq. (5.14). In this case, we have the antisymmetry g (cid:0) + | − (cid:1) ⇆ − g (cid:0) − | + (cid:1) . (5.18)under the simultaneous change k ⇆ k and π ⇆ − π .Using g ( − | + ) given by Eq. (5.14), we find that in the Fermi case, differential mean numberof created particles is N cr n = |C ∆ | . (5.19)In the Bose case, using g ( − | + ) given by Eq. (5.17), we find N cr n = (cid:12)(cid:12)(cid:12) C sc ∆ | χ =0 (cid:12)(cid:12)(cid:12) . (5.20)It is clear that mean numbers N cr n depend on modulus squared of the transversal momentum, p ⊥ . It follows from Eqs. (5.16) and (5.18) that the numbers N cr n are invariant under thesimultaneous change k ⇆ k and π ⇆ − π for fermions and bosons, respectively. Then if k = k , the numbers N cr n appear to be even functions of the longitudinal momentum p x . B. Slowly varying field
The inverse parameters k − , k − represent scales of time duration of the electric fieldin the increasing and decreasing time intervals I and II . In particular, slowly varying fieldscorrespond to small values of k and k , satisfying the conditions min ( h , h ) ≫ max (cid:0) , m /eE (cid:1) , (5.21)28here h and h are defined by Eq. (5.4). In this case, we have a two-parameter reg-ularization for a constant electric field (additional to the above presented one-parameterregularizations by the Sauter-like electric field and the T -constant electric field).Let us analyze how the differential numbers N cr n depend on the quantities p x and π ⊥ .A semiclassical consideration show that N cr n are exponentially small for very large π ⊥ & min (cid:0) eEk − , eEk − (cid:1) . Then the range of fixed π ⊥ is of interest and in the following weassume that condition (4.13) holds true, where in the case under consideration any givennumber K ⊥ satisfies the inequality min ( h , h ) ≫ K ⊥ ≫ max (cid:0) , m /eE (cid:1) . (5.22)By virtue of the symmetry properties of the numbers N cr n discussed above, one can onlyconsider either positive or negative p x . Let us, for example, consider the interval −∞
K , where K is any given large number, K ≫ K ⊥ , is quite simple. In this case,using the appropriate asymptotic expressions of the confluent hypergeometric function onecan see that N cr n are negligibly small. To see this, Eq. (A9) (see Appendix A) is useful inthe range h & − π /k > K , while an expression for large c with a fixed a and h andan expression for large c with fixed a − c and h , given in [49], are useful in the range − π /k ≫ h .We expect a significant contribution for the numbers N cr n in the range h ≥ π /k > − K . (5.23)This range can be divided in four subranges (a) h ≥ π /k > h (cid:20) − (cid:16)p h g (cid:17) − (cid:21) , (b) h (cid:20) − (cid:16)p h g (cid:17) − (cid:21) > π /k > h (1 − ε ) , (c) h (1 − ε ) > π /k > h /g , (d) h /g > π /k > − K , (5.24)where g , g , and ε are any given numbers satisfying the conditions g ≫ , g ≫ , (cid:16)p h g (cid:17) − ≪ ε ≪ .
29e note that τ = − ih / (2 − c ) ≈ h k | π | in the subranges (a), (b), and (c) and τ = ih /c ≈ h k | π | in the whole range (5.23). In the subranges, τ satisfies the inequalities :(a) 1 ≤ τ − < (cid:20) (cid:16)p h g (cid:17) − (cid:21) , (b) (cid:20) (cid:16)p h g (cid:17) − (cid:21) < τ − < (1 + εk /k ) , (c) (1 + εk /k ) < τ − < [1 + k /k (1 − /g )] , (d) [1 + k /k (1 − /g )] < τ − . (1 + k /k ) . (5.25)We see that τ − → and τ − → in the range (a), while | τ − | ∼ in the range (c),and | τ − | ∼ in the ranges (c) and (d). In the range (b) these quantities vary from theirvalues in the ranges (a) and (c).We choose χ = 1 for convenience in the Fermi case. In the range (a) we can use theasymptotic expression for the confluent hypergeometric function given by Eq. (A1) in Ap-pendix A. Using Eqs. (A6) and (A7) (see Appendix A), we can find that the differentialmeans numbers for fermions and bosons in the leading approximation have the same form N cr n = e − πλ [1 + O ( |Z | )] , max |Z | . g − . (5.26)In the range (c), the confluent hypergeometric function Φ ( a , c ; ih ) is approximated byEq. (A8) and the function Φ (1 − a , − c ; − ih ) is approximated by Eq. (A9) given in theAppendix A. Then we find that N cr n = e − πλ (cid:2) O ( |Z | ) − + O ( |Z | ) − (cid:3) , (5.27) max |Z | − . p g /h , max |Z | − . g − . Using asymptotic expression (A1) and taking into account Eqs. (5.26) and (5.27), we ob-tain that in the range (b) the following estimate holds N cr n ∼ e − πλ . In the range (d), theconfluent hypergeometric function Φ ( a , c ; ih ) is approximated by Eq. (A8) and the func-tion Φ (1 − a , − c ; − ih ) is approximated by Eq. (A10) given in Appendix A. Then, in30his range, we obtain the following leading-order approximation for the differential meannumbers N cr n ≈ exp h − πk ( ω − π ) i sinh (2 πω /k ) × sinh [ π ( ω + π ) /k ] in Fermi casecosh [ π ( ω + π ) /k ] in Bose case . (5.28)It is clear that when π ≫ π ⊥ the expressions given by Eqs. (5.28) take the form (5.27), N cr n → e − πλ . Consequently, the result (5.28) is valid in the whole range (5.23). Assuming m/k ≫ , we see that N cr n given by Eqs. (5.28) are negligible in the range π . π ⊥ . Thenone can see that substantial value of N cr n are formed in the range π ⊥ < π eE/k and aregiven the same formula N cr n ≈ exp (cid:20) − πk ( ω − π ) (cid:21) . (5.29)both for bosons and fermions.Considering positive p x > , we can take into account that numbers N cr n are invariantunder the simultaneous exchange k ⇆ k and π ⇆ − π . In this case π is positive andlarge, π > eE/k , while π varies from negative to positive values, − eE/k < π < ∞ . Wefind that a substantial contribution to N cr n are formed in the range − h < π /k < K , (5.30)where K is any given large number, K ≫ K ⊥ . In this range, similarly to the case ofnegative p x , we obtain the following leading-order approximation for the differential meannumbers N cr n ≈ exp h − πk ( ω + π ) i sinh (2 πω /k ) × sinh [ π ( ω − π ) /k ] in Fermi casecosh ( π ( ω − π ) /k ) in Bose case . (5.31)Assuming m/k ≫ , we see that substantial value of N cr n are formed in the range − eE/k < π < − π ⊥ and are given the same formula N cr n ≈ exp (cid:20) − πk ( ω + π ) (cid:21) (5.32)both for bosons and fermions.Consequently, for any given λ satisfying Eqs. (4.13) and (5.22), the quantities N cr n are quasiconstant over the wide range of longitudinal momentum p x . When h , h → ∞ ,differential mean numbers coincide with (3.19) in a constant uniform electric field.31he above analysis shows that dominant contributions for mean numbers of createdparticles by a slowly varying field are formed in ranges of large kinetic momenta and havethere asymptotic forms (5.29) for p x < and (5.32) for p x > . In this case, the range Ω in Eq. (3.21) is realized as π ⊥ < π eE/k for p x < and as − eE/k < π < − π ⊥ for p x > . Therefore, we can represent integral (3.22) as follows ˜ n cr = J ( d ) (2 π ) d − Z √ λ 14 max { ω /ω , ω /ω } . (5.45)We can treat this fact as an indication that the back reaction is big. If so, the concept ofthe external field in the scalar QED is limited by the condition min ( ω /k , ω /k ) & forthe fields under consideration. We do not see similar problem in the spinor QED.34f k = k (in this case ∆ U = ∆ U = ∆ U/ ), we can compare the above results withthe regularization of rectangular steps by the Sauter-like potential, given by Eqs. (3.7) and(3.10). We see that both regularizations are in agreement, for fermions under the condition | ω − ω | ≪ ∆ U , and for bosons under condition (3.9). D. Exponentially decaying field In the examples, considered above, increasing and decreasing phases of the fields arealmost symmetric. Here we consider an essentially asymmetric configuration of the peakfield, when, for example, the field switches abruptly on at t = 0 , that is, k is sufficientlylarge, while the parameter k > is arbitrary, including, for example, the case of smoothswitching off process. Note, that due to the invariance of the mean numbers N cr n under thesimultaneous change k ⇆ k and π ⇆ − π , one can easily transform this situation to thecase with a large k and arbitrary k > .Let us assume that a sufficiently large k satisfies the inequalities ∆ U /k ≪ , ω /k ≪ . (5.46)Then Eqs. (5.19) and (5.20) can be reduced to the following form | ∆ | ≈ | ∆ ap | = e iπν (cid:12)(cid:12)(cid:12)(cid:12)(cid:20) χ ∆ U + ω − ω + k h (cid:18) − 12 + ddη (cid:19)(cid:21) Φ ( a , c ; η ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 . (5.47)Under the condition ∆ U /ω ≪ , (5.48)one can disregard the term χ ∆ U in Eq. (5.47) and set approximately π ≈ p x . Thus, ω ≈ p p x + π ⊥ . In this approximation, leading terms do not contain ∆ U , so that weobtain N cr n ≈ |C ∆ ap | for fermions (cid:12)(cid:12)(cid:12) C sc ∆ ap | χ =0 (cid:12)(cid:12)(cid:12) for bosons . (5.49)In fact, differential mean numbers obtained in these approximations are the same as in theso-called exponentially decaying electric field, given by the potential (5.3). Under condition(5.48), the results presented by Eqs. (5.49) for arbitrary k > are in agreement with onesobtained in Ref. [31]. 35et us consider the most asymmetric case when Eqs. (5.49) hold and when the incre-ment of the longitudinal kinetic momentum due to exponentially decaying electric field issufficiently large ( k are sufficiently small), h = 2∆ U /k ≫ max (cid:0) , m /eE (cid:1) . (5.50)In this case only the range of π ⊥ (4.13) is essential, in which K ⊥ is any given numbersatisfying the condition h ≫ K ⊥ ≫ max (cid:0) , m /eE (cid:1) . (5.51)It should be noted that the distribution N cr n , given by Eqs. (5.49) for this most asymmetriccase, coincides with the one obtained in our work [31]. However, the detailed study of thisdistribution was not performed there. The main contribution from this distribution tothe total number N cr (3.20) was estimated in our recent work [32]. Here we consider adetailed dependence of distribution (5.49) on the physical parameters p x and π ⊥ and thecorresponding consequences to the global quantities N cr and P v . We choose χ = − forconvenience in the Fermi case.In the case of large negative momenta p x , p x < and − π /k > g h (where g is anygiven number, g ≫ ), using an expression for the confluent hypergeometric function withlarge c and fixed a and h , given in [49], one can verify that the mean numbers N cr n arenegligibly small both for fermions and bosons. The same holds true for very large positive p x , such that π /k > K , where K is any given large number, K ≫ K ⊥ . We see that themean numbers N cr n differ from zero only in the range − g h < π /k < K . This rangecan be divided in the following subranges: (a) (1 + ε ) h ≤ − π /k < g h , (b) h (cid:20) (cid:16)p h g (cid:17) − (cid:21) ≤ − π /k < (1 + ε ) h , (c) h (cid:20) − (cid:16)p h g (cid:17) − (cid:21) ≤ − π /k < h (cid:20) (cid:16)p h g (cid:17) − (cid:21) , (d) (1 − ε ) h ≤ − π /k < h (cid:20) − (cid:16)p h g (cid:17) − (cid:21) , (e) h /g < − π /k < (1 − ε ) h , (f) − K < − π /k < h /g , (5.52)where g and ε are any given numbers satisfying the conditions g ≫ and ε ≪ . We36ssume that ε √ h ≫ . Note that in the ranges (5.52) τ = ih /c ≈ h k | π | . Then in theranges from (a) to (e), τ varies from /g to g .In the range (a), the confluent hypergeometric function Φ ( a , c ; ih ) is approximated byEq. (A8) given in Appendix A. In this range the differential mean numbers in the leading-order approximation are N cr n ≈ ω − | p x | ω (cid:2) O (cid:0) |Z | − (cid:1)(cid:3) × ω −| p x || p x | for bosons , (5.53)where max |Z | − ∼ (cid:0) ε √ h (cid:1) − . p g /h . We see from Eq. (5.53) that N cr n → if | p x | ≫ π ⊥ . Note that εeE/k < | p x | < ( g − eE/k . Taking into account the inequality (4.13),we see that the numbers (5.53) are negligibly small if ε √ h ≫ K ⊥ .In the range (c), τ − → and, using Eqs. (A2), (A3), and (A4) given in Appendix Awe find that N cr n = ω + p x ω e − πλ/ (cid:2) O ( |Z | ) − (cid:3) × cosh (cid:0) πλ (cid:1) for fermions π ( ω + p x ) √ eE | Γ ( iλ ) | for bosons , (5.54)where max |Z | − . g − . Note that N cr n given by Eq. (5.54) are finite and restricted, N cr n ≤ for fermions and N cr n . /g for bosons. In the range (b) the distributions N cr n vary betweentheir values in the ranges (a) and (c).In the range (e), parameters η and c are large with a fixed and τ > . In this case,using the asymptotic expression of the confluent hypergeometric function given by Eq. (A9)in Appendix A, we find that N cr n = exp (cid:20) − πk ( ω + π ) (cid:21) (cid:2) O (cid:0) |Z | − (cid:1)(cid:3) , (5.55)where Z is given by Eq. (A2) in the Appendix A, both for fermions and bosons, . Wenote that modulus |Z | − varies from |Z | − ∼ (cid:0) ε √ h (cid:1) − to |Z | − ∼ (cid:2) ( g − √ h (cid:3) − .Approximately, expression (5.55) can be written as N cr n ≈ exp (cid:18) − ππ ⊥ k | π | (cid:19) . (5.56)Note that eE/g < k | π | < (1 − ε ) eE in the range (e) and the distribution N cr n given byEq. (5.56) has the following limiting form: N cr n → e − πλ as k | π | → (1 − ε ) eE . (5.57)37hus, the distribution (3.19) is reproduced in the case of an exponentially decaying electricfield in the wide range of a large increment of the longitudinal kinetic momentum, − π ∼ eE/k . Taking into account the condition ε √ h ≫ , we see that p x / √ eE ≫ in this range.Thus, condition (5.48) holds if ∆ U / √ eE . . (5.58)Under this condition, the form of the distribution N cr n does not depend on the details of theswitching on before the time instant t = 0 . In the range (d), the distributions N cr n vary fromtheir values in the ranges (c) and (e) for fermions and bosons.In the range (f), we can use an asymptotic expression of the confluent hypergeometricfunction for large h at fixed a and c given by Eq. (A10) in Appendix A to get the followingresult: N cr n ≈ exp h − πk ( ω + π ) i sinh (2 πω /k ) × sinh [ π ( ω − π ) /k ] for fermionscosh ( π ( ω − π ) /k ) for bosons (5.59)in the leading-order approximation. The same distribution takes place in a slowly varyingfield for p x > , see Eq. (5.31). In the range (f) the form of N cr n given by Eq. (5.59) doesnot depend on details of switching on at t = 0 if condition (5.58) holds true. For m/k ≫ ,distribution (5.59) is approximated by Eq. (5.55).Note that WKB approximation holds true under the condition ( ω + π ) /k ≫ for N cr n given by Eq. (5.55). In the range (e) N cr n given by (5.55) coincides exactly with an estimation,obtained previously in [51, 52] in the framework of the semiclassical consideration. We stressthat in our consideration (which does not use the WKB) the the approximation (5.55) holdsfor any value of ( ω + π ) /k and exact results given by Eqs. (5.53), (5.54), and (5.59) arequite different from the corresponding semiclassical ones.Now, we can estimate the number density n cr of pairs created by an exponentially de-caying electric field, defined by Eq. (3.20). To this end, we represent the leading terms ofintegral (3.20) as a sum of two contributions, one due to the ranges (e) and (f) and anotherone due to the ranges (b), (c), and (d): n cr ≈ J ( d ) (2 π ) d − Z √ λ As was recently discovered in our work [53], an information derived from considerationsof exactly solvable cases allows one to make some general conclusions about quantum effectsin slowly varying strong fields for which no closed form solutions of the Dirac equation are39nown. Below, we briefly represent such conclusions about an universal behavior of vacuummean values in slowly varying strong electric fields.We note that in all these cases the quantity N cr n is quasiconstant over the wide range ofthe longitudinal momentum p x for any given λ, namely N cr n ∼ e − πλ . Pair creation effectsin such fields are proportional to large increments of the longitudinal kinetic momentum, ∆ U = e | A x (+ ∞ ) − A x ( −∞ ) | . Defining the slowly varying regime in general terms, onecan observe an universal character of vacuum effects caused by strong electric field.We call E ( t ) a slowly varying electric field on a time interval ∆ t if the following conditionholds true: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˙ E ( t )∆ tE ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ , ∆ t/ ∆ t mst ≫ , (6.1)where E ( t ) and ˙ E ( t ) are mean values of E ( t ) and ˙ E ( t ) on the time interval ∆ t , respectively,and ∆ t is significantly larger than the time scale ∆ t mst which is ∆ t mst = ∆ t st max n , m /eE ( t ) o , ∆ t st = h eE ( t ) i − / . (6.2)We are primarily interested in strong electric fields, m /eE ( t ) . . In this case, inequality(6.2) is simplified to the form ∆ t/ ∆ t st ≫ , in which the mass m is absent. In such cases, thepotential of the corresponding electric steps hardly differs from the potential of a constantelectric field, U ( t ) = − eA x ( t ) ≈ U c ( t ) = eE ( t ) t + U , (6.3)on the time interval ∆ t , where U is a given constant. This behavior is inherent to the fieldsof exact solvable cases presented above.If the electric field is not very strong, mean numbers N cr n of created pairs (or distributions)at the final time instant are exponentially small, N cr n ≪ . In this case the probability ofthe vacuum to remain a vacuum and probabilities of particle scattering and pair creationhave simple representations in terms of these numbers, | w n (+ − | | ≈ N cr n , | w n ( −|− ) | ≈ (1 + N cr n ) , P v ≈ − X n N cr n . (6.4)The latter relations are often used in semiclassical calculations to find N cr n and the totalnumber of created pair, N cr = P n N cr n , from the representation of P v given by Schwinger’seffective action. 40owever, when the electric field cannot be considered as a weak one (e.g. in somesituations in astrophysics and condensed matter), the mean numbers N cr n can achieve theirlimited values N cr n → already at finite time instants t and the sum N cr cannot be consideredas a small quantity. Moreover, for slowly varying strong electric fields this sum is proportionalto the large parameter T eff / ∆ t st . In such a case relations (6.4) are not correct anymore.However, as shown next, for arbitrary slowly varying strong electric field one can derive inthe leading-term approximation an universal form for the total density of created pairs.Let us define the range D ( t ) as follows: D ( t ) : h P x ( t ) i < , |h P x ( t ) i| ≫ π ⊥ . (6.5)In this range the longitudinal kinetic momentum h P x ( t ) i = p x − U ( t ) is negative and bigenough. If p x components of the particle momentum belongs to the range D ( t ) , then theparticle energy is in main determined by an increment of the longitudinal kinetic momentum , U ( t ) − U ( t in ) , during the time interval t − t in and h P x ( t ) i = h P x ( t in ) i− [ U ( t ) − U ( t in )] . Notethat D ( t ) ⊂ D ( t ′ ) if t < t ′ . The leading term of the total number density of created pairs, ˜ n cr , is formed over the range D ( t out ) , that is, the range D ( t out ) is chosen as a realization ofthe subrange Ω in Eq. (3.22).In the case when the electric field does not switch abruptly on and off, that is, thefield slowly weakens at t → ±∞ and one of the time instants t in and t out , or both areinfinite t in → −∞ and t out → ∞ , one can ignore exponentially small contributions to ˜ n cr from the time intervals (cid:16) t in , t eff in i and (cid:16) t eff out , t out (cid:17) , where electric fields are much less thanthe maximum field E , E (cid:16) t eff in (cid:17) , E (cid:16) t eff out (cid:17) ≪ E . Thus, in the general case it is enough toconsider a finite interval (cid:16) t eff in , t eff out i . Denoting t = t eff in and t M +1 = t eff out , we divide thisinterval into M intervals ∆ t i = t i +1 − t i > , i = 1 , ..., M , P Mi =1 ∆ t i = t eff out − t eff in . Wesuppose that Eqs. (6.1) and (6.2) hold true for all the intervals, respectively. That allowsus to treat the electric field as approximately constant within each interval, E ( t ) ≈ E ( t i ) ,for t ∈ ( t i , t i +1 ] . Note that inside of each interval ∆ t i abrupt changes of the electric field E ( t ) whose duration is much less than ∆ t i , cannot change significantly the total value of ˜ n cr , since N cr n ≤ for fermions. Using Eq. (4.23) for the case of T -constant field, we can41epresent ˜ n cr as the following sum ˜ n cr = M X i =1 ∆˜ n cr i , ∆˜ n cr i ≈ J ( d ) (2 π ) d − Z eE ( t i )( t i +∆ t i ) eE ( t i ) t i dp x Z √ λ i In which follows we use the example of the T -constant field to consider the contributions Re h j µ ( t ) i c and Re h T µν ( t ) i c to the mean values of the current density h j µ ( t ) i and the EMT h T µν ( t ) i , given by Eqs. (2.44). Note that the mean current density h j µ ( t ) i and the physicalpart of the mean value h T µν ( t ) i are zero for any t < t in . For t > t in , we are interested inthese mean values only for a large time periods ∆ t = t − t in satisfying Eq. (6.1). In thiscase, the longitudinal kinetic momentum belongs to the range (6.5) and distributions N cr n are approximated by Eq. (3.19). Using approximation (6.19), the functions − ϕ n ( t ) , givenby Eq. (4.4), and similar functions + ϕ n ( t ) can be taken in the following form − ϕ n ( t ) = V − / d − CD − − ρ [ − (1 + i ) ξ ] , + ϕ n ( t ) = V − / d − CD ρ [ − (1 − i ) ξ ] . (6.29)In the same approximation, the causal propagator S c ( x, x ′ ) (2.42) can be calculated usingsolutions ± ψ n ( x ) and ± ψ n ( x ) with scalar functions given by Eqs. (6.19) and (6.29) inthe range (6.5). It can be shown that the main contributions to Re h j µ ( t ) i c , h j ( t ) i and Re h T µµ ( t ) i c are formed in the range (6.5) for a large time period ∆ t . It is important thatthese contributions are independent of the interval ∆ t , that is, the densities Re h j µ ( t ) i c , h j ( t ) i , and Re h T µµ ( t ) i c are local quantities describing only vacuum polarization effects.Then we integrate in Eq. (2.42) over all the momenta. Thus, we see that in the case underconsideration, the propagator S c ( x, x ′ ) can be approximated by the propagator in a constantuniform electric field.The propagator S c ( x, x ′ ) in a constant uniform electric field can be represented as the50ock–Schwinger proper time integral: S c ( x, x ′ ) = ( γP + m )∆ c ( x, x ′ ) , ∆ c ( x, x ′ ) = Z ∞ f ( x, x ′ , s ) ds , (6.30)see [12] and [61, 62], where the Fock–Schwinger kernel f ( x, x ′ , s ) reads f ( x, x ′ , s ) = exp (cid:16) i e σ µν F µν s (cid:17) f (0) ( x, x ′ , s ) , f (0) ( x, x ′ , s ) = − eEs − d/ (4 πi ) d/ sinh( eEs ) × exp (cid:20) − i (cid:0) e Λ + m s (cid:1) + 14 i ( x − x ′ ) eF coth( eF s )( x − x ′ ) (cid:21) . Here coth( eF s ) is the matrix with the components [coth( eF s )] µν , F µν = E (cid:0) δ µ δ ν − δ µ δ ν (cid:1) ,and Λ = ( t + t ′ )( x − x ′ ) E/ , see [1, 63].It is easy to see that h j ( t ) i c = 0 , as should be expected due to the translational sym-metry. If d = 3 there is a transverse vacuum-polarization current, h j ( t ) i = ± e π / γ (cid:18) , πm eE (cid:19) E, (6.31)for each ± fermion species, [27], where γ (1 / , x ) is the incomplete gamma function. Notethat the transverse current of created particles is absent, h j ( t ) i = 0 if t > t out . The factorin the front of E in Eq. (6.31) can be considered as a nonequilibrium Hall conductivity forlarge duration of the electric field. In the presence of both ± species in a model, h j ( t ) i = 0 for any t .Using Eq. (6.30), we obtain components of the EMT for the T -constant field in thefollowing form Re h T ( t ) i c = − Re h T ( t ) i c = E ∂ Re L [ E ] ∂E − Re L [ E ] , Re h T ll ( t ) i c = Re L [ E ] , l = 2 , ..., D, (6.32)where L [ E ] = 12 Z ∞ dss tr f ( x, x, s ) , tr f ( x, x, s ) = 2 [ d/ cosh( eEs ) f (0) ( x, x, s ) . (6.33)The quantity L [ E ] can be identified with a non-renormalized one-loop effective Euler-Heisenberg Lagrangian of the Dirac field in an uniform constant electric field E . Note thatcomponents Re h T µν ( t ) i c do not depend of the time duration ∆ t of the T -constant field if ∆ t is sufficiently large. 51his result can be generalized to the case of arbitrary slowly varying electric field . To thisend we divide as before the finite interval (cid:16) t eff in , t eff out i into M intervals ∆ t i = t i +1 − t i > , such that Eq. (6.1) holds true for each of them. That allows us to treat the electric fieldas approximately constant within each interval, E ( t ) ≈ E ( t i ) for t ∈ ( t i , t i +1 ] . In eachsuch an interval, we obtain expressions similar to the ones (6.32) and (6.33), where theconstant electric field E has to be substituted by E ( t i ) . Then components of the EMT forarbitrary slowly varying strong electric field E ( t ) in the leading-term approximation can berepresented as Re h T ( t ) i c = − Re h T ( t ) i c = E ( t ) ∂ Re L [ E ( t )] ∂E ( t ) − Re L [ E ( t )] , Re h T ll ( t ) i c = Re L [ E ( t )] , l = 2 , ..., D, (6.34)where L [ E ( t )] = 12 Z ∞ dss tr ˜ f ( x, x, s ) , tr ˜ f ( x, x, s ) = 2 [ d/ cosh [ eE ( t ) s ] ˜ f (0) ( x, x, s ) , ˜ f (0) ( x, x, s ) = − eE ( t ) s − d/ exp ( − im s )(4 πi ) d/ sinh [ eE ( t ) s ] . (6.35)Note that L [ E ( t )] evolves in time due to the time dependence of the field E ( t ) . The quantity L [ E ( t )] describes the vacuum polarization. The quantities (6.34) are di-vergent due to the real part of the effective Lagrangian (6.35), which is ill defined. Thisreal part must be regularized and renormalized. In low dimensions, d ≤ , Re L [ E ( t )] canbe regularized in the proper-time representation and renormalized by the Schwinger renor-malizations of the charge and the electromagnetic field [1]. In particular, for d = 4 , therenormalized effective Lagrangian L ren [ E ( t )] is L ren [ E ( t )] = Z ∞ ds exp ( − im s )8 π s ( eE ( t ) coth [ eE ( t ) s ] s − s − [ eE ( t ) s ] ) . (6.36)In higher dimensions, d > , a different approach is required. One can give a precisemeaning and calculate the one-loop effective action using zeta-function regularization, seedetails in Ref. [27]. Making the same renormalization for h T µµ ( t ) i c , we can see that for therenormalized EMT the following relations hold true Re h T ( t ) i cren = − Re h T ( t ) i cren = E ( t ) ∂ Re L ren [ E ( t )] ∂E ( t ) − Re L ren [ E ( t )] , Re h T ll ( t ) i cren = Re L ren [ E ( t )] , l = 2 , , . . . , D. (6.37)52n the strong-field case ( m /eE ( t ) ≪ ), the leading contributions to the renormalizedEMT are Re h T µµ ( t ) i cren ∼ [ eE ( t )] d/ , d = 4 k [ eE ( t )] d/ ln [ eE ( t ) /M ] , d = 4 k . (6.38)The final form of the vacuum mean components of the EMT are h T µµ ( t ) i ren = Re h T µµ ( t ) i cren + Re h T µµ ( t ) i p , (6.39)where the components Re h T µµ ( t ) i cren and Re h T µµ ( t ) i p are given by Eqs. (6.37) and (6.23),respectively. For t < t in and t > t out the electric field is absent such that Re h T µµ ( t ) i cren = 0 .On the right hand side of Eq. (6.39), the term Re h T µµ ( t ) i p represents contributionsdue to the vacuum instability, whereas the term Re h T µµ ( t ) i cren represents vacuum polar-ization effects. For weak electric fields, m /eE ≫ , contributions due to the vacuum in-stability are exponentially small, so that the vacuum polarization effects play the principalrole. For strong electric fields, m /eE ≪ , the energy density of the vacuum polarization Re h T ( t ) i cren is negligible compared to the energy density due to the vacuum instability h T ( t ) i p , h T µµ ( t ) i ren ≈ Re h T µµ ( t ) i p . (6.40)The latter density is formed on the whole time interval t − t in , however, dominant con-tributions are due to time intervals ∆ t i with m /eE ( t i ) < and the large dimensionlessparameters q eE ( t i )∆ t i .We note that effective Lagrangian (6.35) and its renormalized form L ren [ E ( t )] coincidewith leading term approximation of derivative expansion results from field-theoretic calcu-lations obtained in Refs. [21, 54, 55] for d = 3 and d = 4 . In this approximation the S (0) term of the Schwinger’s effective action, given by the expansion (6.11), has the form S (0) [ F µν ] = Z dx L ren [ E ( t )] . (6.41)It should be stressed that unlike to the authors of Refs. [21, 54, 55], expression (6.35)and its renormalized form were derived in the framework of the general exact formulationof strong-field QED [6, 23], using QED definition of the mean value of the EMT, given byEq. (2.44). Therefore L ren [ E ( t )] is obtained independently from the derivative expansionapproach and the obtained result holds true for any strong field under consideration. More-over, it is proven that in this case not only the imaginary part of S (0) but its real part as53ell is given exactly by the semiclassical WKB limit. It is clearly demonstrated that theimaginary part of the effective action, Im S (0) , is related to the vacuum-to-vacuum transitionprobability P v and can be represented as an integral of L ren [ E ( t )] over the total field his-tory, whereas the kernel of the real part of this effective action, Re L ren [ E ( t )] , is related tothe local EMT which defines the vacuum polarization. Obtained results justify the deriva-tive expansion as an asymptotic expansion that can be useful to find the corrections formean values of the EMT components. We also note that some authors have argued thatthe locally constant field approximation, which amounts to limiting oneself to the leadingcontribution of the derivative expansion of the effective action, allows for reliable results forelectromagnetic fields of arbitrary strength; cf., e.g., [64, 65]. VII. CONCLUDING REMARKS We have presented in detail consistent QED calculations of zero order quantum effects inexternal electromagnetic field that correspond to the most important three exactly solvablecases of t -electric potential steps, namely, the Sauter-like electric field, the T -constant electricfield, and the exponentially growing and decaying electric fields. In all the cases, we presentsome new important details, unpublished so far. Nontrivial details underlying the derivationof number density of pairs created from vacuum due to the strong Sauter-like case arepresented. Next-to-leading term approximation for the differential mean number of paircreated, N cr n , due to T -constant electric field of long duration is obtained. A detailed studyof differential mean numbers of pair created in the most asymmetric case of the exponentiallygrowing and decaying electric fields is presented. On the base of exactly solvable cases, weconsider in detail distributions N cr n as functions on the particle momenta and establish theranges of dominant contributions for mean numbers of created particles due to a slowlyvarying field. This allows us to to gain new insights on the universal behavior of the vacuummean values in slowly varying intense electric fields. Comparing results for three exactlysolvable cases, one can see the appearance of a large parameter, which is an increment ofthe longitudinal kinetic momentum, and which corresponds to a large number of quantumstates, in which particles can be created. One can define the slowly varying regime in generalterms. Using the cases of the T -constant and exponentially growing and decaying electricfields, we find universal forms of the vacuum mean values of current, EMT components and54he total density of created pairs in the leading-term approximation for any large durationof an electric field. One can find a close relation of obtained universal forms with a leadingterm approximation of derivative expansion results in field-theoretic calculations. In fact,it is the possibility to adopt a locally constant field approximation which makes an effectuniversal. These results allow one to formulate semiclassical approximations, that are notrestricted by smallness of differential mean numbers of created pairs, and could be helpfulfor the development of numerical methods in strong-field QED. Acknowledgement The reported study of T.C.A., S.P.G., and D.M.G. was sup-ported by a grant from the Russian Science Foundation, Research Project No. 15-12-10009. Appendix A: Some asymptotic expansions The asymptotic expression of the confluent hypergeometric function for large η and c with fixed a and τ = η/c ∼ is given by Eq. (13.8.4) in [66] as Φ ( a, c ; η ) ≃ c a/ e Z / F ( a, c ; τ ) , Z = − ( τ − W ( τ ) √ c,F ( a, c ; τ ) = τ W − a D − a ( Z ) + R D − a ( Z ) , R = (cid:0) W a − τ W − a (cid:1) / Z , W ( τ ) = (cid:2) τ − − ln τ ) / ( τ − (cid:3) / (A1)where D − a ( Z ) is the Weber parabolic cylinder function (WPCF) [49]. Using Eq. (A1) wepresent the functions y ( η ) , y ( η ) and their derivatives at t = 0 as y ( η ) (cid:12)(cid:12) t =0 ≃ e ih / ( ih ) − ν (2 − c ) (1 − a ) / e Z / F (1 − a , − c ; τ ) , Z = − ( τ − W ( τ ) √ − c , τ = − ih / (2 − c ) ,∂y ( η ) ∂η (cid:12)(cid:12)(cid:12)(cid:12) t =0 ≃ e ih / ( ih ) − ν (2 − c ) (1 − a ) / e Z / (cid:20) − ih − − c ∂∂τ (cid:21) F (1 − a , − c ; τ ) ; y ( η ) (cid:12)(cid:12) t =0 ≃ e − ih / ( ih ) ν c a / e Z / F ( a , c ; τ ) , Z = − ( τ − W ( τ ) √ c , τ = ih /c ,∂y ( η ) ∂η (cid:12)(cid:12)(cid:12)(cid:12) t =0 ≃ e − ih / ( ih ) ν c a / e Z / (cid:20) − ih + 1 c ∂∂τ (cid:21) F ( a , c ; τ ) . (A2)Assuming τ − → , one has W − a ≈ a − 13 ( τ − , R ≈ a + 1)3 √ c , Z ≈ − ( τ − √ c,∂F ( a, c ; τ ) ∂τ ≈ a D − a ( Z ) + ∂D − a ( Z ) ∂τ + R ∂D − a ( Z ) ∂τ . Z = 0 , in the leading approximation at Z → one obtains that ∂F ( a, c ; τ ) ∂τ ≈ −√ ηD ′− a (0) + O ( η ) ,F ( a, c ; τ ) ≈ D − a (0) + O (cid:0) c − / (cid:1) , (A3)and D − a (0) = 2 − a/ √ π Γ (cid:0) a +12 (cid:1) , D ′− a (0) = 2 (1 − a ) / √ π Γ (cid:0) a (cid:1) , (A4)where Γ( z ) is the Euler gamma function. We find under condition (5.21) that ω , ≈ | π , | (1 + λ/h , ) , a , ≈ (1 + χ ) / iλ/ , − c ≈ − i (cid:18) λ + 2 π k (cid:19) , c ≈ i (cid:18) λ − π k (cid:19) ,τ − ≈ − h (cid:18) i + λ + 2 p x k (cid:19) , τ − ≈ h (cid:18) i − λ + 2 p x k (cid:19) . (A5)Using Eqs. (A2), (A3), and (A5) we represent Eq. (5.19) in the form N cr n = e − πλ/ h | δ | + O (cid:16) h − / (cid:17) + O (cid:16) h − / (cid:17)i ,δ = e iπ/ D − a (0) D ′ a − (0) − e − iπ/ D ′− a (0) D a − (0) . (A6)Assuming χ = 1 for fermions and χ = 0 for bosons, and using the relations of the Eulergamma function we find that δ = exp ( iπ/ iπχ/ e − πλ/ . 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