Pair creation by time-dependent electric fields: Analytic solutions
aa r X i v : . [ h e p - t h ] S e p Pair creation by time-dependent electric fields:Analytic solutions
Iwo Bialynicki-Birula , Lukasz Rudnicki and Albert Wienczek , Center for Theoretical Physics, Polish Academy of SciencesAl. Lotnik´ow 32/46, 02-668 Warsaw, Poland Faculty of Physics, University of Warsaw, Warsaw, PolandE-mail: [email protected]
Abstract.
Exact analytical solutions are presented for the time evolution of thedensity of pairs produced in the QED vacuum by a uniform electric field that isadiabatically switched on starting at −∞ . Pair production is described by the Dirac-Heisenberg-Wigner function introduced before [Phys. Rev. D , 1825 (1991)].The explicit solution is obtained by an extension of the method of the spinorialdecomposition to deal with a time-varying electric field. The main result of this workis that the pair density is an analytic function of the field strength; it can be expandedinto a convergent power series. Therefore, the essential singularity present in theSchwinger formula is to be attributed to the infinitely long duration of the process ofpair creation by a time-independent field.PACS numbers: 12.20.Ds, 11.15.Tk air creation by time-dependent electric fields: Analytic solutions
1. Introduction
The process of pair creation by the time independent and uniform electric fieldas described in our previous publication [1] suffers from a serious drawback. Theinstantaneous switching on of the field leads to unphysical transient effects clearly seenin Fig. 1 of [1]. In the present paper we solve analytically the problem of pair productionby a uniform electric field that is switched on adiabatically .To explain what this means in physical terms, let us consider a large (boundaryconditions are disregarded) capacitor charged so that the electric field is E . We shallcompare two cases. In the first case, the electric field is primordial, it existed always.In the second case, the capacitor has been continuously charged until at t = 0 the fieldreached the desired value E . The creation of pairs in the first case, i.e. for the time-independent electric field, was calculated by Schwinger [2] whose analytical formula hasa characteristic essential singularity as a function of the field strength E .Consider now the observer who at time t = 0 can measure the strength of the electricfield and the number of pairs but does not have information about the past history. Thisobserver can plot the number of pairs as a function of the field strength and find outwhether this dependence shows the singular behavior characteristic for the Schwingersolution. We show with the help of exact analytic solutions that the singularity willnot be found if the field has been turned on adiabatically. In this case the Schwingersingularity is not present. The formulas that describe the total pair production for anadiabatically switched field are analytic in the field strength—they can be expanded inthe field strength into a convergent power series—for every finite value of the parameter b , that sets the time scale of the charging process. The Schwinger singularity in thefield strength appears now as a singularity in b . Thus, the essential singularity found bySchwinger is to be attributed entirely to rather unphysical assumptions that the fieldhas infinite duration or that it has been suddenly switched on.The production of pairs by an adiabatic perturbation may seem impossible sincethere is an energy gap of 2 mc between the vacuum state and the state with one pair.Therefore, it would seem that under an adiabatic perturbation the vacuum will notchange. However, these doubts should disappear when one realizes that pair productionby a constant electric field is due to tunneling. Arbitrarily weak electric field can producean arbitrarily large difference in potential energy for a sufficiently large separation.When this difference in potential energy exceeds the value 2 mc , pairs are created.The problem of pair production will be formulated in terms of a field-theoreticgeneralization of the standard, nonrelativistic, Wigner function. This generalizationwas introduced in [3] and called the Dirac-Heisenberg-Wigner (DHW) function. Thisformulation was further developed in [4]-[12]. It allows for a description of pair creationas a time evolution of the vacuum state. The advantage of using the DHW functionis its applicability to the expectation values of field operators and not only to theone-particle wave functions that are solutions of the Dirac equation. However, theanalytic solutions of the one-particle Dirac equation are still very useful. The method of air creation by time-dependent electric fields: Analytic solutions b —the timedependence is governed by a simple exponential. In the second example the field isswitched on at a decreasing rate—the time dependence is governed by the hyperbolicsecant squared. In both cases the signature of the emerging Schwinger singularity isthe shrinking of the radius of convergence to zero when b tends to zero. We also showthat in the first case the power expansion of the exact solution coincides with thestraightforward perturbation theory.The evolution equation for the DHW function has many solutions, most of themdo not represent physical states. The DHW function shares this property with thestandard Wigner function in nonrelativistic quantum mechanics—not every functionover the phase-space is a Wigner function. For example, the Wigner function cannotbe concentrated too much in position and momentum since this would contradict theuncertainty relations. There is, however, a class of problems for which one can besure that the Wigner function represents a physically admissible state. These are theproblems of time evolution with the initial value being a genuine Wigner function. TheDHW function that is the solution of the evolution equation will correspond to a physicalstate—the one that evolved from the initial state according to the Schr¨odinger equation.This is precisely the method that is used in this work. To describe the time evolutionunder the influence of adiabatically switched on fields we shall always start with theeasily computable exact DHW function—the one that describes the free vacuum state.In nonrelativistic quantum mechanics the Wigner function is defined as a Fourierintegral of a product of wave functions and in many cases this integral can be explicitlyevaluated. In contrast, the calculation of the DHW function in quantum field theoryrequires the summation of contributions from an infinite number of states of virtualparticles in the vacuum. The only case where this can be easily done is the vacuumstate of the free field. For that reason in our analysis of the time evolution we shallchoose the DHW function of the vacuum as the initial value.In general, the evolution equations for the components of the DHW function, evenin the approximation where all radiative corrections are neglected, form a set of 16integro-differential equations and their analytic solution is a hopeless task. However,in the case of a spatially uniform electromagnetic field the evolution equations becometractable since they reduce to linear partial differential equations. These equationsallow for the application of the method of characteristics leading to ordinary differentialequations. In turn, these ordinary differential equations in certain cases can be solvedanalytically by spinorial decomposition introduced in Ref. [1].In the present paper we construct a complete set of solutions of the evolutionequations for the DHW function describing the evolution of the vacuum in the presenceof a uniform but time-dependent electric field. With the use of these solutions we findthe evolution when the initial state is the free-field vacuum. Our exact solutions clarifythe mechanism responsible for pair production and may explain some details of the air creation by time-dependent electric fields: Analytic solutions O (3) symmetrythat enables us to express the solutions of the evolution equations in terms of spinors.The method of spinorial decomposition is extended in this section to cover a time-dependent field. In Sec. 4, using this method, we find an analytic solution of the spinorialequations when the electric field is turned on exponentially. In Sec. 5 we use this solutionto determine the time evolution of the vacuum. In Sec. 6 we show the the consecutiveapproximations to the analytic solution can also be obtained by perturbation theory. InSec. 7 we construct the analytic solution from the solutions of the Dirac equation foundby Sauter [16]. Finally, in Sec. 8 we compare the time evolution of the QED vacuum inboth cases underscoring the main features.
2. Evolution of the DHW functionin a time-varying electric field
The DHW function (which in fact is a 4 × W αβ ( r , p , t ) = − Z d η e − i( p · η / ~ + ϕ ) D Φ (cid:12)(cid:12)(cid:12)h ˆΨ α ( r + η / , t ) , ˆΨ † β ( r − η / , t ) i(cid:12)(cid:12)(cid:12) Φ E , (1)where the square bracket denotes the commutator of the quantized Dirac field operator ˆΨ with its hermitian conjugate and the phase ϕ is the line integral of the vector potential ϕ = e ~ Z / − / d λ η · A ( r + λ η , t ) . (2)The physical meaning of this phase factor is best seen by rewriting Eq. (1) with the useof the gauge covariant shift operator W αβ ( r , p , t ) = − Z d η e − i p · η / ~ D Φ (cid:12)(cid:12)(cid:12)h e η / · D ˆΨ α ( r , t ) , e − η / · D ˆΨ † β ( r , t ) i(cid:12)(cid:12)(cid:12) Φ E , (3)where D = ∇ − ie/ ~ A ( r , t ). The equivalence of this form of W and the previous onefollows from the relationse ± η / · D = exp " − i e ~ Z / d λ η · A ( r ± λ η , t ) e ± η / · ∇ . (4)The phase factor has been introduced in [3] to secure gauge invariance of the DHWfunction. That is why only the electromagnetic fields and not the potentials appear inthe evolution equations. This also means that the argument p of the DHW function isthe kinetic not the canonical momentum.The most general DHW function has 16 real components. The 4 × W αβ can be decomposed into 16 matrices [3]. We found it convenient to use the matricesintroduced originally by Dirac [17], W ( r , p , t ) = [ f + X i =1 ρ i f i + σ · g + X i =1 ρ i σ · g i ] . (5) air creation by time-dependent electric fields: Analytic solutions f ( r , p , t ), the spin density g ( r , p , t ), the electric current density g ( r , p , t ), andthe magnetic moment density g ( r , p , t ). Moreover, as shown in [1, 3], only threeindependent coefficients ( w , w , w ) are needed to express all these quantities, namely − f g g g = w E ⊥ mc c p ⊥ + w E ⊥ c p ⊥ × n mc n + w n , (6)where n is the unit vector in the direction of the electric field, p ⊥ is the component of p perpendicular to the field, and E ⊥ is the transverse energy, E ⊥ = p m c + c p ⊥ . Thefactor -1/2 has been introduced to have the vector ( w , w , w ) normalized to 1.The evolution equations for the three coefficients were derived in [1] and they are:( ∂ τ + E ( τ ) ∂ q ) w w w = 2 q − q − w w w . (7)These equations differ from Eqs. (11) of Ref. [1] only by the time-dependence of theelectric field. We introduced here the dimensionless variables, namely τ = E ⊥ t ~ , q = cp k E ⊥ , E ( τ ) = e ~ cE ( τ ) E ⊥ , (8)where p k is the component of p in the direction of the electric field.We shall expose the group-theoretic content of the evolution equations for the vector w ( q, τ ) by writing them with the use of the spin-one matrices S in the Cartesian basis,( ∂ τ + E ( τ ) ∂ q ) w ( q, τ ) = 2i S · H ( q ) w ( q, τ ) , (9)where H ( q ) = (1 , , q ) , S · H ( q ) = − i q − q − . (10)In this way we may interpret the evolution as a rotation in the three-dimensional vectorspace [18].In order to convert partial differential equations into ordinary differential equationswe use an extension of the method of characteristics to systems of equations. However,this time the conversion goes in opposite direction as compared to our previous work.Namely, we shall eliminate the derivative with respect to q , not the time derivative. Thisis achieved by replacing the vector w ( q, τ ) by a new vector X ( q, τ ) with the momentumvariable q displaced by the vector potential A ( τ ) = e ~ cA ( τ ) /E ⊥ , q q + A ( τ ) , A ( τ ) = − Z ττ d τ ′ E ( τ ′ ) . (11) air creation by time-dependent electric fields: Analytic solutions w ( q, τ ) = exp [ A ( τ ) ∂ q ] X ( q, τ ) . (12)When this expression is inserted into Eqs. (9) we find that the components of this newvector satisfy ordinary differential equations with q playing the role of a fixed parameter,d X ( q, τ )d τ = 2i S · H ( q − A ( τ )) X ( q, τ ) . (13)This equation was obtained with the use of the relationexp [ −A ( τ ) ∂ q ] H ( q ) exp [ A ( τ ) ∂ q ] = H ( q − A ( τ )) . (14)The solution of the initial value problem will proceed by construction of the fundamentalmatrix ˆ F ( q, τ ) for the set of equations (13). This matrix is built (cf., for example,[19]) from a set of three linearly independent solutions ( X ( q, τ ) , X ( q, τ ) , X ( q, τ )) ofEqs. (13), ˆ F ( q, τ ) = X X X X X X X X X . (15)The fundamental matrix normalized to the unit matrix at τ isˆ F N ( q, τ, τ ) = ˆ F ( q, τ ) ˆ F − ( q, τ ) . (16)The solution of the initial value problem for the vector X with an arbitrary initialcondition X ( q, τ ) set at τ = τ is therefore X ( q, τ ) = ˆ F N ( q, τ, τ ) X ( q, τ ) . (17)Our aim is to find the time evolution of the vacuum state. The DHW function forthis state is built from the following set of w i functions [3]: w ( p ) = E ⊥ E p , w ( p ) = 0 , w ( p ) = c p · n E p , (18)where E p = p m c + c p is the total energy. Thus, in our dimensionless units thevector describing the vacuum has the form: V ( q ) = 1 E q (1 , , q ) , (19)where E q = p q represents the dimensionless total energy. Choosing X ( q, τ ) = V ( q ) as the initial state we obtain the time evolved DHW function by inserting theformula (17) into Eq. (12) w ( q, τ ) = ˆ F N ( q + A ( τ ) , τ, τ ) V ( q + A ( τ )) . (20)All we need now to solve the initial value problem is the fundamental matrix. Inthe next section we show how to construct this matrix using the method of spinorialdecomposition. air creation by time-dependent electric fields: Analytic solutions
3. Spinorial decomposition
The set of three differential equations leads to third order equations for each componentstudied in Ref. [6]. As we have shown in [1], one may replace third order equationswith second order equations by employing the method of spinorial decomposition thatexploits the O (3) group symmetry. We shall extend this method here to cover the caseof the time-varying field. To this end, we introduce the following representation forthe components of the vector X in terms of a two-component spinor Ψ and the Paulimatrices, X = Ψ † σ x Ψ , X = Ψ † σ y Ψ , X = Ψ † σ z Ψ . (21)One may check by inspection that the evolution equations (13) for the vector X followfrom the evolution equations for the spinor Ψ and its conjugate,dΨ( q, τ )d τ = i σ · H ( q − A ( τ ))Ψ( q, τ ) , (22 a )dΨ † ( q, τ )d τ = − iΨ † ( q, τ ) σ · H ( q − A ( τ )) . (22 b )These two coupled equationsd ψ − ( q, τ )d τ = i( q − A ( τ )) ψ − ( q, τ ) + i ψ + ( q, τ ) , (23 a )d ψ + ( q, τ )d τ = − i( q − A ( τ )) ψ + ( q, τ ) + i ψ − ( q, τ ) (23 b )can be transformed to the following two separate second order equations for the upper ψ − and lower ψ + components of the spinor, (cid:20) d d τ + 1 + ( q − A ( τ )) ∓ i E ( τ ) (cid:21) ψ ∓ ( q, τ ) = 0 . (24)These equations coincide with the second order equations obtained from the Diracequation in the time-dependent potential A ( τ ) along the z axis after the separationof the spatial variables through the following substitution ψ ( r , t ) = exp(i p · r ) ψ ( t ) . (25)Therefore, known analytic solutions of the reduced Dirac equation (24) will lead toanalytic solutions for the Wigner function. Linearly independent solutions of Eqs. (24)serve as building blocks in the construction of the spinors from which we shall constructthe fundamental matrix (16). The most convenient choice of two spinorial solutionsis when they are orthonormal. Let us suppose that we found one solution Ψ ( q, τ ) ofEqs. (22 a - b ), Ψ ( q, τ ) = f − g ! . (26)The minus sign in the lower component is chosen for future convenience. Then, thesecond solution of Eqs. (22 a - b ) can always be chosen in the formΨ ( q, τ ) = g ∗ f ∗ ! . (27) air creation by time-dependent electric fields: Analytic solutions X k from two orthonormal spinors Ψ and Ψ : X = − (cid:16) Ψ † σ Ψ + Ψ † σ Ψ (cid:17) , (28 a ) X = − (cid:16) − iΨ † σ Ψ + iΨ † σ Φ (cid:17) , (28 b ) X = − (cid:16) Ψ † σ Ψ − Ψ † σ Ψ (cid:17) . (28 c )The orthonormality of these vectors is a result of the following identity valid for everyset of four spinors, (cid:16) Ψ † σ Ψ (cid:17) · (cid:16) Ψ † σ Ψ (cid:17) = 2 (cid:16) Ψ † Ψ (cid:17) (cid:16) Ψ † Ψ (cid:17) − (cid:16) Ψ † Ψ (cid:17) (cid:16) Ψ † Ψ (cid:17) . (29)The fundamental matrix built from orthonormal vectors (28 a - c ) is orthogonal.Therefore, in the formula (16) we may replace the inverse matrix by the transposedmatrix. We have chosen the minus sign in the definition (28 a - c ) to have the vectors( X , X , X ) forming a right-handed set, X · ( X × X ) = 1, so that the determinantof the fundamental matrix is equal to 1 ( ˆ F describes a pure rotation).We may now use the three orthonormal vectors (28 a - c ) to build the matrix ˆ F .The components of this matrix [20] are real and imaginary parts of various quadraticexpressions of the two functions f and g ,ˆ F = −ℜ [ f − g ] ℑ [ f − g ] 2 ℜ [ f g ∗ ] ℑ [ f + g ] ℜ [ f + g ] − ℑ [ f g ∗ ] − ℜ [ f g ] 2 ℑ [ f g ] gg ∗ − f f ∗ . (30)We need a specific form of the function A ( τ ) to proceed any further. This will be donein next sections.
4. Electric field switched on at a constant rate
The adiabatic switching on of the electric field will be described by an exponentialprefactor. Thus, the electric field will be given by the formula E ( τ ) = E exp( bτ ) n with b >
0. It follows from the Maxwell equations that this field is produced by the current b E ( τ ). For sufficiently small values of b this current can be neglected in comparisonwith the current associated with the produced pairs.Assuming that the initial time τ lies in the remote past ( τ = −∞ ), we obtain A ( τ ) = −E b − exp( bτ ) . (31)This choice of the time dependence results in the following second order equations (24)for the upper and lower spinor components: (cid:20) d d τ + 1 + ( q + E b e bτ ) ∓ i E e bτ (cid:21) ψ ∓ ( q, τ ) = 0 . (32) air creation by time-dependent electric fields: Analytic solutions ψ − have the form f ( q, τ ) = exp(i τ E q )e i s/ √ p E q − qE q F (cid:18) i E q − qb , E q b ; − i s (cid:19) , (33 a ) g ∗ ( q, τ ) = − exp( − i τ E q )e − i s/ √ p E q + qE q F (cid:18) − i E q − qb , − E q b ; i s (cid:19) , (33 b )where s = 2 E b − e bτ . The lower components are, in general, linear combinations of both f ∗ ( q, τ ) and g ( q, τ ). However, for our choice (33 a - b ) of the upper components the lowercomponents ψ + contain only one term and the two orthonormal solutions of Eqs. (22 a - b ) are (26) and (27). This is the simplest pair of solutions because each componentcontains only one confluent hypergeometric function. Orthogonality of the two spinorsis self-evident but the check of normalization requires the use of the evolution equations(22 a - b ). Since these equations preserve scalar products of spinors, it is sufficient to checkthe normalization at any value of τ and we choose τ = −∞ or s = 0. At this value, theconfluent hypergeometric functions reduce to 1 and the normalization | f | + | g | = 1follows immediately. It may be worth noticing that the normalization condition for anarbitrary time generates some quadratic identities for the confluent functions that seemto be unknown.To elucidate the connection between the results obtained for the time-independentfield in [1] and the time-dependent field in the present case, we concoct one “super”equation that encompasses these two cases. The evolution equation (22 a - b ) will havethe form: dΨ( q, τ )d τ = i σ · H (cid:18) q + E exp( bτ ) − exp( bτ ) b (cid:19) Ψ( q, τ ) , (34)where τ specifies the initial moment at which Ψ describes the field-free vacuum. Theexact solution of this equation can still be found. All we have to do is to replace q by q − exp( bτ ) /b in (33 a - b ). There are two mutually exclusive limiting procedures thatlead either to the solution of Eqs. (22) in [1] or to the time-dependent solution given by(33 a - b ). The first limit is when τ = 0 and b = 0 and it leads to a solution singular at E = 0. This solution reproduces the Schwinger singularity as shown in [1]. The secondlimit is when τ → −∞ . This solution is analytic in E but it has a singularity when b →
0. Thus, there is no direct connection between these two limiting cases.
5. Time evolution of the QED vacuum
One must be careful with taking the limit τ → −∞ in Eq. (20) because of the oscillatoryterms present in the fundamental matrix even in the limit of vanishing field. However,these terms cancel out when the initial state is the vacuum,lim τ →−∞ ˆ F T ( q, τ ) V ( q ) = (0 , , − . (35) air creation by time-dependent electric fields: Analytic solutions w v ( q, τ ) that evolved from the vacuum at τ = −∞ is obtained by combining the formulas (20), (30), and (35), w v ( q, τ ) = ˆ F ( q + A ( τ ) , τ )(0 , , −
1) = (cid:16) − ℜ ( ˜ f ˜ g ∗ ) , ℑ ( ˜ f ˜ g ∗ ) , | ˜ f | − | ˜ g | (cid:17) , (36)where ˜ f = f ( q + A ( τ ) , τ ) and ˜ g = g ( q + A ( τ ) , τ ). This vector satisfies the evolutionequations (7) and the vacuum initial condition. The validity of these statements followsfrom our general procedure but it can also be checked by a direct (albeit tedious)calculation.The density of pairs n ( q, τ ) produced by the electric field [21] was shown in [3] tobe n ( q, τ ) = 1 − O ( q, τ ) , (37)where O ( q, τ ) is the overlap between the DHW vector at time τ given by (36) and theinitial vacuum vector (19), O ( q, τ ) = V ( q ) · w v ( q, τ ) = q ( | ˜ f | − | ˜ g | ) − ℜ [ ˜ f ˜ g ∗ ] E q . (38)Since both vectors V ( q ) and w v ( q, τ ) are normalized, 1 − O ( q, τ ) is simply one-halfof the squared difference between the initial and the evolved vectors. The relation(37) has been guessed from numerical calculations in Ref. [3]. In the Appendix wegive a derivation of this important formula and show that it is valid for an arbitraryelectromagnetic field. In the present case f vanishes because the charge density in astate that evolved from the vacuum under the action of a uniform field vanishes at alltimes. Therefore, the general formula (A.8) reduces to n ( r , p , t ) = (cid:20) mf ( r , p , t ) + p · g ( r , p , t )2 E p (cid:21) . (39)With the use of Eqs. (6) and (18) the second term can be identified with (minus) theoverlap function and the formula (37) is thus proven.The number of pairs should not change under the simultaneous reversal of thefield and the parallel momentum because it is only the relative orientation of these twovectors that matters. This property directly follows from the fact that under the changeof signs E → −E , q → − q the functions ˜ f and ˜ g are interchanged ˜ f → − ˜ g, ˜ g → − ˜ f .In Fig. 1 we show the pair density as a function of q for different values of the fieldstrength E . In contrast to the results of Ref. [1], there are no transients. The numberof pairs is a smooth function of q .The most striking result is the absence of an essential singularity in the dependenceof the number of produced pairs on the field strength E . All functions appearing in theformula (38), as long as b = 0, can be expanded into a convergent power series in E because the confluent hypergeometric function F ( a, c ; z ) is an analytic function of itsthree arguments. However, it has poles in c at all negative integers − n (including 0).Since in the formulas (33 a - b ) the parameter c is a function of E , c = 1 ± p q − E /b ) b , (40) air creation by time-dependent electric fields: Analytic solutions n H q L - - E Figure 1.
The pair density n ( q, τ ) for the exponentially growing field at time τ = 0and for b = 0 .
05 plotted as a function of the dimensionless momentum q at the values0.1, 0.2, 0.3, and 0.4 of the dimensionless field strength E . n H q L - - E Figure 2.
The analogous plot as in Fig. 1 obtained from the sixth-order perturbativeexpansion. the radius of convergence R of the series in E can be determined from the conditions c = 0 and it reads R = b p q + b / . (41)Thus, the perturbative expansion is convergent only for E smaller than the radius R . Inthe limiting case, when b →
0, the radius of convergence goes to zero as it must becauseof the essential singularity that reappears in this limit.The expansion in E up to the sixth order for τ = 0 gives the following formula: O ( q,
0) = 1 − N E E q (4 + ˜ b ) − N E E q (4 + ˜ b ) (1 + ˜ b ) − N E E q (4 + ˜ b ) (1 + ˜ b ) (4 + 9˜ b ) − N E E q (4 + ˜ b ) (1 + ˜ b ) (4 + 9˜ b ) (1 + 4˜ b ) − N E E q (4 + ˜ b ) (1 + ˜ b ) (4 + 9˜ b ) (1 + 4˜ b ) (4 + 25˜ b ) + O (cid:0) E (cid:1) , (42) air creation by time-dependent electric fields: Analytic solutions b = b/E q and N = 1 / , (43 a ) N = q (7˜ b + ˜ b ) , (43 b ) N = 3 / (cid:16) − q + 112 + ˜ b (1760 q − b (2164 q − b (472 q − b (36 q − (cid:17) , (43 c ) N = q/ (cid:16) ˜ b ( − q + 65472) − ˜ b (110528 q − b (755440 q − b (1332348 q − b (795136 q − b (193144 q − b (24804 q − b (1296 q − (cid:17) , (43 d ) N = 1 / (cid:16) q − q + 3559424+ ˜ b ( − q + 924008448 q − b ( − q + 11574500352 q − b ( − q + 41551564800 q − b (21767894400 q + 52683929664 q − b (151238728256 q − q − b (236416765448 q − q + 2556452997)+ ˜ b (183464805216 q − q + 6447749532)+ ˜ b (78987949264 q − q + 4744001262)+ ˜ b (19825772704 q − q + 1733216492)+ ˜ b (3204676008 q − q + 339299253)+ ˜ b (293855040 q − q + 34399080)+ ˜ b (11664000 q − q + 1458000) (cid:17) . (43 e )In Fig. 2 we show that the six terms present in (42) reproduce reasonably well the exactresults. The structure of the denominators in this formula is a direct consequence of theseries expansion of the confluent hypergeometric function F ( a, c ; z ) = 1 + ac z
1! + a ( a + 1) c ( c + 1) z
2! + a ( a + 1)( a + 2) c ( c + 1)( c + 2) z
3! + . . . (44)In our case c is given by Eq. (40) and the expansion of the expressions 1 / ( n + c ) intopowers of E gives1 n + c = ˜ b (( n + 1)˜ b ∓ n + 1) ˜ b ± q (( n + 1)˜ b ∓ E q (4 + ( n + 1) ˜ b ) E + . . . . (45)Substituting here the consecutive values of n we reproduce the factors in thedenominators of (42).Even though each term of the expansion into powers of E is finite, for b = 0 theseries becomes only an asymptotic one. In order to make this point absolutely clear, air creation by time-dependent electric fields: Analytic solutions q = 0 the first 8 terms of the perturbation series for the number ofproduced pairs for two values of b . n = 110 E − E + 255139802000 E + 438840055497834150480000 E b = 1 − E + 1776480040832261689003353180781584001131645584000000 E − E + 17193628747837682311369175260079760483098557822866919571206822106176531328917776928027793134400000000000 E + O (cid:0) E (cid:1) , (46) n = 18 E + 21128 E + 8691024 E + 33447732768 E + 59697183262144 E b = 0+ 344292919054194304 E + 1463159457604533554432 E + 687874205963671652147483648 E + O (cid:0) E (cid:1) . (47)These expansions leave no doubt that the first series is convergent, at least for E < b = 0corresponds to the time-independent electric field when the Schwinger formula precludesthe possibility of an expansion in powers of E . We shall illustrate these results with thegraphs of the pair density.
6. Perturbation theory
The solution of the evolution equation (13) for the vector X ( q, τ ) in perturbation theoryis most easily obtained by expanding this vector in the eigenbasis of the matrix appearingin (7). This matrix has eigenvalues 0 and ± E q . Its orthonormal eigenvectors are: e ( q ) = 1 E q q , (48 a ) e ± ( q ) = 1 √ E q − q ∓ i E q . (48 b )We shall seek the solutions of equations (13) in the form X ( q, τ ) = x ( q, τ ) e ( q ) + x + ( q, τ ) e + ( q ) + x − ( q, τ ) e − ( q ) . (49)The evolution equations for the expansion coefficients ared x ( q, τ )d τ = − i √ A ( τ ) ( x − ( q, τ ) − x + ( q, τ )) , (50 a )d x + ( q, τ ) dτ = 2i E q x + ( q, τ ) − i ˜ A ( τ ) (cid:16) q x + ( q, τ ) − √ x ( q, τ ) (cid:17) , (50 b )d x − ( q, τ )d τ = − E q x − ( q, τ ) + i ˜ A ( τ ) (cid:16) q x − ( q, τ ) − √ x ( q, τ ) (cid:17) , (50 c ) air creation by time-dependent electric fields: Analytic solutions A ( τ ) = A ( τ ) /E q . In order to prepare these equations for the perturbativesolution we shall multiply the second equation by exp( − E q τ ) and the third equationby exp(2i E q τ ) and then integrate all three equation with respect to τ from −∞ to τ . Assuming that initially at τ = −∞ the system is in the vacuum state, i.e. x ( q, −∞ ) = 1 , x ± ( q, −∞ ) = 0, we obtain after integration by parts the followingset of three coupled integral equations: x ( q, τ ) = 1 − i √ Z τ −∞ d τ ˜ A ( τ ) ( x − ( q, τ ) − x + ( q, τ )) , (51 a ) x + ( q, τ ) = − i Z τ −∞ dτ e E q ( τ − τ ) ˜ A ( τ ) (cid:16) q x + ( q, τ ) − √ x ( q, τ ) (cid:17) , (51 b ) x − ( q, τ ) = i Z τ −∞ dτ e − E q ( τ − τ ) ˜ A ( τ ) (cid:16) q x − ( q, τ ) − √ x ( q, τ ) (cid:17) . (51 c )The solution of these equations by iteration produces the perturbative series in powers ofthe field strength which coincides with the power expansion of the analytic solution. Wechecked that perturbation theory reproduces the formula (42). Note that the function A ( τ ) blows up for b = 0. Hence, the integral equations (51 a - c ) become invalid which isanother manifestation of the Schwinger singularity in the absence of adiabatic switching.
7. Electric field switched on at a decreasing rate
To complete our study we shall show now that the main conclusion of our investigation—the removal of the essential singularity—holds also when the time dependence of theelectric field is different. To this end we use analytic solutions of the Dirac equation ina uniform time-dependent field discovered by Sauter [16] and analyzed in more detailby Sommerfeld [22]. The time dependence of the electric field in the Sauter solutionis E ( τ ) = E sech ( bτ / n so that A ( τ ) = − E (1 + tanh( bτ / t = −∞ .The second order equations (24) for the Sauter field have the following form: " d d τ + 1 + (cid:18) q + 2 E (1 + tanh( bτ / b (cid:19) ∓ i E cosh ( bτ / ψ ∓ ( q, τ ) = 0 . (52)Following Sauter and Sommerfeld we replace the time variable by u = (1+tanh( bτ / / bu (1 − u ) d ψ − ( q, u )d u = i( q + 4 E ub ) ψ − ( q, u ) + i ψ + ( q, u ) , (53 a ) bu (1 − u ) d ψ + ( q, u )d u = − i( q + 4 E ub ) ψ + ( q, u ) + i ψ − ( q, u ) . (53 b )Two orthonormal solutions (26) and (27) of the first order equations (23 a - b ) areconstructed this time from the following two solutions of the second order equation air creation by time-dependent electric fields: Analytic solutions n H q L - - E Figure 3.
The pair density n ( q ) for the Sauter field at time τ = 0 and for b = 0 . q at the values 0.1, 0.2, 0.3, and0.4 of the dimensionless field strength E . for ψ − : f ( q, u ) = u i α (1 − u ) − i β √ p E q − qE q F (cid:18) E b + i α − i β, − E b + i α − i β, α ; u (cid:19) , (54 a ) g ∗ ( q, u ) = − u − i α (1 − u ) i β √ p E q + qE q F (cid:18) − E b − i α + i β, E b − i α + i β, − α ; u (cid:19) , (54 b )where α = p q b , β = p q + 4 E /b ) b . (55)The expression for the number of pairs constructed according to Eqs. (37) and (38) isan analytic function of E but this time the radius of convergence is four times smallerthan the value (41) for the exponential field. For the Sauter field we reproduce theformula for the number of produced pairs at t = + ∞ that were obtained by Narozhnyiand Nikishov [23],2 sinh (cid:0) π E b − π (cid:0) α − ¯ β (cid:1)(cid:1) sinh (cid:0) π E b + π (cid:0) α − ¯ β (cid:1)(cid:1) sinh (2 πα ) sinh (cid:0) π ¯ β (cid:1) , (56)where ¯ β differs from β by having the sign of E reversed.
8. Common properties of the two solutions
In [1] we derived a general expression for the density of pairs n ( q ) created by a constantelectric field during the finite time interval [ τ , τ ]. From this result, in the limit τ → −∞ corresponding to the infinite duration of the pair creation process, we obtainthe following time independent expression found in [11]: n ( q ) = 1 p q (cid:12)(cid:12)(cid:12)(cid:12) f c ( − q ) qp q − q − g c ( − q ) qp q + q (cid:12)(cid:12)(cid:12)(cid:12) , (57) air creation by time-dependent electric fields: Analytic solutions f c ( q ) = e − π/ (8 E ) √ E D − − i / (2 E ) (cid:18) (1 + i) q √E (cid:19) , (58 a ) g c ( q ) = e − π/ (8 E ) √ − i) D − i / (2 E ) (cid:18) (1 + i) q √E (cid:19) . (58 b )and D ν ( z ) is the parabolic cylinder function.In Fig. 4 we compare the pair density n ( q ) for the constant and the exponentialfields as functions of the dimensionless momentum q . The same comparison is shownin Fig. 5 for the constant field and the Sauter field. In both plots we took b = 0 . τ = 0 because at this moment the fieldstrength of the exponential or the Sauter field E is the same as that of the constant field,which was taken as E = 1. The pair density n ( q ) for the exponential and the Sauterfields is given in (37), while f ( q, τ ) and g ( q, τ ) for these fields are given respectively in(33 a - b ) and (54 a - b ).Both plots clearly show the difference between the constant field and the exponentialor the Sauter field. The pair density for all three fields is exactly the same for negativeand small q , but for larger q they are quite different. In the limit when q → ∞ thepair density n ( q ) for the constant field tends to a constant, which is nonanalytic in E = 0. However, for the exponential and the Sauter fields the pair density tends fastto 0. Maximal rate of growth of the particles momenta caused by the exponential orthe Sauter field equals E /b . If we assume that pairs are created with q ≈
0, then thepair density in Figs. 4 and 5 should be negligible for q ≤
10. Both plots satisfy thiscondition. It is also worth noting that for both the exponential and the Sauter field theoscillations of n ( q ) for the constant field (57) are reproduced.The two exactly solvable cases differ considerably but they share common featuresconfirming the main message of our work. We have chosen the parameters in such away that at τ = 0 the electric field in both cases attains the same value E and the rateof growth of the field in the remote past is b . The amplitude of the field is, however,four time larger for the Sauter field, E cosh ( bτ / ≈ E exp( bτ ) . (59)In Fig. 6 we show the time dependence of the electric field in the two cases. There isa direct correspondence in the remote past between the functions that solve the twosets of equations. The confluent hypergeometric functions (33 a - b ) are related to thehypergeometric functions (54 a - b ) by the limiting procedure, F ( α, γ ; z ) = lim β →∞ F ( α, β, γ ; z/β ) . (60)In our case evaluating the limit in (54 a - b ), τ → −∞ , E → ∞ , E e bτ = const = b s, (61)we obtain (33 a - b ). air creation by time-dependent electric fields: Analytic solutions - H q L Figure 4.
The pair density n ( q ) as a function of the dimensionless momentum q forthe constant (upper curve) and the exponential (lower curve) field for the dimensionlesselectric field strength E = 1. The value of the exponential field is taken at τ = 0 andfor the value of b = 0 .
1. Appropriate expressions are given in (37) and (57). - H q L Figure 5.
The pair density n ( q ) as a function of the dimensionless momentum q forthe constant (upper curve) and the Sauter (lower curve) field for the dimensionlesselectric field strength E = 1. The value of the Sauter field is taken at τ = 0 and forthe value of b = 0 .
1. Appropriate expressions for n ( q ) are given in (57) and (37) with f ( q, τ ) and g ( q, τ ) defined in (54 a - b ). air creation by time-dependent electric fields: Analytic solutions - - - - Τ E H Τ L Figure 6.
The field strength E shown as a function of time. The lower curvecorresponds to the exponential dependence and the upper curve to the square of thehyperbolic secans. - - - - Τ n H E , q = L Figure 7.
The pair density n ( q, τ ) for the exponential field (lower curve) and theSauter field (upper curve) plotted as a function of time. Since at τ = 0 both fieldsreach the same value, the number of pairs is almost the same. Since at τ = 0 the electric field attains in both cases the same value E , we expectthat the number of produced pairs will not differ significantly. Indeed, the results ofthe numerical calculations shown in Figs. 7 and 8 fully confirm this expectation. Thenumber of produced pairs is slightly higher for the Sauter field because, as shown inFig. 6, this field dominates over the exponentially growing field.The analytic character of the number of pairs function is seen in Fig. 8 where weplotted the dependence on E in the neighborhood of zero. The parabolic character ofboth curves clearly shows that the expansion in powers of E begins in both cases withthe quadratic term. The difference between the two cases diminishes with decreasing b .
9. Conclusions
We have shown with the help of exact analytic solutions that adiabatic switching of theelectric field removes the essential singularity which is present in the Schwinger formula air creation by time-dependent electric fields: Analytic solutions - - E n H E , q = L Figure 8.
The pair density n ( E ) for the exponentially growing field at the time τ = 0for b = 0 . q = 0 plotted as a function of the dimensionless field strength E forthe exponential field (lower parabola) and for the Sauter field (upper parabola). Forcomparison we also show a typical nonanalytic behavior representing the Schwingersingularity. for pair creation. This observation opened the way to a perturbative treatment whichgives the results coinciding with the power expansion of the exact solution. The methodof spinorial decomposition that enabled us to find exact solutions is quite general andmight be applicable to a class of similar problems. Acknowledgments
All calculations and the figures we produced with the help of Mathematica [24]. Thisresearch was partly supported by the grant from the Polish Ministry of Science andHigher Education for the years 2013–2016.
Appendix
The starting point is the expansion of the Dirac field operator (cf., for example, [25])into creation and annihilation operators (c=1),ˆΨ( r , t ) = X s Z d p (2 π ~ ) / r mE p h u ( p , s )ˆ b ( p , s, t ) e i p · r / ~ + v ( p , r ) ˆ d † ( p , s, t )e − i p · r / ~ i . (A.1)We allowed for an arbitrary time dependence of the creation and annihilation operators,so that the validity of this formula is not restricted to free fields. Next, we invert theFourier transform and use the properties of the spinors u and v to obtainˆ b ( p , s, t ) = r mE p u † ( p , s ) · Z d r (2 π ~ ) / e − i p · r / ~ ˆΨ( r , t ) , (A.2)With the use of the completeness relation for the spinors [25], X s u γ ( p , s ) u † δ ( p , s ) = ( ρ m + ρ σ · p + E p ) γδ m , (A.3) air creation by time-dependent electric fields: Analytic solutions p :ˆ N ( p , t ) = X s ˆ b † ( p , s, t )ˆ b ( p , s, t )= Z d r (2 π ~ ) Z d η e − i p · η / ~ ˆΨ † ( r − η / , t ) ρ m + ρ σ · p + E p E p ˆΨ( r + η / , t ) . (A.4)Now, we replace the product of field operators by the commutator and anticommutatorand use the canonical equal-time anticommutation relationsˆΨ † β ( r − η /
2) ˆΨ α ( r + η /
2) = 12 δ ( η ) δ αβ − h ˆΨ α ( r + η / , ˆΨ † β ( r − η / i . (A.5)We have not assumed that the field operators obey the free Dirac equation since weallowed in Eq. (A.1) for an arbitrary dependence of the creation and annihilationoperators on t . However, in the presence of an external field we must introducemodifications required by gauge invariance. Following the procedure that led tothe formula (3) we replace the shift operators in Eq. (A.4) by their gauge invariantcounterparts (4) and extract the phase factor e − iϕ . The gauge invariant number operatorˆ N g obtained in this way isˆ N g ( p , t ) = X s ˆ b † ( p , s, t )ˆ b ( p , s, t )= Z d r (2 π ~ ) Z d η e − i( p · η / ~ + ϕ ) ˆΨ † ( r − η / , t ) ρ m + ρ σ · p + E p E p ˆΨ( r + η / , t ) . (A.6)The expectation value of ˆ N g ( p , t ) can then be expressed in terms of the DHW function.The quantity of interest is the pair density in phase space n ( r , p , t ) which in the presentcase is equal to the density of electrons. Therefore, in the formula (A.6) we may dropthe integration over r and the normalization of the phase-space volume element (2 π ~ ) to obtain the density of pairs per unit phase-space volume in the form n ( r , p , t ) = 1 + Tr { ( ρ m + ρ σ · p + E p ) W ( r , p , t ) } E p . (A.7)Due to the orthogonality of the Dirac matrices under the trace only the components f , g , and f may contribute and we obtain the final formula n ( r , p , t ) = 1 + mf ( r , p , t ) + p · g ( r , p , t ) + E p f ( r , p , t )2 E p . (A.8)The number of antiparticles calculated in the same manner is¯ n ( r , p , t ) = 1 + mf ( r , p , t ) + p · g ( r , p , t ) − E p f ( r , p , t )2 E p . (A.9)The difference between the two expressions is the charge density n ( r , p , t ) − ¯ n ( r , p , t ) = f ( r , p , t ) , (A.10)as was to be expected. air creation by time-dependent electric fields: Analytic solutions References [1] Bialynicki-Birula I and Rudnicki L 2011
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