Consistency restrictions on maximal electric field strength in QFT
aa r X i v : . [ h e p - t h ] A ug Consistency restrictions on maximal electricfield strength in QFT
S.P. Gavrilov ∗ and D.M. Gitman † November 12, 2018
Abstract
QFT with an external background can be considered as a consistent model onlyif backreaction is relatively small with respect to the background. To find the corre-sponding consistency restrictions on an external electric field and its duration in QEDand QCD, we analyze the mean-energy density of quantized fields for an arbitrary con-stant electric field E , acting during a large but finite time T . Using the correspondingasymptotics with respect to the dimensionless parameter eET , one can see that theleading contributions to the energy are due to the creation of paticles by the electricfield. Assuming that these contributions are small in comparison with the energy den-sity of the electric background, we establish the above-mentioned restrictions, whichdetermine, in fact, the time scales from above of depletion of an electric field due tothe backreaction.PACS numbers: 12.20.Ds,11.15.Tk,11.10.Wx I. It is well-known that QFT in an external background provides an efficient model forthe study of quantum processes in those cases when some part of a quantized field is strongenough to be treated as a classical one. For example, QED with an external electromagneticbackground formally arises from extending the QED Lagrangian by the interaction of thematter current with a given external electromagnetic field A ext µ , which is not quantized.This is naturally implied as a certain approximation. The study of some problems in QEDand QCD with superstrong external backgrounds and their applications to astrophysics andcondensed matter (to graphene physics) has once again raised the question of a consistency ofsuch theories. Obviously, the question must be answered, first of all, in the case of a constantexternal field. Calculations that have an immediate relation to the above-mentioned problemhave first been carried out by Heisenberg and Euler in the case of QED with constant parallelelectric E and magnetic B fields, see [1]. They computed the change of the vacuum energyof spinning particles for arbitrary B and a weak electric field E ≪ E c = m /e ( ℏ = c = 1)which is unable to effectively create pairs from vacuum. They interpreted this change as achange of the energy of the external field itself and, at the same time, as a change of theMaxwell Lagrangian L (0) = (cid:0) E − B (cid:1) / π by a certain addition L (1) . The limiting case ofa strong magnetic field ( B ≫ m /e, E = 0) yields L (1) ≈ − (cid:18) α π ln eBm (cid:19) L (0) , (1)where α = e is the fine-structure constant. This result is in agreement with more advancedcalculations carried out by Ritus [2], who arrived at the conclusion that the loop expansion ∗ Department of General and Experimental Physics, Herzen State Pedagogical University of Russia, Moykaemb. 48, 191186 St. Petersburg, Russia; e-mail: [email protected] † Institute of Physics, University of S˜ao Paulo, CP 66318, CEP 05315-970 S˜ao Paulo, SP, Brazil; e-mail:[email protected] B ≪ F max , F max = m e exp (cid:18) πα (cid:19) ≈ m e . Shabad and Usov [3] have recently established a more rigid limitation for the maximaladmissible strength of magnetic field, B ≪ m /e , having analyzed the structure of QEDof vacuum in this field, taking into account the interaction of virtual electron-positron pairs.The addition L (1) has been generalized in a certain way to an arbitrary constant field(to arbitrary E ) and is now called the Heisenberg-Euler Lagrangian (HEL), for a review,see [4]. However, its physical meaning for a strong electric field is not completely clear. Theproblem has been approached from another angle by Schwinger [5], who supposed that thevariation of an effective Lagrangian of electromagnetic field should arise due to a non-zero(in external fields) in − out vacuum current of charged particles, without any restrictions onthe intensity of electric field; in doing so, he presented its calculation in a constant externalfield, and obtained precisely the HEL. In the general case, the HEL is complex-valued,its imaginary part determines the probability of pair-creation, as has been confirmed byindependent calculations [6, 7]. Considering the in − out vacuum current, Schwinger hasmade it possible to obtain an elegant expression for his effective Lagrangian in terms ofthe causal (Feynman’s) Green function. Nevertheless, such an effective Lagrangian is notrelated to the problem of mean values, in particular, it does not reproduce the change of thevacuum energy of spinning particles for arbitrary E ; the latter problem has to be formulatedindependently as a mean-value problem, and is expressed via noncausal Green’s functions;see [7].In a strong electric field ( E ≫ m /e, B = 0), the real-valued part of HEL, describing theeffects of vacuum polarization, is given by the right-hand side of (1), where B is replacedby E . From this expression one can extract the negative-valued additive correction E (1) =Re L (1) to the classical Maxwell density of energy, E (0) = L (0) . Consequently, in order thatthe total energy density of electric field E = E (0) + E (1) should be zero, the only effects ofvacuum polarization at the value E ∼ F max are sufficient by themselves. On these grounds,in [8] it was suggested that F max should be also a limiting value of electric field. It turns outthat Im L (1) ∼ E , which can be interpreted as an evidences that the vacuum instability isless than the vacuum polarization. This is not true since the vacuum instability is a nonlocaleffect, being directly dependent on the electric field duration.In our opinion, the most adequate object, whose analysis can answer these questions,is the mean value of the energy-momentum tensor of matter, computed with respect tovarious initial states. A detailed calculation of such a mean value in QED in the one-loopapproximation, taking an exact account of the interaction with the electric background, hasbeen given in [9]. Here, we will only use some of these results, in particular, the meanenergy density of matter for large values of strength and duration of the electric field. Weexamine two cases, when the initial state of the quantized Dirac field is vacuum, and is inthermal equilibrium. In addition, we consider the case of charged bosons and QCD withan external chromoelectric field. We demonstrate that under these conditions, the effect ofparticle-creation is precisely the main reason for the change of the energy of matter. Makinga comparison between the change of the energy density of matter and the energy density ofthe external electric field, which is responsible for this change, we obtain restrictions on theintensity of the external field and its duration, which we call consistency restrictions.II. A constant electric field acting during an infinite time creates an infinite number ofpairs from vacuum even in a finite volume. This is why we choose the external backgroundas a constant electric field acting during a finite period of time; we refer to this field asa T -constant field. The finiteness of field duration is a natural regularization in the givenproblem; on the other hand, it is a necessary physical parameter, which subsequently entersthe consistency restriction on the value of the maximal electric field. The T -constant field2urns on at − T / t > t in and turns off at T / t < t out . We choose the nonzero T -constant field potential A ( t ) as a continuous function of the form A ( t ) = − Et for t ∈ [ t , t ], being constant for t ∈ ( −∞ , t ) and t ∈ ( t , + ∞ ) . The effects of particle-creationby the T -constant field have been studied in detail in [10]. In particular, it was shown thatin case T ≫ ( eE ) − / (cid:2) m /eE (cid:3) (2)all the finite effects caused by particle-creation reach their asymptotic values, whereas thedetails involving the form of switching the field on and off can be neglected. In our calcula-tions, we assume this restriction from below for the time T. Taking the time instant t − T / − h ˆ H i of the spinor field oncondition that its state at the initial instant t in → −∞ should be vacuum. The potentialsof the T -constant field do not depend on the spatial coordinates, which implies that h ˆ H i isproportional to the space volume V . In the one-loop approximation, h ˆ H i = wV , with themean energy density w being independent of the spatial coordinates, w = 12 h , in | (cid:2) ψ ( x ) † , H ψ ( x ) (cid:3) | , in i (cid:12)(cid:12)(cid:12)(cid:12) x = t − , (3)where H is the one-particle Dirac Hamiltonian; ψ ( x ) are the operators of the Dirac fieldin the generalized Furry representation (see, e.g., [7]) obeying the Dirac equation with theexternal background; | , in i is the initial vacuum state in the same representation. Theabove-mentioned choice allows one to take a complete account of the pair-creation effectduring the entire time. Also, since the electric field has not yet been switched off, this allowsus to make a complete study of the vacuum polarization effect. Notice that the initialvacuum | , in i is identical with the vacuum of those free particles that correspond to theinitial potential A = ET / w is obviously real-valued. One can see that it can be represented as w = − (cid:20) lim t → t ′ − tr [( ∂ − ∂ ′ ) S in ( x, x ′ )] + lim t → t ′ +0 tr [( ∂ − ∂ ′ ) S in ( x, x ′ )] (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) x = x ′ ,x = t − , (4)where tr [ · · · ] is the trace in the space of 4 × S in ( x, x ′ ) is the so-called in − in Green function, S in ( x, x ′ ) = i h , in | T ψ ( x )¯ ψ ( x ′ ) | , in i = S c ( x, x ′ ) + S p ( x, x ′ ) , (5)where S c ( x, x ′ ) is Feynman’s causal Green function, while the function S p ( x, x ′ ) is a dif-ference of two Green’s functions, satisfying the homogeneous Dirac equation; see [7]. Thefinal vacuum | , out i is the vacuum of free particles in the generalized Furry picture andcorresponds to the constant potential A = − ET / S in into the c - and p -parts is responsible for the separation of w intothe two respective summands w = w c + w p . One can verify that the expression for w c hasa finite limit at T → ∞ , i.e., it permits a transition to the limit of a constant electric field.Then S c can be presented by a proper-time integral, see [11]. Using this expression, one canreadily verify that w c is expressed in terms of the real-valued part of HEL (at B = 0). Thiscontribution is due to vacuum polarization. In a superstrong electric field, it has the form w c = E ∂ Re L (1) ∂E − Re L (1) ≈ − (cid:18) α π ln eEm (cid:19) L (0) . The contribution w p arises due to particle-creation. It is computed as follows. First ofall, using the general theory of particle-creation (see, [7]), one can represent the function3 p ( x, x ′ ) in the form S p ( x, x ′ ) = i X nm − ψ n ( x ) (cid:2) G ( + | − ) G ( − | − ) − (cid:3) † nm + ¯ ψ m ( x ′ ) . (6)Here, { ± ψ n ( x ) } are the so-called in -solutions of the Dirac equation in a T -constant electricfield, their asymptotics at t ≤ t being stationary states of free electrons (+) and positrons( − ) for the Dirac Hamiltonian with the constant potential A = ET /
2. The matrices G ( ± | ± )(being a matrix generalization of the Bogolyubov coefficients) are defined by decompositionsof the so-called out -solutions in the in -solutions: ± ψ ( x ) = + ψ ( x ) G ( + | ± ) + − ψ ( x ) G ( − | ± ) . Here, { ± ψ n ( x ) } are the out -solutions of the Dirac equation in a T -constant electric field;their asymptotics at t ≥ t describe free particles (electrons (+) and positrons ( − )) with anenergy spectrum defined by the Dirac Hamiltonian with the constant potential A = − ET / G ( ± | ± ) are expressed via the inner products of in - and out -solutions, and obeysome unitary relations following from normalization conditions for these solutions.In a T -constant uniform electric field, we can choose the quantum numbers of particlesas n = ( p , r ), where p is the particle momentum and r = ± G ( ± | ± ) are diagonal, G ( ± | ± ) n,n ′ = δ r,r ′ δ p , p ′ g ( ± | ± ) p ,r . Here, we use the standardvolume regularization, so that δ ( p − p ′ ) → δ p , p ′ . In addition, the differential mean numbersof electrons (equal to the corresponding differential mean number of pairs) created fromvacuum with a given momentum p and spin projections r are given by ℵ p ,r = | g ( − | + ) | . It is easy to verify, using (6), that the function S p , which enters the expression for w p at x ≈ x ′ , can be taken there in the form S p ( x, x ′ ) = − i Z d p X r = ± ℵ p ,r (cid:2) + ψ p ,r ( x ) + ¯ ψ p ,r ( x ′ ) − − ψ p ,r ( x ) − ¯ ψ p ,r ( x ′ ) (cid:3) , x ≈ x ′ . For t ≥ t , the out -solutions ± ψ p ,r ( x ) describe free particles with the quantum numbers p , r and energies ε p ,r = q m + p ⊥ + ( eET / − p ) , p ⊥ = ( p , p , . Taking all the aboveinto account, we obtain w p = 14 π Z d p X r = ± ℵ p ,r ε p ,r . (7)This quantity is the mean energy density of pairs created from vacuum. It can be estimatedin the case of strong electric fields, E & E c and a sufficiently large T as follows. As has beendemonstrated in [10], in case the time T is sufficiently large, T ≫ (cid:0) m + p ⊥ + eE (cid:1) ( eE ) − / , and the longitudinal momenta are restricted by the condition | p | ≤ (cid:16) √ eET / − K p (cid:17) √ eE ,where K p is a sufficiently large arbitrary constant, √ eET ≫ K p ≫ (cid:0) m + p ⊥ (cid:1) /eE , thedifferential mean numbers ℵ p ,r have the form ℵ asy p ,r = exp (cid:18) − π m + p ⊥ eE (cid:19) . For any fixed p ⊥ , the function ℵ p ,r is fast-decreasing for | p | > (cid:16) √ eET / − K p (cid:17) √ eE . Forthis reason, we can disregard the contribution to the integral (7) due to the integrationover such momenta p in comparison with the main contribution, which is defined by thedimensionless parameter eET . The latter parameter, in fact, determines a large integrationdomain over p . In its turn, the exponential decrease of ℵ asy p ,r with the grows of p ⊥ allows oneto ignore the contributions to the integral (7) due to a large p ⊥ /eE & √ eET . Consequently,4n order to evaluate the term which leads in √ eET in integral (7) we can replace ℵ p ,r by ℵ asy p ,r under condition (2), while restricting the limits of integration over momenta bythe region | p | ≤ √ eE (cid:16) √ eET / − K (cid:17) , where K is a sufficiently large arbitrary constant, √ eET ≫ K ≫ m /eE . Having calculated the integral (7) over p ⊥ , we obtain the T -leading term in the form w p = eET ℵ , ℵ = e E T π exp (cid:18) − π m eE (cid:19) . (8)III. We now suppose that the energy density of particles w = w p , arising precisely dueto the action of a T -constant electric field, should be essentially smaller than the density ofthe electric field itself, being equal to the classical Maxwell density of energy, E (0) = E / π .Thus, the condition of a smallness of back-reaction is w p ≪ E / π, which, owing to (8),takes the form of a restriction from above on the dimensionless parameter eET : eET ≪ π e exp (cid:18) π m eE (cid:19) . (9)On the other hand, all the asymptotic formulas have been obtained under condition (2),which restricts the mentioned parameter from below, (cid:2) m /eE (cid:3) ≪ eET . Since π / e ≫
1, there exists a region of values of E and T that satisfies both the inequalities. We notethat time scale from above in (9) is more restrictive than the scale derived from the rate ofpair production, see [12].In case the initial state is in thermal equilibrium at temperature θ , the mean energydensity has an additional term w cθ , which represents, in fact, the work of a T -constant fieldon particles from the initial state, as well as the term w pθ w pθ = − π Z d p X r = ± ℵ p ,r n p ,r ( in ) ε p ,r , n p ,r ( in ) = [exp (˜ ε p ,r /θ ) + 1] − , where ˜ ε p ,r = q m + p ⊥ + ( qET / p ) is the energy of a free in - particle. The latterterm determines a temperature-dependent correction to the energy of particles created fromvacuum; see [9]. The energies of particles that contribute to w cθ in the limit of a large T are mostly determined by a large longitudinal kinetic momentum, with the energy being oforder eET , as well as the energies of particles created at θ = 0 in the expression (8) for w p .Given that, however, the density of initial particles is constant, being determined only bythe initial condition, whereas the density of created particles increases in proportion with T .Therefore, at large T and E, w cθ can be neglected in comparison with w p , and w ≃ w p + w pθ . In case the initial state is in thermal equilibrium at low temperatures θ ≪ eET , thecontribution w pθ turns out to be small in comparison with w p . At high temperatures θ ≫ eET, the energy density has the form w = ( eET / θ ) w p . Thus, the restriction (9) is validboth for the vacuum initial state and for a low-temperature initial thermal state. At hightemperatures we have a weaker restriction:( eE ) T θ ≪ π e exp (cid:18) π m eE (cid:19) . (10)Analogously, one can find restrictions for QED with charged bosons in a T -constantelectric field. At low temperatures, we have eET ≪ π Je exp (cid:18) π m eE (cid:19) , J is the number of the spin degrees of freedom ( J = 1 for scalar particles and J = 3 forvector particles). In the case of high temperatures the restriction has a completely differentcharacter than (10), namely, θT ln (cid:16) √ eET (cid:17) ≪ π Je exp (cid:18) π m eE (cid:19) . One can easily extend these results to D + 1 dimensions, using the corresponding N in (8),taken from Eq. (37) in [10].IV. A similar analysis can be performed in the case of QCD with an electric-like non-Abelian external background. Such a background is a part of the known chromoelectricflux-tube model [13]. At present, the chromoelectric field is associated [14] with an effectivetheory, color glass condensate. Here, we shall derive restrictions on the external backgroundwhich allows one to treat particles created from vacuum still as weakly coupled, owing to theproperty of asymptotic freedom in QCD. To this end, we use the results obtained in [15, 16]for QCD with a constant SU (3) chromoelectric field E a ( a = 1 , . . . , T -constant chromoelectric field; see [15]. The same is valid atany finite temperature; therefore, for our purposes it is sufficient to take into account onlythe gluon contribution. It has been demonstrated in [15] that the p ⊥ -distribution density n gluon p ⊥ of gluons produced from vacuum with all the possible values p and the quantumnumbers that characterize the inner degrees of freedom can be presented as follows: n gluon p ⊥ = 14 π X j =1 T q ˜ E ( j ) ℵ ( j ) p , ℵ ( j ) p = exp − π p ⊥ q ˜ E ( j ) ! , (11)where ˜ E ( j ) are positive eigenvalues of the matrix if abc E c for the adjoint representation of SU (3), and q is the coupling constant. The ( j )-terms in (11) can be interpreted as thoseobtained for Abelian-like electric fields ˜ E ( j ) , respectively. Then, the total energy densityof gluons created from vacuum by the field ˜ E ( j ) is determined by integrals of the kind (7).Taking into account that maxima of the fields are restricted by the condition ˜ E ( j ) ≤ √ C ( C = E a E a is a Casimir invariant for SU (3)) and the relation P j =1 ˜ E j ) = 3 C /
2, one canfind that at low temperatures θ ≪ q √ C T the consistency restriction for the dimensionlessparameter q √ C T has the form q p C T ≪ π / q . As in the case of QED, this restriction must be accompanied by a restriction from below,1 ≪ q √ C T , which is related to the fact that all the asymptotic expressions have beenobtained for sufficiently large values of T . Therefore, the T -constant SU (3) chromoelectricfield approximation is consistent during the period when the produced partons can be treatedas weakly coupled, due to the property of asymptotic freedom in QCD. At high temperatures, θ ≫ q √ C T , the consistency restriction is far more rigid: θT ln (cid:16) q p C T (cid:17) ≪ π / q . The above established consistency restrictions determine, in fact, the time scales fromabove of depletion of an electric field due to the backreaction.
Acknowledgement
S.P.G. thanks FAPESP for support and Universidade de S˜aoPaulo for hospitality. D.M.G. acknowledges the permanent support of FAPESP and CNPq.6 eferences [1] W. Heisenberg and H. Euler, Z. Phys. , 714 (1936).[2] V. I. Ritus, Sov. Phys. JETP
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