Nonperturbative QED Effective Action at Finite Temperature
aa r X i v : . [ h e p - t h ] J u l Nonperturbative QED Effective Action at Finite Temperature
Sang Pyo Kim ∗ Department of Physics, Kunsan National University, Kunsan 573-701, Korea andAsia Pacific Center for Theoretical Physics, Pohang 790-784, Korea
Hyun Kyu Lee † and Yongsung Yoon ‡ Department of Physics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea (Dated: May 16, 2018)We propose a novel method for the effective action of spinor and scalar QED at finite temperaturein time-dependent electric fields, where charged pairs evolve in a nonadiabatic way. The imaginarypart of the effective action consists of thermal loops of the Fermi-Dirac or Bose-Einstein distributionfor the initial thermal ensemble, weighted with factors of the Bogoliubov coefficients for quantumeffects. And the real part of the effective action is determined by the mean number of producedpairs and vacuum polarization at zero temperature. In the weak-field limit, the mean number ofproduced pairs is shown twice the imaginary part. We explicitly find the finite temperature effectiveaction in a constant electric field.
PACS numbers: 12.20.-m, 13.40.-f, 12.20.Ds, 11.15.Tk
I. INTRODUCTION
In a strong electromagnetic field the vacuum becomes polarized due to the interaction of the electromagnetic fieldwith virtual charged pairs from the Dirac sea. The effective actions in electromagnetic fields have been continuouslyinvestigated since the early work by Sauter, Heisenberg and Euler, and Weisskopf [1] and later on the proper-time inte-gral for the effective action by Schwinger [2]. The Heisenberg-Euler effective action exhibits both vacuum polarizationand pair production, and has many physical applications (for a review and references, see Refs. [3–5]).The effective actions at zero temperature have been a nontrivial task for general profiles of electromagnetic fields.As a strong electric field always creates pairs from the vacuum, the corresponding effective action contains not onlythe real part responsible for vacuum polarization but also the imaginary part for the decay of vacuum. Thus, thequantum field theory for strong electric fields should properly handle pair creation from the vacuum. The effectiveactions have been found for a pulsed-electric field of Sauter type in the resolvent method [6] and in the evolutionoperator method [7], and the effective action could be found for a spatially localized electric field [8].However, the QED effective action at finite temperature in electric field backgrounds has been an issue of constantinterest and controversy, partly because different formalisms give conflicting results [9–14] and partly because thethermal effects may be important to astrophysical objects involving strong electromagnetic fields. In fact, mostmethods for finite temperature field theory may not be directly applied to electric fields due to pair creation from thevacuum. The one-loop energy momentum tensor of fermions in an initial thermal ensemble was found in a constantelectric field background [15]. Recently the closed-time formalism has been employed to find the QED effectiveaction at finite temperature in 0+1 dimension [16]. The enhancement of pair production by the electric field at finitetemperature is also found [17].The purpose of this paper is two-fold: we first propose a novel method for the effective action at finite temperaturefor time-dependent quantum fields and then find the QED effective action in strong electric field backgrounds. Atzero temperature the effective action is the scattering amplitude between the out-vacuum and the in-vacuum, whichis the expectation value of the evolution operator with respect to the in-vacuum [7]. To extend the in- and out-stateformalism to finite temperature, we first express the evolution operator in terms of the Bogoliubov coefficients andthen find the effective action as the expectation value of the evolution operator with respect to the ‘thermal vacuum’.It turns out that the finite temperature effective action is the trace of the evolution operator weighted with the initialthermal ensemble of fermions or bosons, which is equivalent to the ‘thermal vacuum’ expectation value of the evolutionoperator in thermofield dynamics. The formalism may be applicable to other time-dependent quantum field, whose ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
Hamiltonian H ( t ) evolves in a nonadiabatic way, that is, out of equilibrium, so that e − βH ( t ) is not the density operatorsatisfying the Liouville-von Neumann equation [18].We apply the new method to the QED effective action at finite temperature in time-dependent electric fields. TheQED effective action consists of the zero-temperature part, the part for thermal and vacuum fluctuations, and thefinite temperature part without the electric field. The logarithm of the Bogoliubov coefficient plays a role of complexchemical potential in the complex thermal distribution for the effective action. The real and the imaginary parts ofthe effective action have an expansion in terms of the Fermi-Dirac or Bose-Einstein distribution and the chemicalpotential. Finally we find the effective action in a constant electric field and discuss it for the Sauter-type electricfield.The organization of this paper is as follows. In Sec. II, we propose a new method to find the finite temperatureeffective action for time-dependent systems. The effective action is given by the trace of the evolution operator andthe initial density operator, which is equivalent to the expectation value of the evolution operator with respect to thethermal vacuum of thermofield dynamics. In Sec. III, we find the effective action in spinor and scalar QED in electricfields and then elaborate an expansion scheme in terms of the Fermi-Dirac and Bose-Einstein distributions. In Sec.IV, we apply the formalism to find the effective action in a constant electric field. Finally, we discuss the controversialissue of thermal effects on pair-production rate in Sec. V. II. FINITE TEMPERATURE EFFECTIVE ACTION
We consider both spinor and scalar QED with the time-dependent gauge field A k ( t ), which generates a constantor time-dependent electric field. For a pulse-like electric field acting for a finite period of time, the ingoing and theoutgoing vacua are well-defined at t in = −∞ and t out = ∞ , for which we may choose a gauge A k ( t in ) = 0 such thatthe ingoing vacuum | , t in i is nothing but the Minkowski vacuum | i M . In the case of a constant electric field, we mayuse the asymptotic state as in Ref. [7]. The particle and antiparticle have the momentum k and the spin state σ ,whose annihilation operators are denoted by a k σ, in and b k σ, in at t in = −∞ and a k σ, out and b k σ, out at t out = ∞ , where σ = ± / σ = 0 for scalar QED. Then, the in- and out-vacua are related through the Bogoliubovtransformations [19] a k σ, out = µ k σ a k σ, in + ν ∗ k σ b † k σ, in = U k σ a k σ, in U † k σ ,b k σ, out = µ k σ b k σ, in + ν ∗ k σ a † k σ, in = U k σ b k σ, in U † k σ , (1)where U k σ is the evolution operator, whose form in terms of µ k σ and ν k σ is explicitly given in Ref. [7], and thecoefficients satisfy the relation | µ k σ | + ( − | σ | | ν k σ | = 1 . (2)In the in- and out-state formalism elaborated in Ref. [7], the in- and out-vacua are annihilated by a k σ, in / out and b k σ, in / out . In fact, the in- and out-vacua are the tensor product of the zero-number states for all k and σ . Theevolution operator transforms the in-vacuum to the out-vacuum as | , out i = U | , in i , where U is also the tensorproduct of each U k σ , that is, U = Q k σ U k σ . The zero-temperature effective action per unit volume and per unit timeis obtained from the scattering amplitude [7] e i R d xdt L eff = h , out | , in i = h , in | U † | , in i . (3)Now we extend the zero-temperature effective action to the finite-temperature one for the system with the initialdensity operator, ρ in = Y k σ h e − n k σ βE ( k ,σ ) | n k σ , in ih n k σ , in | i , (4)where β = 1 /k B T , k B being the Boltzmann constant, and E ( k , σ ) is the energy for massive charged particles andantiparticles.[26]In finite temperature field theories for static systems, one employs either the partition function Z ( β ) = Tr( ρ ) orthe thermal expectation value h O i β = Tr( Oρ ) / Tr( ρ ), which is equivalent to the ‘thermal vacuum’ expectation value, h O i β = h , β, in | O | , β, in i [20–22]. However, finite temperature field theories should be modified for time-dependentsystems since they nonadiabatically evolve the initial states. For such time-dependent quantum fields we propose thefinite-temperature effective action e i R d xdt L eff ( T ) = Tr( U † ρ in )Tr( ρ in ) . (5)In fact, the effective action (5) is equivalent to h , β, in | U † | , β, in i for the ‘thermal vacuum’ [20, 21] | , β, in i ≡ Z − / Y k σ hX n k σ e − n k σ βE ( k ,σ ) / | n k σ , in i ⊗ | ˜ n k σ , in i i , (6)where | ˜ n k σ , in i denotes the state for a noninteracting fictitious system of the extended Hilbert space. In fact, Eq. (5)has the correct zero-temperature limit (3). The effective action (5) may be applied to other time-dependent quantumfields as well as QED, which evolve in a nonadiabatic way. III. QED EFFECTIVE ACTION AT T We now advance a method to compute the QED effective action (5) in electric fields. Evaluating Eq. (5), we obtainthe effective action at finite temperature per unit volume and per unit time, L eff ( T, E ) = ( − | σ | i X k σ h − βz k σ + ln(1 + ( − | σ | e − β ( ω k − z k σ ) ) − ln(1 + ( − | σ | e − βω k ) i , (7)where ω k = q m + k ⊥ + ( k k + qA k ) and1 µ ∗ k σ = e βz k σ , ( z k σ = z r ( k , σ ) + iz i ( k , σ )) . (8)The summation is over all possible states such as momenta and spin states. Each term in Eq. (7) has the followinginterpretation: the first term is the effective action L eff ( T = 0 , E ) at zero temperature, the second term is the combinedeffect of thermal and quantum fluctuations, while the last term is the subtraction of the effective action (potentialenergy) L eff ( T, E = 0) at finite temperature without the electric field. From now on we subtract the zero-temperaturepart from the effective action and let ∆ L eff ( T, E ) = L eff ( T, E ) − L eff (0 , E ) . (9)Note that z k σ ( E ), which depends on the electric field E and z k σ (0) = 0, plays a role of complex chemical potential,as will be explained below.Further, we elaborate an expansion scheme for the effective action in terms of the Fermi-Dirac or Bose-Einsteindistribution and z k σ . First, the imaginary part of the effective action (9) can be expanded asIm(∆ L eff ) = ( − | σ | X k σ ∞ X j =1 [( − | σ | n F / B ( k )] j j [( e βz k σ − j + ( e βz ∗ k σ − j ] , (10)where n F / B ( k ) denotes either the Fermi-Dirac distribution n F ( k ) = 1 / ( e βω k +1) for spinor QED or n B ( k ) = 1 / ( e βω k −
1) for scalar QED. Second, the real part of the effective action (9) is given byRe(∆ L eff ) = ( − | σ | X k σ ∞ X j =1 [( − | σ | e − β ( ω k − z r ( k ,σ )) ] j j sin( jβz i ( k , σ )) . (11)The thermal factors in Eqs. (10) and (11) correspond to thermal loops in the diagrammatic representation, which areweighted with factors from quantum fluctuationsNow, we give physical interpretations for the effective action. In the weak-field limit ( qE ≪ m ) where βz i ( k ) ≪ L eff ) ≈ X k σ βz i ( k , σ ) e β ( ω k − z r ( k ,σ )) + ( − | σ | , (12)while the imaginary part (10) approximately leads to2Im(∆ L eff ) ≈ ( − | σ | X k σ | ν k σ | n F/B ( k ) . (13)Thus, the imaginary part may be regarded as the pair-production rate due to thermal and quantum effects. Thethermal effects suppress the fermion pair production due to the Pauli blocking but enhance the boson pair productiondue to the Bose-Einstein condensation, as expected. In Ref. [19], the mean number of produced pairs with a given mo-mentum k at T is given by ¯ N sp ( T ) = P k σ | ν k σ | tanh( βω k /
2) for spinor QED and ¯ N sc ( T ) = P k σ | ν k σ | coth( βω k / N = ( ¯ N ( T ) − ¯ N (0)) /
2, of one species of particle or antiparticle due to thermaleffects approximately satisfies the relation between the mean number and the imaginary part:∆ ¯ N = X k σ | ν k σ | n F/B ( k ) ≈ L eff ) . (14)The relation between the mean number of produced pairs and twice of the imaginary part also holds at T = 0 in theweak-field limit [7].A few comments are in order. The series of the real part (11) may be summed as [23]Re(∆ L eff ) = X k σ arctan h sin( βz i ( k )) e β ( ω k − z r ( k )) + ( − | σ | cos( βz i ( k )) i . (15)Using L eff (0 , E ) = ( − | σ | i P k σ βz k σ from Eq. (7) and the Bogoliubov relation (2), we have the real and imaginaryparts βz r ( k , σ ) = ( − | σ | Im( L eff (0 , E )) = − | σ | − | σ | | ν k σ | ) ,βz i ( k , σ ) = ( − | σ | Re( L eff (0 , E )) . (16)Then the effective action (15) at finite temperature can be written in terms of the mean number, the vacuumpolarization, and the thermal distribution asRe(∆ L eff ) = ( − | σ | X k σ arctan h sin(Re( L eff (0 , E ))) e βω k (1 + ( − | σ | | ν k σ | ) | σ | + ( − | σ | cos(Re( L eff (0 , E ))) i . (17)Another interesting observation is that the mid-term in Eq. (7), W eff ( T, E ) = ( − | σ | i X k σ ln(1 + ( − | σ | e − β ( ω k − z k σ ) ) , (18)is reminiscent of the potential energy [24] and carries both thermal and quantum effects. Equation (18) suggests z k σ as the chemical potential and the variation with respect to z k σ yields the Fermi-Dirac or Bose-Einstein distribution. IV. APPLICATIONS
In this section we find the QED effective action in a constant electric field and discuss a Sauter-type electric field, E ( t ) = E sech ( t/τ ) in Ref. [7]. In the time-dependent gauge, A k ( t ) = − Et for the constant electric field and A k ( t ) = − E τ (1 + tanh( t/τ )) for the Sauter-type electric field, the energy of charged particles explicitly depend ontime. We will take the weak-field limit ( qE ≪ m ), where the real part (12) and the imaginary part (13) of theapproximate effective action can be worked out for the constant electric field and in principle for the Sauter-typeelectric field.In the constant electric field, the state along the direction of the electric field is asymptotically determined, whosemomentum integral gives a factor qE/ (2 π ) [7]. Using the mean number of produced pairs, | ν k σ | = e − π m k ⊥ qE , (19)which is independent of the spin states, the imaginary part (13) is given byIm(∆ L eff ( T, E )) ≈ | σ | (cid:16) qE π (cid:17) e − πm qE ∞ X n =0 ( − | σ | ( n +1) m e − βm ( n +1) πm + βmqE ( n + 1) × h βm ( qE ) (2 πm + βmqE ( n + 1)) − βm ( qE ) (2 πm + βmqE ( n + 1)) + · · · i . (20)The factor in front of the summation is the leading term of the imaginary part at zero temperature. Further, in thelow-temperature limit ( βm ≫ L eff ( T, E )) ≈ | σ | (cid:16) qE π (cid:17) e − πm qE h ( − | σ | m qE e − βm βm + πm qE i . (21)In the special case of thermal effect dominance, neglecting all terms of m/βqE , the first series in Eq. (20) approximatelyleads to Im(∆ L eff ( T, E )) ≈ | σ | (cid:16) qE π (cid:17) e − πm qE h − m βm ln(1 + ( − | σ | e − βm ) i . (22)Similarly, using Re( L eff (0 , E )) in Ref. [7], the real part (12), for instance, of spinor QED is given byRe(∆ L speff ( T, E )) ≈ − qE π m π ∞ X n =0 ( − n ∞ X l =2 l − | B l | (2 l )! (cid:16) qE π (cid:17) l − m l − × h e − βm ( n +1) (cid:16) Ψ(1 , − l, α ) + βm ( n + 1)4 Ψ(3 , − l, α ) − βm ( n + 1)8 Ψ(4 , − l, α ) + · · · (cid:17) − e − βmn (cid:16) Ψ(1 , − l, γ ) + βmn , − l, γ ) − βmn , − l, γ ) + · · · (cid:17) + · · · i , (23)where B l is the Bernoulli number, Ψ denotes the second confluent hypergeometric function [25], and α = βm ( n + 1)2 , γ = βm ( n + 1)2 + πm qE . (24)In the low-temperature limit ( βm ≫ l = 2 in Eq. (23) leads toRe(∆ L speff ( T, E )) ≈ − (2 π ) m (cid:16) qE π (cid:17) h ( −
1) 3 βm ln(1 + e − βm ) i . (25)Here the factor in front of the square bracket is the real part at zero temperature. The real part of effective action inscalar QED may be found in a similar way.Finally, we discuss the Sauter-type electric field. The charged particle has the free energy ω k , in = √ m + k beforethe onset of the electric field while it has ω k , out = p m + k ⊥ + ( k z − qE τ ) after the completion of the interaction.At zero temperature, the mean number of produced pairs, Eqs. (68) and (83), and the vacuum polarization, Eqs.(66) and (80) of Ref. [7], which depend on ω k , in , ω k , out , and λ = p ( qE τ ) − (2 | σ | − /
4, lead to the effectiveaction, Eqs. (12) and (13). To find analytical expressions for the effective action would be more complicated than theconstant electric field, which will be addressed elsewhere.
V. CONCLUSION
In this paper we have advanced a new method for the finite-temperature effective action for time-dependent quantumfields and have studied the one-loop effective action of spinor and scalar QED at finite temperature in a constant ortime-dependent electric fields. External electric fields make the vacuum unstable against pair production, which isa consequence of the out-vacuum differing from the in-vacuum. The instability enforces a careful application of thefinite-temperature field theory to time-dependent quantum fields. The finite-temperature effective action (5) is givenby the trace of the initial thermal ensemble evolved by the time-evolution operator, which is equivalent to the thermalvacuum expectation value of the evolution operator in thermofield dynamics.The imaginary part (10) of the effective action exhibits factorization into thermal factors and quantum factors,which correspond to thermal loops in the diagrammatic representation with vertices of the external electric field. Inthe weak-field limit ( qE ≪ m ), twice of the imaginary part is the mean number of produced pairs, as shown in Eq.(13). However, the thermal and quantum effects are intertwined in the real part of the effective action, Eqs. (11),(15), and (17). In fact, the finite-temperature effective action (17) is determined by the vacuum polarization at zerotemperature, the mean number of produced pairs, and thermal distribution. In the weak-field and lower-temperaturelimits, the leading factors of the real and imaginary parts, (22) and (25), of the effective action in a constant electricfield are proportional to those at zero temperature and the potential energy for the rest mass in spinor and scalarQED.Our results show many interesting aspects. First, the imaginary part (10) of the thermal contribution does notvanish for any non-zero electric field, which implies thermal effects on pair production and thus may resolve thecontroversial issue of thermal effects on pair production: thermal effects are shown to exist in Refs. [10, 13], whilein Refs. [11] no thermal effects are found. Further, in the weak-field limit for small pair production, twice of theimaginary parts (14) are the pair-production rate at T , which was shown in Ref. [19]. Though our in- and out-stateformalism differs from the imaginary-time formalism, the imaginary part (13) in the weak-field limit is the pair-production rate times the Fermi-Dirac or Bose-Einstein distribution, which may correspond to two-loop dominancein Ref. [14]. Second, the Bogoliubov coefficient (8), which is responsible for vacuum polarization at T = 0, plays arole of chemical potential in the effective action (7) and in the potential energy (18) at T . In fact, the variation ofthe effective action with respect to the chemical potential yields the Fermi-Dirac or the Bose-Einstein distribution. Acknowledgments
The authors would like to thank Holger Gies for useful information and comments, Ismail Zahed for useful dis-cussions, and Sergey Gavrilov for useful comments. S. P. K. would like to thank W-Y. Pauchy Hwang for the warmhospitality at National Taiwan University, where part of this paper was written. The work of S. P. K. was supportedby the Korea Research Foundation (KRF) Grant funded by the Korea Ministry of Education, Science and Technology(2009-0075-773) and the work of H. K. L. was supported by the World Class University Program (R33-2008-000-10087-0) of the Korea Ministry of Education, Science and Technology. [1] F. Sauter, Z. Phys. , 742 (1931); W. Heisenberg and H. Euler, Z. Physik , 714 (1936); V. Weisskopf, K. Dan. Vidensk.Selsk. Mat. Fys. Medd. XIV , No. 6 (1936).[2] J. Schwinger, Phys. Rev. , 664 (1951).[3] W. Dittrich and H. Gies, Springer Tracts Mod. Phys. , 1 (2000).[4] G. V. Dunne, “Heisenberg-Euler Effective Lagrangians: Basics and Extensions,” From Fields to Strings: CircumnavigatingTheoretical Physics , edited by M. Shifman, A. Vainshtein, and J. Wheater, (World Scientific, Singapore, 2005), Vol. I, pp.445-522, hep-th/0406216.[5] R. Ruffini, G. Vereshchagin, and S.-S. Xue, Phys. Rep. , 1 (2010).[6] G. V. Dunne and T. Hall, Phys. Rev. D , 105022 (1998).[7] S. P. Kim, H. K. Lee, and Y. Yoon, Phys. Rev. D , 105013 (2008).[8] S. P. Kim, H. K. Lee, and Y. Yoon, “Effective Action of QED in Electric Field Bakgrounds II: Spatially Localized Fields,”[arXiv:hep-th/0910.3363].[9] P. Elmfors, D. Persson, and B.-S. Skagerstam, Phys. Rev. Lett. , 480 (1993); P. Elmfors, P. Liljenberg, D. Persson, andB.-S. Skagerstam, Phys. Rev. D , 5885 (1995).[10] M. Loewe and J. C. Rojas, Phys. Rev. D , 2689 (1992).[11] P. Elmfors and B.-S. Skagerstam, Phys. Lett. B , 141 (1995); Erratum, Phys. Lett. B , 330 (1996).[12] A. K. Ganguly, P. K. Kaw, and J. C. Parikh, Phys. Rev. C , 2091 (1995).[13] J. Hallin and P. Liljenberg, Phys. Rev. D , 1150 (1995).[14] H. Gies, Phys. Rev. D , 105002 (1999); Phys. Rev. D , 085021 (2000).[15] S. P. Gavrilov and D. M. Gitman, Phys. Phys. D , 045017 (2008).[16] A. Das and J. Frenkel, Phys. Lett. B , 195 (2009); Phys. Rev. D , 125039 (2009).[17] A. K. Monin and A. V. Zayakin, JETP Lett. , 709 (2008); A. Monin and M. B. Voloshin, Phys. Rev. D , 025001(2010).[18] S. P. Kim and C. H. Lee, Phys. Rev. D , 125020 (2000).[19] S. P. Kim and H. K. Lee, Phys. Rev. D , 125002 (2007); S. P. Kim, H. K. Lee, and Y. Yoon, Phys. Rev. D , 045024(2009).[20] Y. Takahashi and H. Umezawa, Collect. Phenem. , 55 (1975) [reprinted in Int. J. Mod. Phys. B (1996), 1755].[21] H. Umezawa, Advanced Field Theory: Micro, Macro and Thermal Physics (AIP, New York, 1993).[22] A. Das,
Finite Temperature Field Theory , (World Scientific, Singapore, 1997).[23] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev,
Integrals and Series (Gordon and Breach Science Publishers, TheNetherlands, 1998) Vol. 1, formula 5.4.9-12.[24] J. I. Kapusta,
Finite-temperature field theory (Cambridge University Press, New York, 1989).[25] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev,
Integrals and Series (Gordon and Breach Science Publishers, TheNetherlands, 1998) Vol. 1, formula 2.3.6-9, which reads R ∞ x a − e − px ( x + z ) b dx = Γ( a ) z a − b Ψ( a, a + 1 − b, pz ).[26] The unit system of c = ~ = k B = 1 is used, where qE/m and βmβm