Reply to the "Comment on Fermion production in a magnetic field in a de Sitter Universe"
aa r X i v : . [ g r- q c ] F e b Reply to the ”Comment on Fermion production in a magneticfield in a de Sitter Universe”
Cosmin Crucean and Mihaela-Andreea B˘aloi
West University of Timi¸soara,V. Parvan Avenue 4 RO-300223 Timi¸soara, Romania
Abstract
In this paper we study the problem of gauge invariance of the first order transition amplitudesin de Sitter QED in Coulomb gauge. We consider the gauge transformations which preserve theCoulomb gauge, that contain the gradient of the gauge function. The final results prove that thefirst order transition amplitudes do not change at a gradient transformation of the vector potentialbecause the only allowed transformation is Λ = 0. Our results suggest that the remarks made inthe comment by Nicolaevici and Farkas
Phys. Rev. D , are not directly applicable to the resultsin our paper since their proposed gauge transformations do not preserve the Coulomb gauge. PACS numbers: 04.62.+v . INTRODUCTION The problem of gauge invariance of the transition amplitudes in Minkowski QED was thesubject of an intense debate some decades ago [3–7, 9–11, 19–21, 23, 24]. While some of theauthors argue that the surface terms that appear in the variation of the amplitude in the firstorder, at a gauge transformation of the potential A µ , are vanishing due to the fact that thefields vanish if we impose boundary conditions [7, 11, 19–22], others prove that the problemof gauge invariance of the amplitudes must be discussed in relation with the renormalizationof the Minkowski QED [3–5], since the renormalization constants also depend on the chosengauge. In quantum field theory on curved spacetime [8] the problem of gauge invariance wasnot studied in detail, and the existing results do not allow us to reach definitive conclusionsregarding the gauge dependence of the amplitudes. The perturbative QED on the de Sitterspacetime was constructed in [14] where it is shown that that a mandatory condition forquantifying the whole theory is to choose and fix the Coulomb gauge. Moreover, this seemsto be a specific feature of the de Sitter geometry as long as the massless limit of the freeProca field on this background [17], gives just the free Maxwell field in Coulomb gauge.Another result obtained recently [15], shows that there are indications that the transitionamplitudes in the first order of perturbation theory are gauge dependent in de Sitter space-time. The comment [16] to our paper [2] is a continuation of the result obtained in [15],and the cause of the gauge dependence of the amplitudes was indicated to be contained inthe temporal part of the Dirac current which loses it’s oscillatory behaviour in the infinitefuture. In this Reply we want to reconsider the problem of gauge invariance of transitionamplitudes in de Sitter QED. This is done by discussing the situation from our paper [2],where we use Coulomb gauge and by making some observations about the results obtainedin the comment [16].The paper is organized as follows: in the second section we discuss the problem of thetransition amplitudes when the Coulomb gauge is used. In the third section the problem ofgauge invariance is discussed in relation with the vector potential decomposition in perpen-dicular and parallel components. In section four we discuss the problem of gauge invariancein relation with the renormalization of the theory and our conclusions are presented insection five. 2 I. AMPLITUDES IN COULOMB GAUGE
The line element [1] which describes the de Sitter universe is: ds = dt − e Ht d~x = 1( Hη ) ( dη − d~x ) (1)where H is the expansion factor and H >
0, while η = − H e − Ht is the conformal time.It is known that Minkowski QED is a gauge invariant theory in the sense that anytransformations of the type: A µ → A µ + ∂ µ Λ ψ → e ie Λ ψ (2)leaves invariant the field equations, where Λ is the gauge function which is a scalar functiondependent both on time and spatial coordinates [18, 19, 21–23]. The second transformationfrom (2) refers to the matter fields and this second transformation must be accompaniedby the transformation of the potential in the case when the electromagnetic field is coupledwith a matter field for leaving unchanged the field equations. We will assume the usualboundary conditions in space and time, such that the gauge function vanishes at infinity[20, 22]: Λ → , (3)for t → ∞ , x → ∞ .Before proceeding in our analysis let us make a few observations related to the Coulombgauge. It true that in our paper about the fermion production in the field of the magneticdipole [2], we omit to mention that we use Coulomb gauge. The reasons for using Coulombgauge will be detailed in what follows and an extended discussion can also be found in ourprevious papers [13, 14, 26]. The significance of Coulomb gauge in de Sitter geometry isvery important if we take into account the conformal invariance of the Maxwell equations.Then it seems that the Coulomb gauge is the only gauge which opens the way to conformallyrelate the whole theory of Maxwell field written in the chart with conformal time { η, ~x } , tothe usual electrodynamics from Minkowski spacetime. The Lorentz condition is conformallyinvariant only in Coulomb gauge in de Sitter geometry, and this allows us to obtain thesolutions of the free Maxwell equations in the helicity-momentum basis as in flat spacetheory [26]. The canonical quantization of the free Maxwell field can be performed in3oulomb gauge where the Lorentz condition becomes conformally invariant [26] ,as we pointout above. This means that it is useful to maintain this gauge for constructing the theoryof interacting fields [14]. However there is a lot of work to be done if we want to speakabout measurable quantities in this geometry, but an important result is that we recover inthe limit of zero expansion factor the results from flat space QED. This means that in thelimit of zero expansion factor the transition amplitudes and probabilities computed in deSitter QED in Coulomb gauge, reduce to those from Minkowski theory. How the theory offree Maxwell field look in other gauges in de Sitter geometry it is not known to the best ofour knowledge and therefore further studies must be done for fully understand the theoryof Maxwell field in this geometry. For the above mentioned reasons we restrict ourselves touse the Coulomb gauge for studding the perturbative QED in de Sitter geometry.All the details related to the construction of perturbative QED in de Sitter geometryusing the Coulomb gauge can be found in [14], and we remind here only the main steps.The construction of the de Sitter QED in Coulomb gauge starts with the Lagrangian theorythat gives the field equations and the principal conserved observables of the interactingfields [14]. Then the equal time commutators and anticommutators are postulated and theequation of the time-dependent evolution operator are derived, obtaining the perturbationseries of the scattering operator in terms of free fields. This is generated by the interactionHamiltonian which does not depend on the Coulomb potential. So the Coulomb gauge allowsa natural quantization separating the Coulomb potential [14]. Finally the asymptotic fieldsare defined and we obtain the in − out amplitudes by using the reduction formalism and thescattering operator [14].Thus, one first chooses a gauge for solving the free Maxwell equations, then the quan-tization can be done. The theory of fields interactions is also constructed by choosing agauge since we have to find the solutions for the interacting fields equations, which in deSitter geometry are strongly dependent on gauge [14]. Then the perturbation theory canbe constructed and finally we can obtain the expressions for the transition amplitudes inany order. It is then clear that once a gauge is fixed and the quantization procedure isdone, one could not do a gauge transformation in the transition amplitudes for passing toanother gauge. In these circumstances any gauge transformations related to the electromag-netic potentials that are allowed are just those that preserve the chosen gauge in which thetheory of interacting fields was constructed and the quantization was done. The authors4f the comment [16] to our paper seem to miss this important observation, since the onlyallowed gauge transformations are those that preserve the Coulomb gauge and their discus-sion should consider these transformations only, for commenting on our paper [2]. Instead atthe beginning of the comment the authors leave the impression that they work in Coulombgauge taking into account that they consider A i = 0 , A = 0 and then they choose to madethe discussion in another gauge such that the amplitude is defined in general with A µ . Thiscomment should construct the theory of free electromagnetic field in other gauge, in de Sit-ter geometry, then make the quantization and finally construct the QED in this new gauge.Therefore a comment to our papers [2, 13, 14] should prove the following: first take anothergauge and solve the Maxwell equations and then construct the perturbative QED in thisnew gauge. Then in this new gauge take a potential which give the same dipolar magneticfield (as the one used in our paper [2]), and compute the first order amplitude correspondingto the fermion pair production and finally compare the results from the this new gauge withour results obtained in Coulomb gauge. To conclude, the comment [16] did not prove thatthe amplitudes from our paper [2], are gauge dependent. In what follows we will clarifywhat it means to do a gauge transformation in our case [2].The amplitude of pair production in an external magnetic field from our paper [2], willbe further considered. In [2], the vector potential that describe the magnetic field producedby a dipole was taken as ~A = ~ M × ~x | ~x | e − Ht , A = 0 , (4)where ~ M is the magnetic dipole moment. We also observe that ∇ ~A = 0 , A = 0. Thefirst order transition amplitude corresponding to pair creation in external field, assumingthe minimal coupling, is [2]: A e − e + = − ie Z d x [ − g ( x )] / ¯ U ~p, λ ( x ) ~γ · ~A ( x ) V ~p ′ , λ ′ ( x ) , (5)where U ~p, λ ( x ) , V ~p ′ , λ ′ ( x ) are the solutions of the Dirac equation in momentum basis in deSitter geometry [12].Our analysis in de Sitter case is done preserving the Coulomb gauge, and the same istrue for our paper [2]. Let us denote by ∇ α the covariant derivative, with α = 0 , , , ∂ i A i = 0 from Minkowski space is replaced by the vanishing of the covariantderivative in de Sitter case ∇ i ( √− gA i ) = 0 [14, 26], but it is sufficient to apply the covariant5erivative only on A i since √− g depends only on time and we obtain: ∇ i A i = 0 = ∂ i A i + Γ iiα A α = ∂ i A i + Γ iij A j + Γ ii A (6)Considering the de Sitter line element (1), with g = 1 , g ij = − δ ij e Ht , we obtain thatΓ iij = 0 , Γ ii = H and the above equation becomes: ∇ i A i = ∂ i A i + HA = 0 , (7)since A = 0 this implies ∂ i A i = 0.It is known that further gauge transformation that preserve the Coulomb gauge conditioncan be made, and these gauge transformations which contain the components of A µ can bewritten as: A i → A i + ∂ i Λ A → A + ∂ Λ , (8)since A = 0 we observe that the condition ∂ t Λ = 0 is mandatory (this conclusion can bereached following similar arguments like in the Minkowski theory, see [10]).Let us apply the covariant derivative to the potential transformation given in equation(8): ∇ i A i → ∇ i A i + ∇ i ( ∂ i Λ) (9)and compute ∇ i ( ∂ i Λ) to obtain: ∇ i ( ∂ i Λ) = ∂ i ∂ i Λ + Γ iij ∂ j Λ + Γ ii ∂ Λ = ∂ i ∂ i Λ + H∂ t Λ . (10)Since we impose the condition ∇ i A i = 0, we observe that we must take ∇ i ( ∂ i Λ) = 0 forpreserving the Coulomb gauge and finally obtain the equation for Λ in de Sitter geometry: ∂ i Λ − He Ht ∂ t Λ = 0 . (11)A similar situation is encountered in Minkowski QED, where for preserving the Coulombgauge the condition ∂ i Λ = 0 is imposed [20, 22, 23, 25]. In equation (11), we observethat Λ is a time independent function, i.e. ∂ t Λ = 0, as shown above and the first term ofthe equation reproduce the situation from flat space case, giving for the gauge function anequation of the Laplace type [20, 22, 23, 25]: △ Λ = 0 . (12)6rom the analysis above we observe that in the Coulomb gauge the gauge function Λdepends only on spatial coordinates. But the above equation (12) has a nice property: itssolutions are unique [23]. In other words if we can find a solution to Laplace equationwhich satisfies the boundary conditions then it is clear that this is the only solution. Thephysical criterion that the gauge function must accomplish, is for it to vanish when thespatial distances and the time become infinite, where the Dirac fields and the potential usedin our calculations also vanish [19, 20, 22, 23]. In these circumstances the unique solutionthat accomplishes these criteria is [18, 22, 23]:Λ = 0 (13)and no gauge arbitrary remains in this case. So our transition amplitudes of fermion pro-duction in the magnetic field on de Sitter spacetime obtained in [2], do not change if weadd to the potential the gradient of the gauge function as long as we impose to remain inthe Coulomb gauge. To the best of our knowledge how physics looks in de Sitter geometryif we chose other gauges is not studied in literature. Instead it seems that for the momentthe Coulomb gauge is the only gauge in which one could construct the theory of free elec-tromagnetic field imposing then the canonical quantization and further making a coherentperturbative QED [14].An interesting observation about our vector potential is that we know it’s divergence andcurl and in addition we know that it vanishes when the spatial distances become infinite.Then a vector field which vanishes at infinity is completely specified once its divergence andits curl are given ( ∇ ~A = 0 , ∇ × ~A = ~B ) [22]. There are no additional terms due to thetransformation (8) in our amplitude of fermion production in magnetic field, and this is theresult of the use of Coulomb gauge and of the use of boundary conditions in space and time.So we prove that any gradient transformation leaves the transition amplitude invariant inCoulomb gauge. Or in other words the only gauge transformation allowed which preservethe Coulomb gauge is Λ = 0, as in the Minkowski QED [23].7 II. GAUGE INVARIANCE AND VECTOR POTENTIAL DECOMPOSITION
Another way to tell the above story is as follows. Consider that the vector potential isdecomposed in longitudinal and transversal parts [27]: ~ A = ~ A ⊥ + ~ A k (14)such that [27], ∇ ~ A ⊥ = 0 , ∇ × ~ A k = 0, which is known since the magnetic field is purelytransversal ~B k = 0. Then the gauge transformation: ~ A → ~ A + ∇ Λ (15)will give the transformation rules for the longitudinal and transversal components [27]: ~ A ⊥ → ~ A ⊥ ,~ A k → ~ A k + ∇ Λ . (16)We observe from the above equation that the transversal component of the potential vectoris gauge invariant [27] and only the longitudinal component transforms. In our calculationswe use only the transversal component ~A = ~ A ⊥ , as given in equation (4). Moreover in theCoulomb gauge ~A k = 0 [27], and only the transversal components are not vanishing.Finally, we conclude that working in the Coulomb gauge, only with the transversal com-ponents of the potential vector one can obtain results which are gauge independent in deSitter QED. For studying the theory of free electromagnetic field and perturbative QED andthen compute the amplitudes one needs to fix a gauge. IV. GAUGE INVARIANCE AND RENORMALIZATION
Since we know from flat space QED that for computing observable quantities one needs todo the renormalization of the theory, which depends on gauge [3–6, 10], the same observationcould be also valid in de Sitter QED. Then the discussion of gauge invariance in de SitterQED is not at all an easy task since a complete proof of the gauge independence/dependenceof the amplitudes/probabilities could depend on the renormalization of the theory, which isan issue not clear at this moment in this geometry. It is known from flat space QED thatthe unrenormalized S -matrix elements obtained from Feynman diagrams always appear in8hysical-scattering amplitudes multiplied by the renormalization constants Z and Z . Aswe know Z is gauge invariant but Z is a gauge dependent quantity [3–6]. This means thatthe unrenormalized transition amplitudes, could be, gauge dependent in order to secure thegauge invariance of the product between the renormalization constants and the unrenor-malized matrix elements [3–6]. So a proof of the gauge invariance could be carried out onthe renormalized amplitudes/probabilities in de Sitter QED. For that we must study theMaxwell and Dirac propagators including their radiative corrections. The above programmust be completed, and only then a definite conclusion could be addressed properly aboutthe gauge dependence of the amplitudes in de Sitter QED. V. CONCLUSIONS
The final conclusion is that the comment [16], needs to be considered with care, andthe authors do not present valid arguments in what regards the gauge dependence of theamplitude from our paper [2]. In our paper we work in Coulomb gauge in which the onlyallowed gauge transformations are Λ = 0. In quantum field theory the standard procedureis to establish the gauge first, and only afterwards perform the quantization. Once a gaugeis fixed and the quantization procedure is done, one could not do a gauge transformation inthe transition amplitudes which alter the chosen gauge.Another important observation is that the analysis of amplitudes variation in other gaugesand with given external electromagnetic fields must be done in de Sitter geometry in orderto understand the problem of gauge invariance, but there are no concrete results at thepresent time in the literature.Since the authors of the comment do not say or prove how the analytical results orphysical interpretation of our results will be modified by their result, we disagree with theimplication of the comment [16] that the analysis in our paper [2] and our previous papers[13, 14], are physically incorrect or there are ambiguities about the quantities that werecomputed.
Acknowledgements
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