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Some Remarks on Nonlinear Electrodynamics

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Some Remarks on Nonlinear Electrodynamics Hindawi Publishing Corporation Advances in High Energy Physics Volume 2016, Article ID 2463203, 10 pages http://dx.doi.org/10.1155/2016/2463203 Research Article Some Remarks on Nonlinear Electrodynamics Patricio Gaete Departamento de F´ısica and Centro Cient´ıfico-Tecnologico´ de Valpara´ıso, Universidad Tecnica´ Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile Correspondence should be addressed to Patricio Gaete; [email protected] Received 13 October 2015; Revised 15 December 2015; Accepted 27 December 2015 Academic Editor: Elias C. Vagenas Copyright © 2016 Patricio Gaete. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3. By using the gauge-invariant, but path-dependent, variables formalism, we study both massive Euler-Heisenberg-like and Euler- Heisenberg-like electrodynamics in the approximation of the strong-field limit. It is shown that massive Euler-Heisenberg-type electrodynamics displays the vacuum birefringence phenomenon. Subsequently, we calculate the lowest-order modifications to the interaction energy for both classes of electrodynamics. As a result, for the case of massive Euler-Heisenbeg-like electrodynamics 3 (Wichmann-Kroll), unexpected features are found. We obtain a new long-range (1/ -type) correction, apart from a long-range 5 (1/ -type) correction to the Coulomb potential. Furthermore, Euler-Heisenberg-like electrodynamics in the approximation of the 5 strong-field limit (to the leading logarithmic order) displays a long-range (1/ -type) correction to the Coulomb potential. Besides, for their noncommutative versions, the interaction energy is ultraviolet finite. 1. Introduction Euler-Heisenberg Lagrangian. Incidentally, as explained in [5], it is of interest to notice that the Euler-Heisenberg result The phenomenon of vacuum polarization in quantum elec- extends the Euler-Kockel calculation (in the constant back- trodynamics (QED), arising from the polarization of virtual ground field limit), which contains nonlinear corrections in electron-positron pairs and leading to nonlinear interactions powers of the field strengths, whereas the Uehling and Serber between electromagnetic fields, remains as exciting as in the result contains corrections linear in the fields (but nonlinear early days of QED [1–5]. An example that illustrates this is the in the space-time dependence of the background fields). We scattering of photons by photons, which despite remarkable further mention that, as in the Euler-Heisenbeg case, Born- progress has not yet been confirmed [6–10]. Along the same Infeld (BI) electrodynamics also contains similar nonlinear line, we also recall that alternative scenarios such as Born- corrections to Maxwell theory from a classical point of Infeld theory [11], millicharged particles [12], or axion-like view, as is well known. Nevertheless, BI electrodynamics is particles [13–15] may have more significant contributions to distinguished, since BI-type effective actions arise in many photon-photon scattering physics. different contexts in superstring theory [16, 17]. In addition to Interestingly, it should be recalled here that the physical Born-Infeld theory, other types of nonlinear electrodynamics effect of vacuum polarization appears as a modification in have been discussed in the literature [18–23]. the interaction energy between heavy charged particles. In In this perspective, we also point out that extensions of fact, this physical effect changes both the strength and the theStandardModel(SM)suchasLorentzinvarianceviolating structural form of the interaction energy. This clearly requires scenarios and fundamental length have become the focus the addition of correction terms in the Maxwell Lagrangian of intense research activity [24–31]. This has its origin in to incorporate the contributions from vacuum polarization the fact that the SM does not include a quantum theory of process. Two important examples of such a class of con- gravitation, so as to circumvent difficulties theoretical in the tributions are the Uehling and Serber correction and the quantum gravity program. Within this context quantum field Wichmann-Kroll correction, which can be derived from the theories allowing noncommuting position operators have 2 Advances in High Energy Physics been studied by using a star-product (Moyal product) [32– 2. Brief Review on the Uehling Potential 37]. In this connection, it becomes of interest, in particular, to recall that a novel way to formulate noncommutative As already expressed, we now reexamine the interaction quantum field theory has been proposed in [38–40]. The energy for Maxwell theory with an additional term corre- key ingredient of this development is to introduce coherent sponding to the Uehling correction (Uehling electrodynam- states of the quantum position operators [41], where a ics). This would not only provide the setup theoretical for our modified form of heat kernel asymptotic expansion which subsequentwork,butalsofixthenotation.Todothatwewill does not suffer from short distance divergences has been calculate the expectation value of the energy operator in |Φ⟩ ⟨⟩ obtained. We also point out that an alternative derivation of the physical state , which we will denote by Φ.We the coherent state approach has been implemented through start off our analysis by considering the effective Lagrangian a new multiplication rule which is known as Voros star- density [48]: product [42]. Anyhow, physics turns out to be independent 1 ] ofthechoiceofthetypeofproduct[43].Itisworthyto L =− (1 − ΔM) , 4 ] 3 (1) note here that this type of noncommutativity (coherent state approach) leads to a smearing effect which is equivalent to where that encountered in a class of nonlocal theory. In other words, noncommutativity is just a subclass of possible nonlocal ∞ 2 2 1 2 √ 4 deformation [44, 45]. More recently, this new approach has M (,) = ∫ (1 + ) 1− , (2) 2 (+Δ) been successfully extended to black holes physics [46], also 4 in connection to holographic superconductors via AdS-CFT with Δ≡ .ItshouldbenotedthatM contains the effect [47]. of vacuum polarization to first order in the fine structure Inspired by these observations, the purpose of this paper 2 constant, =/ℏ,and, istheelectronmass.Wemay is to extend our previous studies [18, 19] on nonlinear electro- parenthetically note here that the presence of Δ in (2) does dynamics to the case when vacuum polarization corrections not offer problems. On the one hand, as we will explain below are taken into account. The preceding studies were done using we restrict ourselves to the static case. On the other hand, in the gauge-invariant but path-dependent variables formalism, order to compute the interaction energy, we will make a series where the interaction potential energy between two static expansion at leading order in . charges is determined by the geometrical condition of gauge Before going on, two remarks are pertinent at this point. invariance. One important advantage of this approach is First, the modification of Coulomb’s law in (1) follows from that it provides a physically based alternative to the usual the weak-field limit of the one-loop effective action of Wilson loop approach. Accordingly, we will work out the quantum electrodynamics (QED). Indeed, as was explained static potential for electrodynamics which include, apart from in [48], this modification can be written as the Maxwell Lagrangian, additional terms corresponding 1 to the Uehling, massive Euler-Heisenberg-like, and Euler- L = ∫ () Π (,) ] () 44, wf 2 ] (3) Heisenberg electrodynamics in the approximation of the strong-field limit (to the leading logarithmic order) and for Π 2 their noncommutative versions. Our results show a long- where wf denotes weak field, and ] is the usual order- 5 range 1/ -type correction to the Coulomb potential for both polarization tensor of QED. As is well known, in momentum massive Euler-Heisenberg-like and Euler-Heisenberg elec- space, this tensor is given by trodynamics in the approximation of the strong-field limit Π () =(2 − )Π(2), (to the leading logarithmic order). Interestingly enough, for ] ] ] (4) massive Euler-Heisenbeg-like electrodynamics (Wichmann- 3 while the momentum space spectral representation of the Kroll), we obtain a new long-range 1/ correction to 2 polarization function Π( ) reads the interaction energy. Nevertheless, for their noncom- mutative versions, the static potential becomes ultraviolet ∞ () 1 Π (2) =− 2 ∫ , finite. 2 (5) 3 42 + The organization of the paper is as follows: In Section 2, 2 we reexamine Uehling electrodynamics in order to establish with () = (1 + 2 /)√1−42/. Next, due to the tensor a framework for the computation of the static potential. In structure of Π](), Lwf canthenbeexpressedinagauge- Section 3 we consider Euler-Heisenberg-like (with a mass invariantway,thatis,intermsof].Asaconsequenceofthis, term)electrodynamicsandshowthatityieldsbirefringence, (3) reduces to the modification of Coulomb’s law appearing computing the interaction energy for a fermion-antifermion in (1). We mention in passing that in [48] the signature of the pair and its version in the presence of a minimal length. metric is different from that used in this paper. In Section 4, we repeat our analysis for Euler-Heisenberg Second, it should be noted that the theory described by electrodynamics
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