# Remarks on nonlinear electrodynamics

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## Abstract

We consider both generalized Born–Infeld and exponential electrodynamics. The field energy of a point-like charge is finite only for Born–Infeld-like electrodynamics. However, both Born–Infeld-type and exponential electrodynamics display the vacuum birefringence phenomenon. Subsequently, we calculate the lowest-order modifications to the interaction energy for both classes of electrodynamics, within the framework of the gauge-invariant path-dependent variables formalism. These are shown to result in long-range (\(1/r^5\)-type) corrections to the Coulomb potential. Once again, for their noncommutative versions, the interaction energy is ultraviolet finite.

## Keywords

Interaction Energy Lagrangian Density Primary Constraint Black Hole Physic Dirac Bracket## 1 Introduction

Amongst the most interesting of the phenomena predicted by quantum electrodynamics (QED) we may quote the photon–photon scattering in vacuum arising from the interaction of photons with virtual electron–positron pairs [1, 2, 3, 4]. However, despite remarkable progress [5, 6, 7, 8, 9], this prediction has not yet been confirmed. Nevertheless, this remarkable quantum characteristic of light remains a fascinating and challenging topic of research. In fact, it is conjectured that alternative scenarios such as Born–Infeld theory [10], millicharged particles [11] or axion-like particles [12, 13, 14] may have more significant contributions to photon–photon scattering physics.

It is worth recalling, at this stage, that Born–Infeld (BI) electrodynamics was proposed in 1934 in order to remove the singularities associated with charged point-like particles. Also, similarly to Maxwell electrodynamics, Born–Infeld electrodynamics displays no birefringence in vacuum. At the same time, BI electrodynamics is distinguished, since BI-type effective actions arise in many different contexts in superstring theory [15, 16]. Additionally, nonlinear electrodynamics (BI) has also been investigated in the context of gravitational physics [17, 18]. Actually, in addition to Born–Infeld theory, other types of nonlinear electrodynamics have been studied in the context of black hole physics [19, 20, 21, 22].

Meanwhile, recent experiments related to photon–photon interaction physics [5, 6, 7, 8, 9] have shown that the electrodynamics in vacuum is a nonlinear theory. Therefore, different models of nonlinear electrodynamics of the vacuum deserve additional attention on the physical consequences presented by a particular nonlinear electrodynamics. In effect, our purpose here is to examine the properties of both Born–Infeld-like electrodynamics and exponential electrodynamics.

On the other hand, we also recall that extensions of the Standard Model (SM), such as Lorentz invariance violating scenarios and scenarios based on a fundamental length, have been subject to intensive investigations over the past years [23, 24, 25, 26, 27]. The main reason for this is that the SM does not include a quantum theory of gravitation, as well as the need to understand and to overcome theoretical difficulties in quantum gravity research. An attempt along this direction has been to consider quantum field theories allowing noncommuting position operators [28, 29, 30, 31, 32, 33], where this noncommutativity is an intrinsic property of space-time. These studies were first made by using a star product (Moyal product). However, in recent years, a novel way to formulate noncommutative quantum field theory (or quantum field theory in the presence of a minimal length) has been initiated in [34, 35, 36], which define the fields as mean values over coherent states of the noncommutative plane. Later, it has been shown that the coherent state approach can be summarized through the introduction of a new multiplication rule which is known as the Voros star-product [37]. Evidently, physics turns out be independent from the choice of the type of product [38]. Subsequently, this new approach has been applied extensively to black holes physics [39].

With these considerations in mind, in a previous work [40], we have considered logarithmic electrodynamics, for which the field energy of a point-like charge is finite, as happens in the case of the usual Born–Infeld electrodynamics. We have also shown that, contrary to the latter, logarithmic electrodynamics displays the phenomenon of birefringence. Further, we have computed the lowest order to the interaction energy for both logarithmic electrodynamics and for its noncommutative version, by using the gauge-invariant but path-dependent formalism. Our calculation has shown a long-range correction to the Coulomb potential for logarithmic electrodynamics. Interestingly enough, for its noncommutative version, the static potential becomes ultraviolet finite. From such a perspective, and given the experiments related to photon–photon interaction physics [5, 6, 7, 8, 9], the present work is an extension of our previous study [40]. To do this, we shall work out the static potential for both Born–Infeld-like and exponential electrodynamics, using the gauge-invariant but path-dependent variables formalism, which is an alternative to the Wilson loop approach.

Let us also mention here that Lagrangian densities of nonlinear extensions of electrodynamics such as Born–Infeld-like electrodynamics, whose Lagrangian density is built up with an arbitrary power of the electromagnetic invariants, have been considered in the context of black hole physics [19, 41]. In the context of single layer graphene, the effective action for the \((2+1)\) relativistic quantum electrodynamics is governed by a power \(3/4\) [42, 43]. In addition, exponential electrodynamics has also been considered in the physics of black holes [19].

Our work is organized according to the following outline: in Sect. 2, we consider Born–Infeld-like electrodynamics, show that it yields birefringence, compute the interaction energy for a fermion–antifermion pair and its version in the presence of a minimal length. In Sect. 3, we repeat our analysis for exponential electrodynamics. Finally, in Sect. 4, we present our final remarks.

In our conventions the signature of the metric is \((+1,-1,-1,-1)\).

## 2 Born–Infeld-like model

The physical picture is that for \(0 < p < 1\), though the charged particle is point-like, its charge somehow spreads in a small region and it is screened by the polarization that results from the quantum effects, which all sums up to produce the effective Lagrangian (1).

Again, as in the case of both Born–Infeld and logarithmic electrodynamics [40, 54], we have a finite electric field at the origin, we find that the interaction energy between two test charges at leading order in \(\beta \) is not finite at the origin. We should point out that this is not a contradiction. Actually, according to the result given by Eq. (10), the electric field and its corresponding potential are finite all over the space. This is an exact result. On the other hand, (40) and (45) are expressions for the interparticle potential obtained upon a \(1/\beta \)-expansion and truncated at order \(1/\beta ^{2}\). So, the singularity at \(r=0\) is just a result of cutting off the potential expression and keeping only the term in \(1/\beta ^{2}\), instead of computing the complete analytic expression as in Eq. (10).

We would like to point out that we are not here going to calculate the electrostatic energy stored in a region corresponding to the Compton wavelength of the electron, \(m_e^{ - 1}\), because we know that the electron mass does not originate from its electrostatic field; it rather comes from the Yukawa coupling between the electron and the Higgs fields and the spontaneous breaking down of the \( SU_L \left( 2 \right) \times U_Y \left( 1 \right) \)-symmetry to the electromagnetic \(U(1)\); actually, \( m_e = y_e \left\langle H \right\rangle \), where \(y_e\) is the electron’s Yukawa coupling and \( \left\langle H \right\rangle \) the order parameter of the breaking of electroweak symmetry.

## 3 Exponential electrodynamics

The physical understanding for a non-regular solution for \(r=0\) is that exponential electrodynamics is actually a power series expansion in \({\mathcal F}\) and \({\mathcal G}^2\) and, then, as we have discussed in Sect. 2, positive powers in \({\mathcal F}\) and \({\mathcal G}\) do not yet lead to a finite field on the charge’s position. We should point out that the Born–Infeld case (\(p = 1/2\)) comes out from the vacuum polarization as a quantum effect of virtual pair production and annihilation, which is responsible for the screening of a charge in a polarized vacuum. For \(0 < p < 1\), the regime of screening is still valid. However, \(p > 1\) is outside this regime and this is why the case of the exponential electrodynamics does not exhibit a regular electrostatic field at the charge’s position. We point out, in this context, the work of Ref. [55], where a quartic model in \(F_{\mu \nu }\) is considered and, though the field is not regular for \(r=0\), the finiteness of the field energy is ensured.

## 4 Final remarks

In summary, within the gauge-invariant but path-dependent variables formalism, we have considered the confinement versus screening issue for both Born–Infeld-like electrodynamics and exponential electrodynamics. Once again, a correct identification of physical degrees of freedom has been fundamental for understanding the physics hidden in gauge theories. We should highlight the identical behaviors of the potentials associated to each of the models. Interestingly enough, their noncommutative version displays an ultraviolet finite static potential. The above analysis reveals the key role played by the new quantum of length in our analysis. In a general perspective, the benefit of considering the present approach is to provide unifications among different models, as well as exploiting the equivalence in explicit calculations, as we have illustrated in the course of this work.

Finally, recently an up-dated upper bound for the electron’s electric dipole moment (EDM) has been published in [7]. Since the understanding of this property involves CP-violation, we believe it would be a viable task to include a CP-violating term given by \({\mathcal G}\), or an odd power of \({\mathcal G}\), and to compute how it may yield an asymmetric charge distribution around the spin of the electron. This, in turn, should induce a contribution to the electron’s EDM in the framework of the Born–Infeld model. To do that, it is clearly important to also know the magnetic field that appears as an effect of the nonlinearity in the case of a point charge [56]. We are presently pursuing this investigation, and we hope to report on it soon.

## Notes

### Acknowledgments

P. G. was partially supported by Fondecyt (Chile) Grant 1130426, DGIP (UTFSM) internal project USM 111458. P. G. also wishes to thank the Field Theory Group of the CBPF for hospitality and PCI/MCT for support.

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