# On generalized logarithmic electrodynamics

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## Abstract

The generalized logarithmic electrodynamics with two parameters \(\beta \) and \(\gamma \) is considered. The indexes of refraction of light in the external magnetic field are calculated. In the case \(\beta =\gamma \) we come to results obtained by Gaete and Helayël-Neto (Eur Phys J C 74:2816, 2014). The bound on the values of \(\beta \), \(\gamma \) was obtained from the Biréfringence Magnétique du Vide (BMV) experiment. The symmetrical Belinfante energy-momentum tensor and dilatation current are found.

## Keywords

External Magnetic Field Classical Electrodynamic Electric Permittivity Linear Birefringence Uniform Magnetic Induction## 1 Introduction

Nonlinear classical electrodynamics in vacuum is of interest because of the one-loop quantum corrections in QED which give non-linear terms [1, 2]. In addition, to solve the problem of singularity of a point-like charge giving an infinite electromagnetic energy, Born and Infeld (BI) [3, 4] introduced a new parameter with the dimension of length. BI non-linear electrodynamics results in a finite electromagnetic energy of point-like particles. Other examples of non-linear electrodynamics were introduced in [5, 6, 7, 8, 9]. In the vacuum, in the presence of strong external magnetic field, non-linear effects can be observed in experiments. Thus, PVLAS [10] and BMV [11] experiments can give the bounds on dimensional parameters introduced in non-linear electrodynamics. In this letter we calculate the values of indexes of refraction of light in the external magnetic field in generalized logarithmic electrodynamics and estimate the bound on the values of the parameters \(\beta \), \(\gamma \) from the BMV experiment.

The Heaviside–Lorentz system with \(\hbar =c=\varepsilon _0=\mu _0=1\) and Euclidian metric are explored.

## 2 The model of generalized logarithmic electrodynamics

## 3 Vacuum birefringence

## 4 The energy-momentum tensor and dilatation current

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## 5 Conclusion

We have considered the model of generalized logarithmic electrodynamics with two parameters \(\beta \) and \(\gamma \). At \(\beta \,=\,\gamma \) we arrive at logarithmic electrodynamics considered in [7]. We show that at \(\beta =\gamma \) in the approximation \( B_0^2/\beta ^2\ll 1\), \( B_0^2/\gamma ^2\ll 1\) the phenomenon of birefringence is absent. Classical electrodynamics with QED corrections gives the effect of birefringence and, therefore, the case \(\beta \ne \gamma \) is important. From the BMV experiment we have obtained the bound on the parameters \(\beta \) and \(\gamma \). Another bound on the parameter \(\beta \) from the point of view that the electron mass has pure electromagnetic nature was proposed in [16]. The scale symmetry is broken and dilatation current was found in the model under consideration.

## Footnotes

## Notes

### Acknowledgments

I wish to thank P. Gaete and J. Helayël-Neto for communications that allowed me to make corrections in the first version of this letter.

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## Copyright information

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