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Singular electromagnetic fields in nonlinear electrodynamics with a constant background field

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Abstract

When exploring equations of nonlinear electrodynamics in effective medium formed by mutually parallel external electric and magnetic fields, we come to special static axial-symmetric solutions of two types. The first are comprised of fields referred to as electric and magnetic responses to a point-like electric charge when placed into the medium. In electric case, this is a field determined by the induced charge density. In magnetic case, this is a field carrying no magnetic charge and determined by an induced current. Fields of second type require presence of pseudoscalar constants for their existence. These are singular on the axis drawn along the external fields. In electric case this is a field of an inhomogeneously charged infinitely thin thread. In magnetic case this is the magnetic monopole with the Dirac string supported by solenoidal current. In both cases the necessary pseudoscalar constant is supplied by field derivatives of nonlinear Lagrangian taken on external fields. There is also a magnetic thread solution dual to electric thread with null total magnetic charge.

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Notes

  1. See also Ref. [42, 43] for recent studies on vacuum birefringence effects near magnetars.

  2. There are also other nonlinear models that guarantee convergence [49] of the field energies of monopoles (even if their fields are singular), the Euler-Heisenberg Lagrangian taken to quadratic order of its expansion in powers of the field invariants belonging to this class. Sufficient conditions to be imposed on the growth of nonlinearity with the field may be found in [50].

  3. Other manifestations of the magneto-electric effect were treated in [72,73,74] as nonlinear response.

  4. The most common example of such theory is provided by the famous Euler-Heisenberg Lagrangian taken for \(\mathfrak {L}\left( X\right) \), which is the approximation of the effective Lagrangian of Quantum Electrodynamics fit for slow-varying fields (it is known in literature as calculated with the accuracy of one and two electron-positron loops).

  5. Derivation of Eq. (3) from the action (2) is described in detail in Refs. [72, 73, 77, 79].

  6. as in QED, whose effective Lagrangians are proportional to the fine structure constant \(\alpha =e^{2}/4\pi \).

  7. This is only for brevity. To be precise, we had to say “electric field has the induced charge as its source”. We shall take the liberty to apply such abuse of terminology to magnetic fields as well.

  8. For the angular variables, we use the convention in which the differential volume element is \(d{\textbf{r}}=r^{2}\sin \theta drd\theta d\varphi \).

  9. Beyond pure electrodynamics, the cosmological pseudoscalar field, responsible for the expansion of Universe, may be cosidered as forming the necessary, almost constant, background. In this respect, Ref. [93] (and references therein) should be paid attention, where magnetic solutions are studied under such Lorentz-violating conditions.

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Acknowledgements

TC Adorno acknowledges the support from the XJTLU Research Development Funding, award no. RDF-21-02-056, and DM Gitman thanks CNPq for permanent support.

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Authors and Affiliations

  1. Department of Physics, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road, Suzhou Dushu Lake Science and Education Innovation District, Suzhou Industrial Park, Suzhou, 215123, People’s Republic of China

    T. C. Adorno

  2. P. N. Lebedev Physical Institute, 53 Leninskiy prospekt, Moscow, Russia, 119991

    D. M. Gitman & A. E. Shabad

  3. Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, São Paulo, S.P., 05508-090, Brazil

    D. M. Gitman

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  1. T. C. Adorno
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Correspondence to T. C. Adorno.

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Adorno, T.C., Gitman, D.M. & Shabad, A.E. Singular electromagnetic fields in nonlinear electrodynamics with a constant background field. Eur. Phys. J. Plus 138, 1036 (2023). https://doi.org/10.1140/epjp/s13360-023-04655-1

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