# Electromagnetic fields with vanishing quantum corrections

###### Abstract

We show that a large class of null electromagnetic fields are immune to any modifications of Maxwell’s equations in the form of arbitrary powers and derivatives of the field strength. These are thus exact solutions to virtually any generalized electrodynamics containing both non-linear terms and higher derivatives. This result holds not only in a flat or (anti-)de Sitter background, but also in a larger subset of Kundt spacetimes.

PACS 03.50.De, 03.50.Kk, 11.10.Lm, 04.40.Nr

## 1 Introduction

Early attempts to modify Maxwell’s theory in order to cure the divergent electron’s self-energy date back to Mie’s work of 1912 [1]. A physically more satisfactory non-linear theory was subsequently proposed by Born and Infeld [2, 3]. From a quantum perspective, non-linear deviations from Maxwell’s theory also appear in the effective Lagrangian derived by Heisenberg and Euler from QED, relevant to light-by-light scattering and electron-positron pair production [4, 5, 6]. Both the Born-Infeld and Heisenberg-Euler theories are special cases of non-linear electrodynamics (NLE), a more general theory defined by a Lagrangian which depends (in principle arbitrarily) on the two algebraic invariants and [7]. The resulting field equations take the form

(1) |

where and , are specific scalar functions of and . Starting from the 80s, interest in NLE has been revived by the observation that the Born-Infeld Lagrangian emerges as an effective low-energy Lagrangian in string theory (cf., e.g., [8] for a review and references).

When it comes to finding exact solutions, non-linear theories such as NLE are obviously much more difficult to deal with. It thus perhaps came as a surprise when Schrödinger pointed out that all null fields (defined by ) which solve Maxwell’s theory also automatically solve any non-linear electrodynamics (in vacuum) [9, 10]. In fact, Schrödinger’s observation can readily be extended to theories more general than NLE, by letting in (1) be any 2-form constructed algebraically and polynomially from . In a nutshell, the proof of this fact boils down to observing that, if is null, any such can only be a linear combination with constant coefficients of and (quadratic and higher terms vanish thanks to the antisymmetry of ), and therefore obeys Maxwell’s equations if does (although NLE was originally formulated in a flat spacetime, the same argument holds also in curved backgrounds). Null fields are thus solutions displaying theory-independent properties. Recall that they are also of interest from a physical viewpoint, since they characterize electromagnetic plane waves [6, 11] as well as the asymptotic behaviour of radiative systems [12], and they approximate the field produced by high-energy sources [13, 11, 14].

It would thus be desirable to identify classes of exact solutions possessing similar “universal” properties in a context more general than NLE, i.e., including higher derivatives theories. These would naturally have broader physical applications, since effective Lagrangians depending on the derivatives of the field strength are relevant both to classical field theories [15, 16] and to string theory [8]. However, the simple argument we have presented above breaks down if contains derivatives , so that Schrödinger’s observation does not readily extend to higher-derivative electrodynamics.

It is the purpose of the present contribution to show that, nevertheless, a large family of universal Maxwell fields exists that solve any theory defined by a Lagrangian constructed out of and its derivatives of arbitrary order. More precisely, we will identify universal solutions of the system (1), where the second equation holds simultaneously for all -forms constructed polynomially from and its covariant derivatives (Maxwell’s theory of course corresponds to the special choice ). Because contains derivatives, in order to be able to prove this fact we will need to place some restrictions also on the background spacetime. However, the most standard case of flat or (A)dS spacetime will be included in our result. Throughout the paper, we will restrict ourselves to the sourcefree Maxwell equations in a four-dimensional spacetime.

## 2 The solutions

Technically, our result may be stated as follows: in a Kundt spacetime of Petrov type III and aligned traceless-Ricci type III, any aligned null that solves Maxwell’s equations is universal. More explicitly, this means that the spacetime can be written as

(2) |

where

(3) | |||

(4) | |||

(5) |

The function is complex and arbitrary, and all the remaining functions are real with and arbitrary and the constraints

(6) | |||

(7) |

where . Then any of the form

(8) |

where is arbitrary, solves identically the equations (1), with as described above. These spacetimes include, in particular, Minkowski and (A)dS, as well as various gravitational and electromagnetic waves propagating on those backgrounds [17, 18].

Let us sketch a proof of the above statement (more technical details will be presented elsewhere along with some generalizations). The first of (1) is obviously satisfied because it takes the same form as in Maxwell’s theory. The strategy is thus to prove that, under the above assumptions, all the possible that one can construct are automatically divergencefree (or zero). The Kundt assumption means that the spacetime admits a geodesic null vector field with vanishing expansion, shear and twist, which can be written as

(9) |

Additionally, the conditions on the curvature mean that the Riemann tensor is of the form,

(10) |

where the ellipsis denote terms of negative boost-weight, i.e., such that no non-zero scalar invariant can be constructed out of those algebraically (their explicit form is not important here). Thanks to [19], the assumptions ensure that all the scalar invariants constructed out of and vanish identically. A result of [20] and the antisymmetry of imply that must be a linear combination with constant coefficients of , and (suitable contractions of) and (roughly speaking, quadratic and higher terms would contain at least two non-contracted s, thus giving zero by antisymmetry). Now, the terms and are harmless since they are divergencefree by assumption. The remaining terms need contractions with the metric tensor in order to produce an object of rank . In flat space covariant derivative commute, so that all possible terms can be rewritten in such a way as to contain (derivatives of) the divergence of or and the proof is already complete. In a curved spacetime this is not true, since shifting indices produces terms linear in the curvature tensor, via the generalized Ricci identity (schematically, for a tensor one has – cf., e.g., [21]). However, repeated use of the latter together with the Weitzenböck identity for -forms [22] leads, thanks to (10), to the same conclusion. Although this idea is straightforward, the intermediate steps are lengthy and require several technicalities (see [19] and references therein) necessary to prove certain special differential properties of the Riemann tensor that hold in the spacetimes (2) (the simplest example of those being ) and that enable one, ultimately, to reorder the covariant derivatives in a desired way.

## 3 Conclusions

The electromagnetic fields described in this paper represent a large family of universal vacuum solutions, thus being relevant to virtually any classical theory of electrodynamics. Interestingly, they can also be generalized to null -forms in arbitrary dimensions, which are of more direct interest for supergravity and string theory applications. These and other extensions will deserve a separate investigation – previous related results in special cases include [6, 23, 24]. Finally, it should be observed that the obtained solutions do not exhaust the universal class. For example, it is easy to see that, within NLE, Schrödinger’s observation extends to all Maxwell fields for which and are constant (not necessarily zero), thus describing, for example, uniform electric and magnetic fields (at least in a flat background). More generally, any covariantly constant is universal, even if not null. However, the latter solutions clearly describe electromagnetic fields physically different from the null ones.

## Acknowledgments

We thank Sigbjørn Hervik for discussions. The stay of M.O. at Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile has been supported by CONICYT PAI ATRACCIÓN DE CAPITAL HUMANO AVANZADO DEL EXTRANJERO Folio 80150028.

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