Completing Born and Infeld's 1934 nonlinear field theory: deciding the action by its consequences, extending it to gravitation, and integrating it with quantum theory

Born and Infeld's 1934 paper 'Foundations of the new field theory' introduced the nonlinear electrodynamics now called the Born–Infeld theory: from a generally covariant determinantal action it derives the Lagrangian $L=\sqrt{1+F-G^2}-1$ (with $F$ and $G$ the two Lorentz invariants of the electromagnetic field), which bounds the electric field by an absolute field constant $b$ and makes the self-energy of a point charge finite. The paper explicitly left three questions open: (1) the new Lagrangian differs from Born's earlier action only by the term $G^2$ — appreciable only near singularities — and the authors leave the choice between the two to be decided by consequences; (2) gravitation is ignored, the metric being fixed to its flat special-relativistic values; (3) quantization is postponed, the goal being a full microscopic theory of particles and photons. Theoretical consequences — electric–magnetic duality invariance, absence of birefringence (no splitting of light polarizations in a background field), causal propagation, and the derivation of the full action in string theory — favor the determinant action over its $G=0$ reduction; but the decision between the two actions near field singularities, a consistent Born–Infeld-type gravitational completion, and a quantum treatment beyond effective-theory status all remain open, with active work continuing through 2026.

Background

Classical electrodynamics (Maxwell's theory) with point charges assigns each charged particle an infinite electrostatic self-energy: the field-energy integral $\int E^2\,dV/8\pi$ diverges at the location of the charge. In the early 1930s Max Born and Leopold Infeld pursued a 'unitarian' alternative in which only the electromagnetic field is fundamental, particles are singularity structures of the field, and mass is derived from field energy; earlier attempts (extended-electron models and Mie's nonlinear theory) had failed. Their program is a nonlinear electrodynamics (NED): a modification of Maxwell's theory in which the Lagrangian is a nonlinear function of the field, so that the field equations are nonlinear in the field strength. The foundational paper (1934) starts from a nonsymmetric rank-2 tensor $a_{kl}=g_{kl}+f_{kl}$, where $g$ is symmetric and identified with the metric and $f$ is antisymmetric (the electromagnetic field tensor), and takes as action density the determinantal form $\mathfrak{L}=\sqrt{-|g_{kl}+f_{kl}|}-\sqrt{-|g_{kl}|}$ (following Eddington), which is invariant under arbitrary coordinate transformations. In terms of the two Lorentz invariants of the field — $F=\tfrac{1}{2}f_{kl}f^{kl}$ and the pseudoscalar $G=\tfrac{1}{4}f^{*}_{kl}f^{kl}$, with $f^{*}$ the dual tensor (in three-vector language these are combinations of $\mathbf{E}$ and $\mathbf{B}$, roughly $B^2-E^2$ and $\mathbf{B}\cdot\mathbf{E}$) — the Lagrangian is $L=\sqrt{1+F-G^2}-1$. The weak-field limit is Maxwell's theory; the field equations keep Maxwell's form but with a field-dependent constitutive relation (a nonlinear 'dielectric constant'). For a static point charge the displacement field keeps the Coulomb form $D_r=e/r^2$, but the field-dependent dielectric function suppresses the electric field, which is bounded by $b$ and attains that maximum at the charge's location, so the self-energy integral converges: the paper reports $U=1.2361\,e^2/r_0$ in terms of the length parameter $r_0=\sqrt{e/b}$, with the values $b=9.18\times10^{15}$ esu (electrostatic units of field) and $r_0=2.28\times10^{-13}$ cm for the electron. Two features of the 1934 foundation matter for the open question. First, the new Lagrangian differs from the one Born had postulated in his immediately preceding paper in the same Proceedings ('On the quantum theory of the electromagnetic field', Proc. R. Soc. A 143, 410) only by the term $G^2$; since that term is quartic in the field, it is negligible except in the immediate neighborhood of singularities, and the paper explicitly leaves it to the consequences to decide which action is correct. Second, the paper confines itself to classical theory: gravitation is ignored (the metric is the special-relativistic one even inside the electron), and quantization is postponed — the derivation of the classical nonlinear field law is deliberately separated from the project of building the microscopic theory of particles and photons that motivated the whole program. Later developments that frame the current status: electric–magnetic duality invariance (invariance under interchange of electric and magnetic fields, studied by Schrödinger in 1935 and others); birefringence (the splitting of light polarizations in a background field, absent in the Born–Infeld theory but generic in other NEDs); the Dirac–Born–Infeld (DBI) action, which is the effective action of open strings and of D-branes (higher-dimensional solitonic objects in string theory); and Born–Infeld-type gravity, determinantal gravitational actions in the Eddington tradition (Deser–Gibbons 1998; Eddington-inspired Born–Infeld gravity from 2010, a modified theory of gravity formulated in the Palatini approach, where metric and connection are varied independently).

Problem Statement

The 1934 foundations paper ends with its program deliberately incomplete in three linked respects: can the nonlinear field theory be completed as a physical theory — and can the correct form of its action be decided by consequences rather than postulate? Concretely: (1) Action choice: the determinantal action $L=\sqrt{1+F-G^2}-1$ and its $G=0$ reduction (Born's earlier action) differ only by the quartic term $G^2$ and coincide whenever $G=0$ (in particular for the purely electrostatic point charge); the paper leaves it to future investigation to determine which of the two proves itself better in the immediate neighborhood of the field singularities. (2) Gravitational extension: the theory is built on a generally covariant determinantal action but is evaluated with a flat metric; incorporate gravitation and obtain a consistent coupled theory. (3) Quantum integration: adapt the nonlinear field law to quantum theory and integrate it into a microscopic theory of particles and photons — the original motivation being the infinite self-energy of point charges and the then-unexplained particle and nuclear phenomena. In modern terms the surviving core is: (a) determine which of the two candidate actions — the 1934 determinantal action or its $G=0$ reduction — proves itself better in the near-singular field regime where they differ; empirically, only lower bounds on the absolute field constant $b$ are known and no deviation from Maxwell theory or quantum electrodynamics has been observed; (b) construct a consistent Born–Infeld-type gravitational theory (the desiderata proposed by Deser and Gibbons in 1998 remain unsatisfied by any accepted exemplar, and Eddington-inspired Born–Infeld gravity and its descendants are still being constrained and debugged); (c) settle the quantum status of the nonlinear law beyond its emergence as the effective action of open strings and D-branes, i.e. beyond effective-theory status.

The three parts (a)–(c) are jointly posed: for (a), which of the two actions is correct, the operative regime being the one in which the two actions actually differ, i.e. configurations with $G=\mathbf{E}\cdot\mathbf{B}\neq0$ in which the quartic term $G^2$ is not negligible (the 1934 paper's 'immediate neighborhood of the field singularities' is its loose heuristic for where that term could matter: the isolated static electron has $G=0$ throughout, so the two actions coincide there and proximity to a purely electrostatic singularity does not discriminate them). The consequences credited in the opening paragraph with favouring the determinant action — duality invariance, absence of birefringence, causal propagation, and the string-theory derivation of the full action — are properties of the action as a whole. Part (b) asks for an explicit Born–Infeld-type action coupling the determinantal field action to gravitation in which the gravitational action is itself of the determinantal (Born–Infeld / Eddington / Deser–Gibbons) type, with the curvature entering the determinant — a theory whose curvature does not enter a determinant, such as plain Einstein–Hilbert gravity with minimally coupled Born–Infeld matter, is a different theory, not this one — and which meets the consistency requirements sought by Deser and Gibbons: a curvature expansion beginning with the Einstein term and free of quadratic curvature terms (so that it propagates gravitons but no ghosts), with a static charged solution whose field remains bounded. Part (c) asks for a settlement of the quantum status of the nonlinear field law beyond effective-theory status: whether a stated quantization scheme exists that constitutes a genuinely non-effective microscopic theory of the charges and photons — defining the theory at all scales, with no externally imposed cutoff or truncation, and not obtained as the low-energy limit of a different fundamental theory, rather than an effective theory — whose classical limit reproduces that law and which yields a prediction in a named observable regime that current bounds on the absolute field constant $b$ leave open.

References

1. M. Born and L. Infeld, 'Foundations of the new field theory', Proceedings of the Royal Society of London. Series A 144, 425–451 (1934). DOI 10.1098/rspa.1934.0059, https://doi.org/10.1098/rspa.1934.0059 (source paper)
2. M. Born and L. Infeld, 'Foundations of the New Field Theory', Nature 132, 1004 (1933). DOI 10.1038/1321004b0, https://www.nature.com/articles/1321004b0 (preliminary announcement)
3. M. Born, 'On the quantum theory of the electromagnetic field', Proceedings of the Royal Society of London. Series A 143, 410–437 (1934). DOI 10.1098/rspa.1934.0010, https://doi.org/10.1098/rspa.1934.0010 (source of the earlier $G=0$ action)
4. M. Born and L. Infeld, 'On the quantization of the new field equations I', Proceedings of the Royal Society of London. Series A 147, 522–546 (1934). DOI 10.1098/rspa.1934.0234, https://doi.org/10.1098/rspa.1934.0234
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