The Effective Radius of an Electric Point Charge in Nonlinear Electrodynamics
Tengyang Liu, Yisong Yang
TL;DR
This work addresses the classical challenge of modeling a pointlike electron with finite self-energy by introducing nonlinear perturbations of Maxwell electrodynamics. It develops two families, non-polynomial and polynomial perturbations, that preserve finite energy for a point charge while allowing the effective radius $a$ to be tuned to arbitrarily small values via a coupling parameter $\gamma$. The non-polynomial family yields finite energy with vanishing $a$ but does not recover Maxwell theory in the weak-coupling limit, whereas the polynomial family restores Maxwellian UV behavior while still achieving small $a$. Across both families, monopoles and dyons are energetically excluded, and in the strong nonlinear regime the free charge and self-energy inside any finite region vanish, offering a classical mechanism for the apparent invisibility of electron substructure. Collectively, these results broaden the landscape of nonlinear electrodynamics and provide a classical framework linking finite-energy point charges to experimentally unobservable substructures.
Abstract
Motivated by the century-old problem of modeling the electron as a pointlike particle with finite self energy, we develop a new class of nonlinear perturbations of Maxwell's electrodynamics inspired by, but distinct from, the Born--Infeld theory. A hallmark of our construction is that the effective radius of an electric point charge can be reduced arbitrarily by tuning a coupling parameter, thereby achieving scales far below the Born--Infeld bound and consistent with the experimentally undetected size of the electron. The models preserve finite self energy for point charges while energetically excluding monopoles and dyons, a robustness that appears intrinsic to this class of nonlinear theories. Two complementary behaviors are uncovered: In the non-polynomial perturbations, the Maxwell limit is not recovered as the coupling vanishes, whereas in polynomial models the self energy diverges correctly, meaning that the Maxwellian ultraviolet structure is reinstated. A further subtlety emerges in the distinction between the prescribed source charge, imposed through the displacement field, and the measurable free charge arising from the induced electric field. In particular, the free charge and the self energy contained within any ball around the point charge tend to zero in the strong-nonlinearity or zero effective-radius limit, rendering a pointlike structure locally undetectable, both electrically and energetically. These findings highlight how nonlinear field equations reconcile theoretical prescription with experimental measurement and suggest a classical rationale for the effective invisibility of the electron substructure.