The classical point-electron in Colombeau's theory of nonlinear generalized functions

TL;DR

This paper uses Colombeau's algebra of generalized functions to rigorously analyze the electromagnetic fields of a point-electron, enabling the calculation of physically significant quantities like mass and spin, which are divergent in classical theory.

Contribution

It introduces a Colombeau algebra framework to handle singularities in electromagnetic fields, allowing for consistent calculation of self-energy, spin, and other quantities of a point-electron.

Findings

Mass and spin are divergent but renormalizable to match quantum theory.

Total self-force and self-momentum are zero, ensuring stability.

Delta-function terms in fields are sources of mass and spin.

Abstract

The electric and magnetic fields of a pole-dipole singularity attributed to a point-electron-singularity in the Maxwell field are expressed in a Colombeau algebra of generalized functions. This enables one to calculate dynamical quantities quadratic in the fields which are otherwise mathematically ill-defined: The self-energy (i.e., `mass'), the self-angular momentum (i.e., `spin'), the self-momentum (i.e., `hidden momentum'), and the self-force. While the total self-force and self-momentum are zero, therefore insuring that the electron-singularity is stable, the mass and the spin are diverging integrals of delta-squared-functions. Yet, after renormalization according to standard prescriptions, the expressions for mass and spin are consistent with quantum theory, including the requirement of a gyromagnetic ratio greater than one. The most striking result, however, is that the electric and magnetic fields differ from the classical monopolar and dipolar fields by delta-function terms which are usually considered as insignificant, while in a Colombeau algebra these terms are precisely the sources of the mechanical mass and spin of the electron-singularity.