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

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 ensuring that the electron singularity is stable, the mass and spin are diverging integrals of delta(2)-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 1. 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. (C) 2008 American Institute of Physics. [DOI: 10.1063/1.2982236]