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

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 illdefined: 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 δ 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 one. The most striking result, however, is that the electric and magnetic fields differ from the classical monopolar and dipolar fields by δ-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.

A concise introduction to Colombeau generalized functions and their applications in classical electrodynamics

“…The detailed proof of (9.4), which is given in [16], is not very complicated but somewhat lengthy. However, it is intuitively evident that the G identity (9.4) is simply the equivalent modulo O( q ), ∀q ∈ N, of the distributional identity…”

“…Due to (9.4), the functions F (r, θ, φ) ∈ F(R 3 ) are well defined at r = 0 when multiplied by ϒ or any of its derivatives. This enables us to integrate the G functions ϒ/r n , ϒ /r n , etc, which arise when calculating the electromagnetic fields of point charges and their derivatives as well as any algebraic combination of them [16]. We will however content ourselves with just a few basic integration formulae, which are proved in the appendix, i.e., T (0) + C [1] T (0), (9.12) where F (r) ∈ F and T (r) ∈ D. As expected, the G functions ϒ and ϒ have properties similar to the distributions H and δ.…”

Generalized two-body self-consistent theory of random linear dielectric composites: an effective-medium approach to clustering in highly-disordered media

“…One possible definition of distribution products, due to Colombeau [70,71], is as equivalence classes, whose representative chosen for calculations must be inferred from the context. This concept allows one to work properly with objects such as δ 2 , which is proportional to δ, but with nonstandard (infinite) proportionality constant [71].…”

“…We note that, in principle, a more fundamental treatment of the d-dimensional case would require acknowledging that the Dirac contribution to the Green function (3.2) stems from a representation (using a superscript to emphasize the dimension) [71]…”

A Paradox of Unity

“…the multiplication among distributions [43][44][45][46][47][48][49][50][51][52], the division of distributions [53], the random square root of the Dirac delta [54] and the square of the Dirac delta [55] as already discussed above. A brief summary of the problem of the Dirac delta squared is given in [56].…”