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Regularizations of point charges, the Liénard-Wiechert potential, and the electron self-energy

Guenther Hoermann, Nathalie Tassotti

Abstract

We apply Colombeau-type regularization to the electromagnetic field of a point-charge and show how the Liénard-Wiechert potential can be derived from a generalized function based on the geometry of Minkowski space. Furthermore, for a charged particle in its rest frame, we discuss the electric monopole, magnetic dipole, electron singularity, and self-energy.

Regularizations of point charges, the Liénard-Wiechert potential, and the electron self-energy

Abstract

We apply Colombeau-type regularization to the electromagnetic field of a point-charge and show how the Liénard-Wiechert potential can be derived from a generalized function based on the geometry of Minkowski space. Furthermore, for a charged particle in its rest frame, we discuss the electric monopole, magnetic dipole, electron singularity, and self-energy.

Paper Structure

This paper contains 7 sections, 7 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Liénard-Wiechert potential
  3. Set-up and notation
  4. Implicit differentiation involving the retarded proper time
  5. The d'Alembertian of $\Phi$
  6. Regularization of the electromagnetic self-interaction energy of a point particle at rest
  7. Proof of equality of $\Upsilon$ with the Heaviside function

Key Result

Lemma 2.3

It holds that which means in detail $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (14)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Theorem 2.5
  • Remark 2.6
  • Proposition 2.7
  • Remark 2.8
  • Proposition 3.1
  • ...and 4 more