Delta Interactions and Electrodynamics of Point Particles

TL;DR

This paper explores the mathematical relationship between point particle interactions, the Maxwell-Lorentz system, and point interactions in wave equations, revealing how classical radiation reaction equations emerge from singular perturbations.

Contribution

It establishes a rigorous connection between point perturbations of the Laplacian and the Maxwell-Lorentz system, deriving classical equations from a wave operator with point interactions.

Findings

Classical Abraham-Lorentz-Dirac equation derived as a limit of the wave equation with point interactions.

Hamiltonian structure of the limit model is identified.

Reduced dynamics analyzed on the stable manifold in absence of external forces.

Abstract

We report on some recent work of the authors showing the relations between singular (point) perturbation of the Laplacian and the dynamical system describing a charged point particle interacting with the self-generated radiation field (the Maxwell-Lorentz system) in the dipole approximation. We show that in the limit of a point particle, the dynamics of the system is described by an abstract wave equation containing a selfadjoint operator $H_m$ of the class of point interactions; the classical Abraham-Lorentz-Dirac third order equation, or better its integrated second order version, emerges as the evolution equation of the singular part of the field and is related to the boundary conditions entering in the definition of the operator domain of $H_m$. We also give the Hamiltonian structure of the limit model and, in the case of no external force, we study the reduced dynamics on the linear stable manifold.