Vacuum local and global electromagnetic self-energies for a point-like and an extended field source

TL;DR

This paper investigates the singular behavior of electromagnetic energy densities around point-like and extended sources, revealing how divergences arise and are resolved through proper mathematical treatment and source extension.

Contribution

It provides a detailed analysis of the singularities in electromagnetic self-energies for point-like sources and demonstrates how extending the source removes these divergences.

Findings

Point-like sources cause divergences in local energy densities.

Proper mathematical treatment includes delta functions and derivatives.

Extending the source eliminates divergences and resolves inconsistencies.

Abstract

We consider the electric and magnetic energy densities (or equivalently field fluctuations) in the space around a point-like field source in its ground state, after having subtracted the spatially uniform zero-point energy terms, and discuss the problem of their singular behavior at the source's position. We show that the assumption of a point-like source leads, for a simple Hamiltonian model of the interaction of the source with the electromagnetic radiation field, to a divergence of the renormalized electric and magnetic energy density at the position of the source. We analyze in detail the mathematical structure of such singularity in terms of a delta function and its derivatives. We also show that an appropriate consideration of these singular terms solves an apparent inconsistency between the total field energy and the space integral of its density. Thus the finite field energy stored in these singular terms gives an important contribution to the self-energy of the source. We then consider the case of an extended source, smeared out over a finite volume and described by an appropriate form factor. We show that in this case all divergences in local quantities such as the electric and the magnetic energy density, as well as any inconsistency between global and space-integrated local self-energies, disappear.