Action of the conformal group on steady state solutions to Maxwell's equations and background radiation

TL;DR

This paper explores how the conformal group acts on steady state solutions of Maxwell's equations in a compactified Minkowski space, revealing a detailed representation structure with physical implications.

Contribution

It explicitly decomposes the conformal group's action into four irreducible representations and analyzes their properties and physical significance.

Findings

Representation decomposes into four irreducible components

Each component has a defined inner product and frequency spectrum

Representations exhibit positive/negative energy and quaternionic structure

Abstract

The representation of the conformal group (PSU(2,2)) on the space of solutions to Maxwell's equations on the conformal compactification of Minkowski space is shown to break up into four irreducible unitarizable smooth Fr\'echet representations of moderate growth. An explicit inner product is defined on each representation. The frequency spectrum of each of these representations is analyzed. These representations have notable properties; in particular they have positive or negative energy, they are of type $A_{\frak q}(\lambda)$ and are quaternionic. Physical implications of the results are explained.