Semidirect product of CCR and CAR algebras and asymptotic states in quantum electrodynamics

TL;DR

This paper constructs a C*-algebra combining CCR and CAR algebras to model asymptotic fields in quantum electrodynamics, revealing universal charge quantization and analyzing covariant representations with implications for the energy-momentum spectrum.

Contribution

It introduces a semidirect product algebra framework for CCR and CAR algebras in QED, exploring its representations and spectral properties.

Findings

Universal charge quantization emerges from the algebra structure.

Vacuum representations are necessarily nonregular with respect to electromagnetic fields.

Constructed covariant irreducible representations are close to vacuum and regular.

Abstract

A C*-algebra containing the CCR and CAR algebras as its subalgebras and naturally described as the semidirect product of these algebras is discussed. A particular example of this structure is considered as a model for the algebra of asymptotic fields in quantum electrodynamics, in which Gauss' law is respected. The appearence in this algebra of a phase variable related to electromagnetic potential leads to the universal charge quantization. Translationally covariant representations of this algebra with energy-momentum spectrum in the future lightcone are investigated. It is shown that vacuum representations are necessarily nonregular with respect to total electromagnetic field. However, a class of translationally covariant, irreducible representations is constructed excplicitly, which remain as close as possible to the vacuum, but are regular at the same time. The spectrum of energy-momentum fills the whole future lightcone, but there are no vectors with energy-momentum lying on a mass hyperboloid or in the origin.