The universal algebra of the electromagnetic field III. Static charges and emergence of gauge fields

TL;DR

This paper develops a universal algebraic framework for the electromagnetic field, incorporating static charges and gauge invariance, revealing hidden degrees of freedom and the role of gauge fields in quantum electrodynamics.

Contribution

It introduces a universal C*-algebra for gauge invariant operators that includes static charges and demonstrates how gauge fields encode non-observable degrees of freedom.

Findings

String-localized operators induce outer automorphisms of the algebra.

Gauge invariant operators encode information about non-observable gauge fields.

Explicit implementations of automorphisms using the Gupta-Bleuler formalism.

Abstract

A universal C*-algebra of gauge invariant operators is presented, describing the electromagnetic field as well as operations creating pairs of static electric charges having opposite signs. Making use of Gauss' law, it is shown that the string-localized operators, which necessarily connect the charges, induce outer automorphisms of the algebra of the electromagnetic field. Thus they carry additional degrees of freedom which cannot be created by the field. It reveals the fact that gauge invariant operators encode information about the presence of non-observable gauge fields underlying the theory. Using the Gupta-Bleuler formalism, concrete implementations of the outer automorphisms by exponential functions of the gauge fields are presented. These fields also appear in unitary operators inducing the time translations in the resulting representations of the universal algebra.