Gauss's law, the manifestations of gauge fields, and their impact on local observables

TL;DR

This paper explores how gauge fields and Gauss's law influence local observables in electromagnetic theory, revealing the role of automorphisms and algebra enlargements in representing external charges and their effects.

Contribution

It introduces an algebraic framework that incorporates external charges via automorphisms and algebra enlargements, clarifying gauge invariance and charge representation.

Findings

Automorphisms induced by external charges are outer automorphisms.

Enlarged algebra includes gauge fields and external charge algebra.

States with zero global charge are locally disjoint from the vacuum.

Abstract

Within the framework of the universal algebra of the electromagnetic field, the impact of globally neutral configurations of external charges on the field is analyzed. External charges are not affected by the field, but they induce localized automorphisms of the universal algebra. Gauss's law implies that these automorphisms cannot be implemented by unitary operators involving only the electromagnetic field, they are outer automorphisms. The missing degrees of freedom can be incorporated in an enlargement of the universal algebra, which can concretely be represented by exponential functions of gauge fields and an abelian algebra describing the external charges. In this manner, gauge fields manifest themselves in the framework of gauge invariant observables. The action of the automorphisms on the vacuum state gives rise to representations of the electromagnetic field with vanishing global charge, which are locally disjoint from the vacuum representation. This feature disappears in the enlarged universal algebra of the electromagnetic field. The energy content of the states is well defined in both cases and bounded from below. The passage from these globally neutral states to charged states and the determination of their energy content are also being discussed.