The universal C*-algebra of the electromagnetic field II. Topological charges and spacelike linear fields

TL;DR

This paper investigates the conditions under which topological charges emerge in the universal C*-algebra of the electromagnetic field, highlighting the role of non-linearity and spacelike linearity in various representations.

Contribution

It demonstrates that non-trivial topological charges require non-linear dependence on test functions and constructs examples of regular vacuum representations with such charges.

Findings

Topological charges appear only with non-linear field dependence.

Examples of vacuum representations with topological charges are constructed.

Topological charges are discussed in the context of multiple electromagnetic fields and non-abelian gauge theories.

Abstract

Conditions for the appearance of topological charges are studied in the framework of the universal C*-algebra of the electromagnetic field, which is represented in any theory describing electromagnetism. It is shown that non-trivial topological charges, described by pairs of fields localised in certain topologically non-trivial spacelike separated regions, can appear in regular representations of the algebra only if the fields depend non-linearly on the mollifying test functions. On the other hand, examples of regular vacuum representations with non-trivial topological charges are constructed, where the underlying field still satisfies a weakened form of "spacelike linearity". Such representations also appear in the presence of electric currents. The status of topological charges in theories with several types of electromagnetic fields, which appear in the short distance (scaling) limit of asymptotically free non-abelian gauge theories, is also briefly discussed.