Topological Dirac variables in Abelian $U(1)$ theory

TL;DR

This paper explores topological Dirac variables in Abelian $U(1)$ theory, revealing that standard QED is the trivial topological sector and identifying monopole modes for non-zero topological charge.

Contribution

It constructs topological Dirac variables for Abelian $U(1)$ theory, extending the concept from non-Abelian models and clarifying the topological structure of QED.

Findings

QED corresponds to the topologically trivial sector ($n=0$).

Non-zero topological charge ($n eq 0$) yields Dirac monopole modes.

Both sectors can be quantized via Hamiltonian reduction using Dirac variables.

Abstract

In this study we, remembering the experience with topological Dirac variables in the non-Abelian Yang-Mills-Higgs (YMH) model with vacuuum BPS monopole solutions, attempt to construct similar for the Abelian $U(1)$ model. We show that QED, as one understands it commonly, is only the topologically trivial sector ($n=0$) of this Abelian $U(1)$ model. For $n\neq 0$ one gets Dirac monopole modes. In both the cases, $n=0$ and $n\neq 0$, the theory can be quantized via the Hamiltonian reduction in terms of Dirac variables.