Contour gauges, canonical formalism and flux algebras

TL;DR

This paper explores the mathematical structure of contour gauges, their relation to flux algebras, and implications for gauge theories like electromagnetism and Yang-Mills, highlighting new methods for defining charges and addressing topological obstructions.

Contribution

It introduces a general framework for contour gauges based on region contractions, defines non-abelian flux and local charges, and analyzes gauge invariance and topological issues in gauge theories.

Findings

Defined a class of contour gauges via region contractions.

Constructed non-abelian flux and charge algebra.

Identified topological obstructions to Gauss law and proposed solutions.

Abstract

A broad class of contour gauges is shown to be determined by admissible contractions of the geometrical region considered and a suitable equivalence class of curves is defined. In the special case of magnetostatics, the relevant electromagnetic potentials are directly related to the ponderomotive forces. Schwinger's method of extracting a gauge invariant factor from the fermion propagator could, it is argued, lead to incorrect results. Dirac brackets of both Maxwell and Yang-Mills theories are given for arbitrary admissible space-like paths. It is shown how to define a non-abelian flux and local charges which obey a local charge algebra. Fields associated with the charges differ from the electric fields of the theory by singular topological terms; to avoid this obstruction to the Gauss law it is necessary to exclude a single, gauge fixing curve from the region considered.