Non-Abelianizable First Class Constraints

TL;DR

This paper investigates the conditions under which first class constraints in gauge theories can be Abelianized, revealing that non-Abelianizable constraints lead to a vanishing Faddeev-Popov determinant, impacting gauge fixing.

Contribution

It provides necessary and sufficient conditions for Abelianizability of first class constraints, linking topological and local structure considerations, and examines implications for gauge fixing.

Findings

Non-Abelianizable constraints cause the Faddeev-Popov determinant to vanish.

Necessary conditions are derived from topological properties of gauge groups.

Explicit analysis performed on an SO(3) gauge invariant model.

Abstract

We study the necessary and sufficient conditions on Abelianizable first class constraints. The necessary condition is derived from topological considerations on the structure of gauge group. The sufficient condition is obtained by applying the theorem of implicit function in calculus and studying the local structure of gauge orbits. Since the sufficient condition is necessary for existence of proper gauge fixing conditions, we conclude that in the case of a finite set of non-Abelianizable first class constraints, the Faddeev-Popov determinant is vanishing for any choice of subsidiary constraints. This result is explicitly examined for SO(3) gauge invariant model.