Gauge-Invariant Coordinates on Gauge-Theory Orbit Space

TL;DR

This paper introduces a gauge-invariant coordinate system for non-Abelian gauge theories using a dual field, enabling analysis of the orbit space's geometry and implications for the mass gap problem.

Contribution

It develops a new gauge-invariant field as a coordinate on gauge orbit space, generalizes the Poincare'-Hodge formula, and explores geometric and measure-theoretic properties in Yang-Mills theories.

Findings

The new field is dual to the gauge field.

The metric and curvature of orbit space are formally non-negative.

The measure on wave functionals resembles known measures in 2+1 dimensions.

Abstract

A gauge-invariant field is found which describes physical configurations, i.e. gauge orbits, of non-Abelian gauge theories. This is accomplished with non-Abelian generalizations of the Poincare'-Hodge formula for one-forms. In a particular sense, the new field is dual to the gauge field. Using this field as a coordinate, the metric and intrinsic curvature are discussed for Yang-Mills orbit space for the (2+1)- and (3+1)-dimensional cases. The sectional, Ricci and scalar curvatures are all formally non-negative. An expression for the new field in terms of the Yang-Mills connection is found in 2+1 dimensions. The measure on Schroedinger wave functionals is found in both 2+1 and 3+1 dimensions; in the former case, it resembles Karabali, Kim and Nair's measure. We briefly discuss the form of the Hamiltonian in terms of the dual field and comment on how this is relevant to the mass gap for both the (2+1)- and (3+1)-dimensional cases.