New Gauge Invariant Variables for Yang-Mills Theory

TL;DR

This paper introduces a new set of gauge invariant variables for 3+1 dimensional SU(2) Yang-Mills theory, providing a geometric interpretation and addressing the electric field energy calculation within the canonical formalism.

Contribution

It defines gauge invariant variables with a geometric interpretation in Yang-Mills theory, resolving issues related to the electric field energy calculation.

Findings

Established a GL(3) covariance in the basic equations

Provided a well-defined transformation of variables using perturbation theory

Clarified the geometric structure of the gauge invariant variables

Abstract

A new set of gauge invariant variables is defined to describe the physical Hilbert space of $d = 3 + 1$ $SU(2)$ Yang-Mills theory in the fixed-time canonical formalism. A natural geometric interpretation arises due to the $GL(3)$ covariance found to hold for the basic equations and commutators of the theory in the canonical formalism. We emphasize, however, that we are not interested in and do not consider the coupling of the theory to gravity. We concentrate here on a technical difficulty of this approach, the calculation of the electric field energy. This in turn hinges on the well-definedness of the transformation of variables, an issue which is settled through degenerate perturbation theory arguments.