Generalized coordinates on the phase space of Yang-Mills theory

TL;DR

This paper demonstrates that complex Wilson loop variables serve as complete generalized coordinates on the physical phase space of SU(2) Yang-Mills theory, with implications for gauge theories and general relativity.

Contribution

It constructs a natural one-to-one map linking the physical phase space of Yang-Mills theory to complexified configuration space, establishing Wilson loops as a complete set of generalized coordinates.

Findings

Complex Wilson loop variables form a complete set of coordinates.

A natural map connects Yang-Mills phase space to complexified configuration space.

Implications discussed for gauge theory and general relativity.

Abstract

We study the suitability of complex Wilson loop variables as (generalized) coordinates on the physical phase space of $SU(2)$-Yang-Mills theory. To this end, we construct a natural one-to-one map from the physical phase space of the Yang-Mills theory with compact gauge group $G$ to a subspace of the physical configuration space of the complex $G^\C$-Yang-Mills theory. Together with a recent result by Ashtekar and Lewandowski this implies that the complex Wilson loop variables form a complete set of generalized coordinates on the physical phase space of $SU(2)$-Yang-Mills theory. They also form a generalized canonical loop algebra. Implications for both general relativity and gauge theory are discussed.