Poisson Algebra of Wilson Loops in Four-Dimensional Yang-Mills Theory

TL;DR

This paper develops a Poisson algebra framework for Wilson loops in four-dimensional Yang-Mills theory, revealing a Lie algebra structure for U(N) and a quadratic algebra for SU(N), with implications for the phase space geometry.

Contribution

It introduces a novel Poisson algebra of gauge invariant Wilson loop observables in Yang-Mills theory, providing new insights into their algebraic structure and phase space geometry.

Findings

For U(N), the algebra is a Lie algebra.

For SU(N), the algebra is quadratic.

Partial results suggest phase space as a co-adjoint orbit.

Abstract

We formulate the canonical structure of Yang--Mills theory in terms of Poisson brackets of gauge invariant observables analogous to Wilson loops. This algebra is non--trivial and tractable in a light--cone formulation. For U(N) gauge theories the result is a Lie algebra while for SU(N) gauge theories it is a quadratic algebra. We also study the identities satsfied by the gauge invariant observables. We suggest that the phase space of a Yang--Mills theory is a co--adjoint orbit of our Poisson algebra; some partial results in this direction are obtained.