The Hamiltonian Analysis for Yang-Mills Theory on $R\times S^2$

TL;DR

This paper develops a gauge-invariant Hamiltonian formalism for pure Yang-Mills theory on ${ m R} imes S^2$, analyzing its configuration space, mass gap, and vacuum structure, with results connecting smoothly to the ${ m R}^3$ case.

Contribution

It introduces a novel Hamiltonian approach for Yang-Mills on ${ m R} imes S^2$ using gauge-invariant parametrization, extending previous ${ m R}^3$ analyses to curved spatial manifolds.

Findings

Derived the volume measure on the physical configuration space.

Analyzed the nonperturbative mass gap.

Computed the leading term of the vacuum wave functional.

Abstract

Pure Yang-Mills theory on ${\mathbb R} \times S^2$ is analyzed in a gauge-invariant Hamiltonian formalism. Using a suitable coordinatization for the sphere and a gauge-invariant matrix parametrization for the gauge potentials, we develop the Hamiltonian formalism in a manner that closely parallels previous analysis on ${\mathbb R}^3$. The volume measure on the physical configuration space of the gauge theory, the nonperturbative mass-gap and the leading term of the vacuum wave functional are discussed using a point-splitting regularization. All the results carry over smoothly to known results on ${\mathbb R}^3$ in the limit in which the sphere is de-compactified to a plane.