Global Gauge Symmetries and Spatial Asymptotic Boundary Conditions in Yang-Mills theory

TL;DR

This paper rigorously derives the physical gauge group in Yang-Mills theory, clarifying the role of boundary conditions and extending the analysis to Yang-Mills-Higgs theory with phase-dependent boundary effects.

Contribution

It provides a rigorous derivation of the physical gauge group in Yang-Mills theory and explores boundary conditions in the Higgs phase.

Findings

The physical gauge group is derived for Abelian and non-Abelian theories.

Boundary conditions and gauge groups differ between unbroken and broken phases.

Restriction to boundary-preserving transformations follows from the structure of the state space.

Abstract

In Yang-Mills theory on a Euclidean Cauchy surface, the physical gauge group is often taken to be $\mathcal{G}^I/\mathcal{G}^\infty_0$, where $\mathcal{G}^I$ consists of boundary-preserving gauge transformations asymptoting to a constant, and $\mathcal{G}^\infty_0$ consists of transformations generated by the Gauss law constraint. We rigorously derive this physical gauge group for both Abelian and non-Abelian theories. A key result is that restricting to $\mathcal{G}^I$ follows from the structure of the instantaneous state space on which the instantaneous Lagrangian is defined. We extend our analysis to Yang-Mills-Higgs theory, showing that boundary conditions and the physical gauge group differ between the unbroken and broken phases.