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

In Yang-Mills theory on a Euclidean Cauchy surface, the physical gauge group is often taken to be ${\mathcal{G}}^I\mathrm{/}{\mathcal{G}}_0^{\mathrm{\infty }}$, where ${\mathcal{G}}^I$ consists of boundary-preserving gauge transformations asymptoting to a constant, and ${\mathcal{G}}_0^{\mathrm{\infty }}$ 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.