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

, where

\mathcal{G}^I 𝒢 I

consists of boundary-preserving gauge transformations asymptoting to a constant, and

\mathcal{G}^∞_0 𝒢 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 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.