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/\mathcal{G}^∞_0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi mathvariant="script">𝒢</mml:mi> <mml:mi>I</mml:mi> </mml:msup> <mml:mi>/</mml:mi> <mml:msubsup> <mml:mi mathvariant="script">𝒢</mml:mi> <mml:mn>0</mml:mn> <mml:mi>∞</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> , where \mathcal{G}^I <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi mathvariant="script">𝒢</mml:mi> <mml:mi>I</mml:mi> </mml:msup> </mml:math> consists of boundary-preserving gauge transformations asymptoting to a constant, and \mathcal{G}^∞_0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msubsup> <mml:mi mathvariant="script">𝒢</mml:mi> <mml:mn>0</mml:mn> <mml:mi>∞</mml:mi> </mml:msubsup> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi mathvariant="script">𝒢</mml:mi> <mml:mi>I</mml:mi> </mml:msup> </mml:math> 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.