Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back

Abstract

I develop a theory of symplectic reduction that applies to bounded regions in Yang-Mills theory and electromagnetism. In this theory superselection sectors for the electric-flux play a central role. Within a given superselection sector, the symplectic structure of the reduced Yang-Mills theory can always be defined without inclusion of new degrees of freedom, but is a priori not unique. I then consider the possibility of defining a symplectic structure on the union of all superselection sectors. This in turn requires including additional boundary degrees of freedom aka "edge modes." However, unless the new edge modes model a material physical system located at the boundary of the region, I argue that the the phase space extension by edge modes is also inherently ambiguous. In both the superselection and edge mode frameworks, the ambiguity can be understood as a residual gauge-fixing dependence due to the presence of boundaries -- a result that resonates with findings in QED with asymptotic boundaries. To conclude, I will compare and contrast the superselection and edge mode frameworks.