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

TL;DR

This paper develops a gauge-invariant symplectic reduction framework for bounded regions in Yang-Mills and electromagnetism, emphasizing superselection sectors and boundary edge modes, and analyzing their physical and mathematical implications.

Contribution

It introduces a gauge-covariant symplectic reduction based on superselection sectors, clarifies the role of flux rotations, and discusses the ambiguity of boundary edge modes in Yang-Mills theory.

Findings

Superselection sectors define natural symplectic structures without new degrees of freedom.

Flux rotations are not dynamical symmetries as they affect system energy.

Including boundary edge modes introduces gauge-breaking ambiguities unless modeling physical boundary matter.

Abstract

I develop a theory of symplectic reduction that applies to bounded regions in Yang-Mills theory and electromagnetism. In this theory gauge-covariant superselection sectors for the electric flux through the boundary of the region play a central role: within such sectors, there exists a natural, canonically defined, symplectic structure for the reduced Yang-Mills theory. This symplectic structure does not require the inclusion of any new degrees of freedom. In the non-Abelian case, it also supports a family of Hamiltonian vector fields, which I call "flux rotations," generated by smeared, Poisson-non-commutative, electric fluxes. Since the action of flux rotations affects the total energy of the system, I argue that flux rotations fail to be dynamical symmetries of Yang-Mills theory restricted to a region. I also 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 phase space extension by edge modes is inherently ambiguous and gauge-breaking.