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 electromagnetism and Yang–Mills theories. 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, I argue that a commonly used phase space extension by edge modes is inherently ambiguous and gauge-breaking.