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 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.

arXiv commentPublished version: entirely self-contained; now includes explicit reference to Marsden-Weinstein-Meyer symplectic reduction (Sec. 1.5 + App. 1) and an improved discussion of superselection (Sect. 1.6 and 5.1). SciPost is a non-profit, two-way open access publication portal featuring peer-witnessed refereeing

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