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.

Quantum reference frame transformations as symmetries and the paradox of the third particle

“…[63], while gauge symmetries do not change the physics of a given system, they alter the way that the system interacts with other systems. This observation is at the heart of the recent pivot to edge modes in gauge theory and gravity [64][65][66][67][68][69][70][71] and our resolution of the paradox of the third particle in Section 4 can also be viewed in this light. In case (iii), the external relatum would be best described as an external classical reference frame, for example the laboratory of an agent experimenting with S. This illustrates that to consider a system "in isolation" in the sense of Assumption 1 does not imply that the system S is literally a physically isolated system.…”

“…The paradox of the third particle and our resolution can be viewed as a finite-dimensional analog of the problem of boundaries and edge modes in gauge theory and gravity [64][65][66][67][68][69][70][71]. Boundaries in space or spacetime usually break gauge-invariance and constitute challenges for gauge-invariant observables.…”

“…As we will see, the core of the problem is how to embed the two-particle observables into the set of threeparticle observables, and the key will be to do so in a relational manner. This bears some resemblance to the issue of boundaries and edge modes in gauge theory and gravity [64][65][66][67][68][69][70][71], which is related to the question of how to embed the gauge-invariant observables of neighbouring subregions in spacetime into the set of gauge-invariant observables associated with the union ('gluing') of these subregions.…”

Note on the bundle geometry of field space, variational connections, the dressing field method, & presymplectic structures of gauge theories over bounded regions

“…Even though one could compare the outputs of both methods in common examples, e.g. in the YM case as was done in [9] or [13], we aim for a more ambitious goal: we want to confront these two approaches in their most general versions. To do so, we must first conduct the most general analysis possible of the relevant presymplectic structure of gauge theories coupled to matter, restricting ourselves -for reasons to be clarified later -to theories that are strictly gauge invariant.…”

“…In the wake of renewed interest in this issue, over the past few years two strategies have been proposed to deal with this boundary problem in Yang-Mills theory and General Relativity: the "edge modes" strategy as introduced by Donnelly & Freidel in [5], and the use of variational connections on field space as advocated by Gomez & Riello first in [6] and further developed in [7][8][9] (see also [10][11][12][13]). Both essentially aim at providing a modified presymplectic structure that descends onto M S and that we will call basic for reasons to be made clear in due time.…”