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.
Highlights
In this article I have studied the construction of a symplectic structure on the reduced phase space of YM theory in the presence of boundaries
I have done so both within covariant superselection sectors for the electric flux, and in a larger context where new “edge-mode” dof conjugate to the fluxes are included in an extended phase space
In the non-Abelian case, this is achieved after one realizes that a canonical completion of the “radiative” symplectic structure (ι∗ΩH ) exists which equips the Coulombic electric field with its own symplectic structure. This completion is based on the Kirillov–Konstant– Souriau construction and leads to non-commutative electric fluxes whose boundary smearings generate—if is flat—physical transformations of the underlying system
Summary
1.1 Context and motivations Building on [1,2,3,4,5], in this article I elaborate and present—in a self-contained manner—a theory of symplectic reduction for Yang–Mills gauge theories over finite and bounded regions. 1.4 Flux rotations In non-Abelian theories, the enlargement of the notion of superselection sector to its covariant counterpart introduces the possibility of “rotating” f within its conjugacy class [ f ] without altering neither ρ nor A These transformations—that I name flux rotations—produce genuinely new field configurations, for they alter, via the Gauss constraint, the Coulombic potential and the energy content of R. This could have been expected because the more naive reduction procedure gets rid of the pure-gauge dof as well as of the Coulombic dof which are conjugate to them; but since f is imprinted precisely in the Coulombic part of the electric field, getting rid of it means that f drops from the candidate symplectic 2-form This is not a problem in Abelian theories, where f is fixed once and for all in a given superselection sector. It is time to delve into the details
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