The quasilocal degrees of freedom of Yang-Mills theory

Abstract

Gauge theories possess nonlocal features that, in the presence of boundaries, inevitably lead to subtleties. We employ geometric methods rooted in the functional geometry of the phase space of Yang-Mills theories to: (1) characterize a basis for quasilocal degrees of freedom (dof) that is manifestly gauge-covariant also at the boundary; (2) tame the non-additivity of the regional symplectic forms upon the gluing of regions; and to (3) discuss gauge and global charges in both Abelian and non-Abelian theories from a geometric perspective. Naturally, our analysis leads to splitting the Yang-Mills dof into Coulombic and radiative. Coulombic dof enter the Gauss constraint and are dependent on extra boundary data (the electric flux); radiative dof are unconstrained and independent. The inevitable non-locality of this split is identified as the source of the symplectic non-additivity, i.e. of the appearance of new dof upon the gluing of regions. Remarkably, these new dof are fully determined by the regional radiative dof only. Finally, a direct link is drawn between this split and Dirac's dressed electron.

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

“…Thus, the actual basic presymplectic 2-form is the d-exact part of the covariant derivative of the presymplectic potential θ. The above formula generalises the remark already made by Gomes & Riello in the YM case - [9], corollary 3.2 and section 3.4, see also [13] end of section 3.1. Manifestly, for a flat connections ω the situation is degenerate,…”

“…These ambiguity relations could also be interpreted as reflecting gluing properties: if one imagines that observers on regions Σ and Σ separated by a boundary ∂Σ use different variational connections to build their respective basic presymplectic structures, then (4.8)-(4.10) -with Σ on the left-hand side replaced by Σ -are gluing relations between these structures. Thus understood, the above results generalise the discussion of section 6.7 in [9] on gluings of basic Yang-Mills presymplectic potentials built via Singer-deWitt connections.…”

“…added arbitrarily, but built from the field content of the theorywhich is in line with the philosophy using variational connections as advocated e.g. in [9]. So, they are indeed field-dependent dressing fields as described here: compare eq.…”

“…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.…”