The quasilocal degrees of freedom of Yang-Mills theory

TL;DR

The paper develops a geometric, gauge-covariant framework for quasilocal Yang–Mills degrees of freedom in bounded regions, using the Singer–DeWitt functional connection to split field-space fluctuations into radiative and Coulombic components. This split, together with a careful treatment of boundaries, yields a covariant definition of regional charges and a robust gluing formalism that shows global (regionally reduced) symplectic forms are not simply the sum of regional parts; new radiative degrees of freedom emerge at interfaces, fully determined by regional radiatives. In Abelian theory, charges organize into superselection sectors and flux data can be completed via a covariant KKS structure on flux orbits, while in non-Abelian theory reducible configurations and their stabilizers complicate global charge definitions, motivating a stratified, sector-wise approach. The SdW-based dressing perspective connects to Dirac’s dressing and clarifies how dressed fields capture gauge-invariant content while still encoding physical charges and boundary data. Overall, the work clarifies the nonlocal, relational nature of YM quasilocal dof, links to Dirac dressing, and provides a principled framework for entanglement, energy accounting, and topological modes in gauge theories with boundaries.

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.

Figures (8)

• Figure 1: A pictorial representation of the configuration space ${\mathcal{A}}$ seen as a principal fibre bundle, on the right. We have highlighted a generic configuration $A$, its (gauge-transformed) image under the action of $R_g:A\mapsto A^g$, and its orbit $\mathcal{O}_A \cong {\mathcal{G}}$. We have also represented the quotient space of 'gauge-invariant configurations' ${\mathcal{A}}/{\mathcal{G}}$. On the left hand side of the picture, we have "zoomed into" a representation of $A$ and $A^g$ as two gauge-related local sections of a connection $\omega$ on $P$, the finite dimensional principal fibre bundle with structural group $G$ over $R$. The principal fibre bundle picture of ${\mathcal{A}}$ will be partially revisited in section \ref{['sec:charges']}---see figure \ref{['fig8']}.
• Figure 2: A pictorial representation of the split of ${\rm T}_A {\mathcal{A}}$ into a vertical subspace $V_A$ spanned by $\{\xi_A^\sharp, \xi\in{\mathrm{Lie}({\mathcal{G}})}\}$ and its horizontal complement $H_A$ defined as the kernel at $A$ of a functional connection $\varpi$. With dotted lines, we represent a different choice of horizontal complement associated to a different choice of $\varpi$.
• Figure 3: Pictorial representation of anholonomic horizontal plances in $\cal A$, corresponding to a non-vanishing curvature $\mathbb F\neq0$.
• Figure 4: A graphical representation in $\mathrm T{\mathcal{A}}$ of $\mathbb E_{\text{rad}}$ and $\mathbb E_{\text{Coul}}$ as vectors in the $\mathbb G$-orthogonal complements of $V_A$ and $H_A$, respectively. Notice that only with the SdW choice $\varpi = \varpi_{\text{SdW}}$, one has $H_A = V_A^\perp$ and therefore $\mathbb E_{\text{rad}} \in H_A^{\text{SdW}}$ and $\mathbb E_{\text{Coul}} \in V_A$; that is, only with this choice do the pictures on the right and left align---see section \ref{['sec:SdWCoul']}.
• Figure 5: In this representation ${\mathcal{A}}$ is the page's plane and the orbits are given by concentric circles. The field $A$ is generic, and has a generic orbit, $\mathcal{O}_A$. The field $\tilde{A}$ has a nontrivial stabilizer group (i.e. it has non-trivial reducibility parameters), and its orbit $\mathcal{O}_{\tilde{A}}$ is of a different dimension than $\mathcal{O}_A$. The projection of $\tilde{A}$ on ${\mathcal{A}}/{\mathcal{G}}$ therefore sits at a qualitatively different point than that of $A$ (a lower-dimensional stratum of ${\mathcal{A}}/{\mathcal{G}}$). Exclusion of the reducible configuration $\tilde{A}$ gives rise to a fibre bundle structure over ${\mathcal{A}}\setminus \{\tilde{A}\}$; here $\sigma$ represents a section of ${\mathcal{A}}\setminus \{\tilde{A}\}$. Locally, the concept of section can be generalized to include reducible configurations such as $\tilde{A}$, thus leading to the notion of "slice". This is briefly reviewed in appendix \ref{['app:slice']}.

Theorems & Definitions (63)

• Definition 2.1: Functional connection GomesHopfRiello
• Remark 2.2: On nonlocality
• Definition 2.3: Horizontal differential
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Definition 2.6: Functional curvature Singer:1981xwGomesHopfRiello
• Proposition 2.7
• proof