Edge modes as reference frames and boundary actions from post-selection

TL;DR

Edge modes in gauge theories are interpreted as dynamical reference frames for finite regions, enabling a regional covariant phase space via a post-selection procedure on a global solution space. The authors develop a systematic algorithm to induce boundary actions and boundary conditions that render the regional variational problem well-defined, recasting edge modes as frame-dressed or relational observables. They identify three types of boundary symmetries—frame reorientations, meta-symmetries, and boundary gauge symmetries—whose status depends on the chosen boundary conditions and illustrate the construction in Maxwell, Abelian Chern-Simons, and Yang-Mills theories. The work connects covariant phase space methods with quantum reference-frame ideas and provides a framework for extending edge-mode concepts to gravitational settings and quantum gravity contexts.

Abstract

We introduce a general framework realizing edge modes in (classical) gauge field theory as dynamical reference frames, an often suggested interpretation that we make entirely explicit. We focus on a bounded region $M$ with a co-dimension one time-like boundary $Γ$, which we embed in a global spacetime. Taking as input a variational principle at the global level, we develop a systematic formalism inducing consistent variational principles (and in particular, boundary actions) for the subregion $M$. This relies on a post-selection procedure on $Γ$, which isolates the subsector of the global theory compatible with a general choice of gauge-invariant boundary conditions for the dynamics in $M$. Crucially, the latter relate the configuration fields on $Γ$ to a dynamical frame field carrying information about the spacetime complement of $M$; as such, they may be equivalently interpreted as frame-dressed or relational observables. Generically, the external frame field keeps an imprint on the ensuing dynamics for subregion $M$, where it materializes itself as a local field on the time-like boundary $Γ$; in other words, an edge mode. We identify boundary symmetries as frame reorientations and show that they divide into three types, depending on the boundary conditions, that affect the physical status of the edge modes. Our construction relies on the covariant phase space formalism, and is in principle applicable to any gauge (field) theory. We illustrate it on three standard examples: Maxwell, Abelian Chern-Simons and non-Abelian Yang-Mills theories. In complement, we also analyze a mechanical toy-model to connect our work with recent efforts on (quantum) reference frames.

Figures (6)

• Figure 1: Two types of transformations which are indistinguishable from their action on particles in region $M$, but distinguishable relative to the dynamical frame $R_1$. The vector field ${\mathcal{X}}_\alpha$ (top) moves both the particles in $M$ and the frame $R_1$ by the same distance, therefore does not affect relative distances and corresponds to a gauge direction. The symmetry (frame reorientation) generator ${\mathcal{Y}}_\rho$ (bottom) only acts on $R_1$, thereby affects relative distances to the reference frame: it generates a physical transformation.
• Figure 2: Pictorial illustration of the frame change mapping the relational observables $\{ Q_{i\vert R_1} \}$ to $\{ Q_{i\vert R_2} \}$. It is implemented by a field-dependent reference frame transformation, namely a translation by the relative distance $U_{R_2} - U_{R_1}$.
• Figure 3: The spacetime region $M$ and its complement. From a system of Wilson lines $\{ \gamma_x \}$ anchored at asymptotic source $\{ s_x \}$, we construct the dynamical edge field $U[\bar{A}]$, which provides a reference frame for the time-like boundary $\Gamma$. It allows us to decompose gauge-invariant observables with support on both $M$ and $\bar{M}$, such as the Wilson loop $W$, into composites of regional gauge-invariant observables.
• Figure 4: Wilson loop $W=\mathrm{Tr}\left(\bar{H}_{xy}[\bar{A}]^{-1}\,H_{xy}[A]\right)$, with support on both $M$ and $\bar{M}$.
• Figure 5: Two systems of paths $\gamma^{(1)}$ and $\gamma^{(2)}$, supporting distinct families of Wilson lines. They generate two distinct group-valued reference frames $U_1[\bar{A}]$ and $U_2[\bar{A}]$ for the boundary $\Gamma$.