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.
Highlights
It is well known that gauge field theories defined on manifolds with boundaries can support the emergence of dynamical edge modes
We introduce a general framework realizing edge modes in gauge field theory as dynamical reference frames, an often suggested interpretation that we make entirely explicit
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
Summary
It is well known that gauge field theories defined on manifolds with boundaries can support the emergence of dynamical edge modes. This will allow us to rederive the structure of Donnelly-Freidel edge modes from first principles, while making their interpretation as reference frames completely transparent; in particular, we will show that they are dynamical reference frames in the same sense as they appear in the recent quantum reference frame literature [41–54] By investigating their interplay with generic boundary conditions, we will elucidate how they can sometimes (but not always) support a non-trivial algebra of boundary symmetries acting on the physical phase space. We will investigate general boundary conditions, that can be obtained by application of a linear canonical transformation of some set of gauge-invariant local canonical Darboux coordinates for the presymplectic current on Γ This will allow us to observe that, within the same global theory space, the physical role of edge modes for the post-selected dynamics in subregion M can be contingent on the choice of boundary conditions. Theory, which we take as a main basic example to illustrate our formalism throughout the paper, and only work out in full detail in that section; Abelian Chern-Simons theory, a interesting example in which the edge modes construction gives rise to an infinite-dimensional algebra of boundary symmetries; and Yang-Mills theory, as a way to illustrate non-Abelian features of the general formalism
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