Edge States in Gauge Theories: Theory, Interpretations and Predictions

TL;DR

This paper investigates boundary-localized observables in gauge theories, their dependence on boundary conditions, and their physical implications, including experimental tests in models like superconductors and topological solitons.

Contribution

It provides a comprehensive analysis of edge observables in gauge theories, linking boundary conditions to physical phenomena and experimental parameters.

Findings

Edge observables can exist regardless of spacetime dimension.

Boundary conditions influence the existence and properties of edge states.

Explicit spectrum dependence on physical parameters is demonstrated in the Higgs model.

Abstract

Gauge theories on manifolds with spatial boundaries are studied. It is shown that observables localized at the boundaries (edge observables) can occur in such models irrespective of the dimensionality of spacetime. The intimate connection of these observables to charge fractionation, vertex operators and topological field theories is described. The edge observables, however, may or may not exist as well-defined operators in a fully quantized theory depending on the boundary conditions imposed on the fields and their momenta. The latter are obtained by requiring the Hamiltonian of the theory to be self-adjoint and positive definite. We show that these boundary conditions can also have nice physical interpretations in terms of certain experimental parameters such as the penetration depth of the electromagnetic field in a surrounding superconducting medium. The dependence of the spectrum on one such parameter is explicitly exhibited for the Higgs model on a spatial disc in its London limit. It should be possible to test such dependences experimentally, the above Higgs model for example being a model for a superconductor. Boundary conditions for the 3+1 dimensional $BF$ system confined to a spatial ball are studied. Their physical meaning is clarified and their influence on the edge states of this system (known to exist under certain conditions) is discussed. It is pointed out that edge states occur for topological solitons of gauge theories such as the 't Hooft-Polyakov monopoles.