Diffeomorphism covariant representations of the holonomy-flux star-algebra

TL;DR

This paper characterizes all diffeomorphism covariant representations of the Sahlmann algebra in loop quantum gravity, showing they are derived from a natural measure on the space of generalized connections, extending previous U(1) results.

Contribution

It proves that all diffeomorphism covariant representations of the Sahlmann algebra are given by a specific measure, generalizing prior results to a broader setting.

Findings

All such representations are given by a natural measure.

They are characterized by Sahlmann's decomposition.

The result extends to non-Abelian gauge groups.

Abstract

Recently, Sahlmann proposed a new, algebraic point of view on the loop quantization. He brought up the issue of a star-algebra underlying that framework, studied the algebra consisting of the fluxes and holonomies and characterized its representations. We define the diffeomorphism covariance of a representation of the Sahlmann algebra and study the diffeomorphism covariant representations. We prove they are all given by Sahlmann's decomposition into the cyclic representations of the sub-algebra of the holonomies by using a single state only. The state corresponds to the natural measure defined on the space of the generalized connections. This result is a generalization of Sahlmann's result concerning the U(1) case.