Diffeomorphism covariant representations of the holonomy-flux star-algebra
Andrzej Okolow, Jerzy Lewandowski
TL;DR
This work addresses the classification of diffeomorphism covariant representations of the holonomy-flux $\star$-algebra in loop quantum gravity. It defines precise covariance notions for Sahlmann’s algebra, and uses Sahlmann’s decomposition into cyclic holonomy representations together with the Ashtekar-Lewandowski measure to constrain the structure of representations. The main result shows that for $\Sigma=\mathbb{R}^d$ and a compact connected gauge group $G$, all measure components $\mu_\nu$ in the decomposition must be the Ashtekar-Lewandowski measure $\mu_{\rm AL}$, and the flux-correction terms are real-valued; hence the representations reduce to the standard AL-type form up to real flux corrections. A companion finding proves that if the flux operators act symmetrically, the underlying measure must be $\mu_{\rm AL}$, reinforcing the uniqueness of the AL framework under these symmetry assumptions. This generalizes Sahlmann’s $U(1)$ result to non-Abelian groups and provides a principled path to classifying diffeomorphism-covariant representations via the AL measure.
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.