Diffeomorphism covariant representations of the holonomy-flux ⋆-algebra

Abstract

Recently, Sahlmann (2002 Preprint gr-qc/0207111) proposed a new, algebraic point of view on the loop quantization. He brought up the issue of a ⋆-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 subalgebra 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 (2002 Preprint gr-qc/0207112) concerning the U(1) case.