Diffeomorphism Covariant Representations of the Holonomy-flux ⋆-algebra

Recently, Sahlmann [1] 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 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 [2] concerning the U(1) case.