Automorphism Covariant Representations of the Holonomy-flux * -algebra

We continue an analysis of representations of cylindrical functions and fluxes which are commonly used as elementary variables of Loop Quantum Gravity. We consider an arbitrary principal bundle of a compact connected structure group and following Sahlmann’s ideas [2] define a holonomy-flux ∗-algebra whose elements correspond to the elementary variables. There exists a natural action of automorphisms of the bundle on the algebra; the action generalizes the action of analytic diffeomorphisms and gauge transformations on the algebra considered in earlier works. We define the automorphism covariance of a ∗-representation of the algebra on a Hilbert space and prove that the only Hilbert space admitting such a representation is a direct sum of spaces L given by a unique measure on the space of generalized connections. This result is a generalization of our previous work [5] where we assumed that the principal bundle is trivial, and its base manifold is R.