- qc / 0 40 51 19 v 2 1 1 Fe b 20 05 Automorphism covariant representations of the holonomy - flux ∗ - algebra

We continue the 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 [1], 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; this 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 the spaces L, given by a unique measure on the space of generalized connections. This result is a generalization of our previous work [2] where we assumed that the principal bundle is trivial and its base manifold is R.