Automorphism covariant representations of the holonomy-flux *-algebra

TL;DR

This paper investigates automorphism covariant representations of the holonomy-flux *-algebra in Loop Quantum Gravity, establishing that such representations are uniquely characterized by a specific measure on the space of generalized connections.

Contribution

It generalizes previous results by removing assumptions of trivial bundles and base manifolds, proving uniqueness of automorphism covariant representations in a broader setting.

Findings

Automorphism covariant representations are unique up to a measure on generalized connections.

The only Hilbert space supporting such a representation is a direct sum of L^2 spaces.

The result extends previous work to arbitrary principal bundles with compact connected structure groups.

Abstract

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 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^2 given by a unique measure on the space of generalized connections. This result is a generalization of our previous work (Class. Quantum. Grav. 20 (2003) 3543-3567, gr-qc/0302059) where we assumed that the principal bundle is trivial, and its base manifold is R^d.