The kinematical frame of Loop Quantum Gravity I

Abstract

In loop quantum gravity in the connection representation, the quantum configuration space $\overline{\mathcal{A}/\mathcal{G}}$, which is a compact space, is much larger than the classical configuration space $\mathcal{A}/$ of connections modulo gauge transformations. One finds that is homeomorphic to the space $Hom($. We give a new, natural proof of this result, suggesting the extension of the hoop group ${\mathcal{L}}_{\ast }$ to a larger, compact group $\mathcal{M}({\mathcal{L}}_{\ast })$ that contains as a dense subset. This construction is based on almost periodic functions. We introduce the Hilbert algebra $L_2(\mathcal{M}($ of $\mathcal{M}({\mathcal{L}}_{\ast })$ with respect to the Haar measure $\xi$ on $\mathcal{M}({\mathcal{L}}_{\ast })$. The measure is shown to be invariant under 3-diffeomorphisms. This is the first step in a proof that $L_2(\mathcal{M}({\mathcal{L}}_{\ast }))$ is the appropriate Hilbert space for loop quantum gravity in the loop representation. In a subsequent paper, we will reinforce this claim by defining an extended loop transform and its inverse.

31 pages, no figures