The kinematical frame of Loop Quantum Gravity I

Abstract

In loop quantum gravity in the connection representation, the quantum configuration space $\bar{\mathcal{A}/\mathcal{G}}$, which is a compact space, is much larger than the classical configuration space $\mathcal{A}/% \mathcal{G}$ of connections modulo gauge transformations. One finds that $% \bar{\mathcal{A}/\mathcal{G}}$ is homeomorphic to the space $Hom(% \mathcal{L}_{\ast},G))/Ad$. 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 $% \mathcal{L}_{\ast}$ as a dense subset. This construction is based on almost periodic functions. We introduce the Hilbert algebra $L_{2}(\mathcal{M}(% \mathcal{L}_{\ast}))$ of $\mathcal{M}(\mathcal{L}_{\ast})$ with respect to the Haar measure $\xi $ on $\mathcal{M}(\mathcal{L}_{\ast})$. The measure $% \xi $ 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.