The kinematical frame of Loop Quantum Gravity

In loop quantum gravity in the connection representation, the quantum configuration space A/G, which is a compact space, is much larger than the classical configuration space A/G of connections modulo gauge transformations. One finds that A/G is homeomorphic to the space Hom(L∗, G))/Ad. We give a new, natural proof of this result, suggesting the extension of the hoop group L∗ to a larger, compact group M(L∗) that contains L∗ as a dense subset. This construction is based on almost periodic functions. We introduce the Hilbert algebra L2(M(L∗)) of M(L∗) with respect to the Haar measure ξ on M(L∗). The measure ξ is shown to be invariant under 3-diffeomorphisms. This is the first step in a proof that L2(M(L∗)) 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. [email protected] [email protected]