The kinematical frame of Loop Quantum Gravity I

2001/12/26 by Andreas Doering, Doering, Andreas, Hans F. de Groote +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph) #gr-qc #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.gr-qc/0112072

31 pages, no figures

arxiv created 2001/12/26 · arxiv updated 2009/11/30

Abstract

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.

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