The kinematical frame of Loop Quantum Gravity I

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dc.creatorDoering, Andreas
dc.creatorde Groote, Hans F.
dc.date2001-12-26
dc.date.accessioned2026-07-07T03:26:33Z
dc.date.available2026-07-07T03:26:33Z
dc.descriptionIn 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 $ξ$ on $\mathcal{M}(\mathcal{L}_{\ast})$. The measure $% ξ$ 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.
dc.description31 pages, no figures
dc.identifierhttps://arxiv.org/abs/gr-qc/0112072
dc.identifierhttp://arxiv.org/abs/gr-qc/0112072
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/34087
dc.subjectGeneral Relativity and Quantum Cosmology
dc.subjectMathematical Physics
dc.titleThe kinematical frame of Loop Quantum Gravity I
dc.typetext

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