2009
DOI: 10.4171/jncg/30
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On spectral triples in quantum gravity II

Abstract: A semifinite spectral triple for an algebra canonically associated to canonical quantum gravity is constructed. The algebra is generated by based loops in a triangulation and its barycentric subdivisions. The underlying space can be seen as a gauge fixing of the unconstrained state space of Loop Quantum Gravity. This article is the second of two papers on the subject.

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Cited by 30 publications
(113 citation statements)
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“…This result shows that this construction encodes information about the differential structure of the manifold M . This is in stark contrast to the measures used in for example loop quantum gravity and previously by ourselves, where the smooth connections have zero measure.…”
Section: A Hilbert Space Representation Of Qhd(m)contrasting
confidence: 73%
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“…This result shows that this construction encodes information about the differential structure of the manifold M . This is in stark contrast to the measures used in for example loop quantum gravity and previously by ourselves, where the smooth connections have zero measure.…”
Section: A Hilbert Space Representation Of Qhd(m)contrasting
confidence: 73%
“…This algebra, which was first introduced and studied in [] can be shown to be metric independent. For details on the HD(M) algebra we refer the reader to .…”
Section: The Quantum Holonomy‐diffeomorphism Algebramentioning
confidence: 99%
“…It is the Dirac operator used as a basic construction block of the spectral triple of Aastrup, Grimstrup, and Nest in loop quantum gravity to model tetrads. 1,10 The Dirac operator, D 1/3 , with parameter t = 1/3 is the cubic Dirac operator of Kostant, 9 whose square, we will see in a moment, has the simple property of consisting only of its degree two term and its degree zero term. Therefore, the family of Dirac operators D t interpolates the most important Dirac operators one considers on a Lie group.…”
Section: One-parameter Family Of Dirac Operators D Tmentioning
confidence: 99%
“…As a step to quantizing gravity in the Ashtekar framework, there are recent developments of describing such an extended space of connections using a spectral triple in noncommutative geometry [1,8], which captures the geometry of the space of generalized connections as operators on a Hilbert space. While the geometries of the base manifold and the space of G-connections on it are in theory retained, the construction of the spectral triple is considered too discrete to practically allow one to reconstruct the base manifold and its G-connections.…”
Section: Introductionmentioning
confidence: 99%
“…In another description, one uses a finite set of curves to probe the space of G-connections to obtain a finite dimensional manifold that depend on the sets of curves. By successively refining the finite sets of curves, one obtains a pro-manifold that extends the original space of connections to the so-called space of generalized connections [4].As a step to quantizing gravity in the Ashtekar framework, there are recent developments of describing such an extended space of connections using a spectral triple in noncommutative geometry [1,8], which captures the geometry of the space of generalized connections as operators on a Hilbert space. While the geometries of the base manifold and the space of G-connections on it are in theory retained, the construction of the spectral triple is considered too discrete to practically allow one to reconstruct the base manifold and its G-connections.…”
mentioning
confidence: 99%