Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity

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dc.creatorFreidel, L.
dc.creatorMajid, S.
dc.date2005-12-30
dc.date2007-12-22
dc.date.accessioned2026-07-07T11:27:21Z
dc.date.available2026-07-07T11:27:21Z
dc.descriptionWe show that the $\star$-product for $U(su_2)$, group Fourier transform and effective action arising in [1] in an effective theory for the integer spin Ponzano-Regge quantum gravity model are compatible with the noncommutative bicovariant differential calculus, quantum group Fourier transform and noncommutative scalar field theory previously proposed for 2+1 Euclidean quantum gravity using quantum group methods in [2]. The two are related by a classicalisation map which we introduce. We show, however, that noncommutative spacetime has a richer structure which already sees the half-integer spin information. We argue that the anomalous extra `time' dimension seen in the noncommutative geometry should be viewed as the renormalisation group flow visible in the coarse-graining in going from $SU_2$ to $SO_3$. Combining our methods we develop practical tools for noncommutative harmonic analysis for the model including radial quantum delta-functions and Gaussians, the Duflo map and elements of `noncommutative sampling theory'. This allows us to understand the bandwidth limitation in 2+1 quantum gravity arising from the bounded $SU_2$ momentum and to interpret the Duflo map as noncommutative compression. Our methods also provide a generalised twist operator for the $\star$-product.
dc.description53 pages latex, no figures; extended the intro for this final version
dc.identifierhttps://arxiv.org/abs/hep-th/0601004
dc.identifierhttp://arxiv.org/abs/hep-th/0601004
dc.identifierClass.Quant.Grav.25:045006,2008
dc.identifierdoi:10.1088/0264-9381/25/4/045006
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/196123
dc.subjectHigh Energy Physics - Theory
dc.subjectGeneral Relativity and Quantum Cosmology
dc.subjectQuantum Algebra
dc.titleNoncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity
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