2006/01/31 by Laurent Freidel, L. Freidel, Shahn Majid +1 · 79 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Black Holes and Theoretical Physics #Canonical quantum gravity #Euclidean quantum gravity #Noncommutative algebraic geometry #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Noncommutative quantum field theory #Quantum differential calculus #Quantum gravity #Quantum spacetime #gr-qc #hep-th #math.QA
paper · pdf · doi:10.1088/0264-9381/25/4/045006
published in Classical and Quantum Gravity 25(4), 045006 (IOP Publishing) · 53 pages latex, no figures; extended the intro for this final version
arxiv created 2007/12/22 · openalex publication_date 2008/01/31 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/08
We show that the ⋆-product for U ( su 2 ), group Fourier transform and effective action arising in Freidel and Livine (2005 Preprint hep-th/0502106) 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 the 2+1 Euclidean quantum gravity using quantum group methods in Batista and Majid (2003 J. Math. Phys. 44 107–37). The two are related by a classicalization 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 renormalization 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 the 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 generalized twist operator for the ⋆-product.