Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity

We show that the star-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 SU2 momentum and to interpret the Duflo map as noncommutative compression. Our methods also provide a generalized twist operator for the star-product.