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

We show that the ⋆-product for U(su2), 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 SU2 to SO3. 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 ⋆-product.