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

TL;DR

The work addresses how a noncommutative geometric description can arise as an effective theory for Euclidean 2+1 gravity and presents a coherent framework that ties the $\star$-product on $U(\mathfrak{su}_2)$ to a bicovariant differential calculus and noncommutative harmonic analysis. It introduces a classicalisation map $\phi$ that links noncommutative plane waves to group-labeled classical waves, revealing that half-integer spin data live in the noncommutative extension via an extra RG-like direction from $SU(2)$ to $SO(3)$. The paper develops radial harmonic analysis, a radial sampling theorem, and a Duflo map-based compression, clarifying bandwidth limits due to bounded $SU(2)$ momentum and enabling noncommutative simulations in physics of 2+1 gravity. It further extends the framework to a 4D setting with a twist operator, connecting renormalisation group ideas with noncommutative geometry and providing a practical toolkit for noncommutative sampling, Gaussian functionals, and radial projections within the gravity-inspired model.

Abstract

We 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.