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

TL;DR

This paper develops noncommutative harmonic analysis tools for 2+1 quantum gravity, linking quantum group structures, the Duflo map, and sampling theory to better understand the model's geometric and algebraic properties.

Contribution

It introduces a classicalisation map connecting different quantum gravity formalisms and develops practical tools like the Duflo map and noncommutative sampling theory for the model.

Findings

Compatibility of $ riangleright$-product with noncommutative calculus

Identification of the Duflo map as noncommutative compression

Bandwidth limitation due to bounded $SU_2$ momentum

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.