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arXiv · hep-th/0601004

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

L. Freidel·S. Majid
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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.

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L. Freidel, S. Majid. 2007-12-22. Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity. https://doi.org/10.1088/0264-9381%2F25%2F4%2F045006

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