Non-commutative waves for gravitational anyons

TL;DR

This paper explores the representation theory of quantum doubles related to 2+1 dimensional Lorentz groups, revealing non-commutative wave structures and deformed symmetries relevant to quantum gravity.

Contribution

It introduces a covariant field framework on curved momentum space using group Fourier transforms, connecting quantum group representations with non-commutative geometry in (2+1)D gravity.

Findings

Representation theory expressed via covariant infinite-component fields.

Derivation of star product on Minkowski space through deformation quantisation.

Identification of non-commutative wave equations linked to quantum Lorentz symmetry.

Abstract

We revisit the representation theory of the quantum double of the universal cover of the Lorentz group in 2+1 dimensions, motivated by its role as a deformed Poincar\'e symmetry and symmetry algebra in (2+1)-dimensional quantum gravity. We express the unitary irreducible representations in terms of covariant, infinite-component fields on curved momentum space satisfying algebraic spin and mass constraints. Adapting and applying the method of group Fourier transforms, we obtain covariant fields on (2+1)-dimensional Minkowski space which necessarily depend on an additional internal and circular dimension. The momentum space constraints turn into differential or exponentiated differential operators, and the group Fourier transform induces a star product on Minkowski space and the internal space which is essentially a version of Rieffel's deformation quantisation via convolution.