Non-commutative waves for gravitational anyons

Sergio Inglima, Bernd Johannes Schroers*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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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 Poincaré 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.

Original languageEnglish
Pages (from-to)1433–1471
Number of pages39
JournalLetters in Mathematical Physics
Early online date27 Dec 2018
Publication statusPublished - 1 Jun 2019


  • 3d quantum gravity
  • Anyons
  • Group Fourier transform
  • Non-commutative spacetime
  • Quantum double

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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