Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 053, 23 pages      arXiv:1402.7039      https://doi.org/10.3842/SIGMA.2014.053
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Towards Non-Commutative Deformations of Relativistic Wave Equations in 2+1 Dimensions

Bernd J. Schroers a and Matthias Wilhelm b
a) Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK
b) Institut für Mathematik und Institut für Physik, Humboldt-Universität zu Berlin, IRIS-Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany

Received February 28, 2014, in final form May 09, 2014; Published online May 20, 2014

Abstract
We consider the deformation of the Poincaré group in 2+1 dimensions into the quantum double of the Lorentz group and construct Lorentz-covariant momentum-space formulations of the irreducible representations describing massive particles with spin 0, 1/2 and 1 in the deformed theory. We discuss ways of obtaining non-commutative versions of relativistic wave equations like the Klein-Gordon, Dirac and Proca equations in 2+1 dimensions by applying a suitably defined Fourier transform, and point out the relation between non-commutative Dirac equations and the exponentiated Dirac operator considered by Atiyah and Moore.

Key words: relativistic wave equations; quantum groups; curved momentum space; non-commutative spacetime.

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