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.

pdf (514 kb)   tex (34 kb)

References

  1. Achúcarro A., Townsend P.K., A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett. B 180 (1986), 89-92.
  2. Amelino-Camelia G., Doubly-special relativity: facts, myths and some key open issues, Symmetry 2 (2010), 230-271, arXiv:1003.3942.
  3. Amelino-Camelia G., Freidel L., Kowalski-Glikman J., Smolin L., The principle of relative locality, Phys. Rev. D 84 (2011), 084010, 13 pages, arXiv:1101.0931.
  4. Arzano M., Latini D., Lotito M., Group momentum space and Hopf algebra symmetries of point particles coupled to 2+1 gravity, arXiv:1403.3038.
  5. Atiyah M.F., Moore G.W., A shifted view of fundamental physics, arXiv:1009.3176.
  6. Bais F.A., Muller N.M., Topological field theory and the quantum double of SU(2), Nuclear Phys. B 530 (1998), 349-400, hep-th/9804130.
  7. Bais F.A., Muller N.M., Schroers B.J., Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity, Nuclear Phys. B 640 (2002), 3-45, hep-th/0205021.
  8. Barut A.O., Raczka R., Theory of group representations and applications, 2nd ed., World Scientific Publishing Co., Singapore, 1986.
  9. Batista E., Majid S., Noncommutative geometry of angular momentum space U(su(2)), J. Math. Phys. 44 (2003), 107-137, hep-th/0205128.
  10. Binegar B., Relativistic field theories in three dimensions, J. Math. Phys. 23 (1982), 1511-1517.
  11. Born M., A suggestion for unifying quantum theory and relativity, Proc. R. Soc. Lond. Ser. A 165 (1938), 291-303.
  12. de Sousa Gerbert P., On spin and (quantum) gravity in 2+1 dimensions, Nuclear Phys. B 346 (1990), 440-472.
  13. Drinfel'd V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
  14. Dupuis M., Girelli F., Livine E., Spinors and Voros star-product for group field theory: first contact, Phys. Rev. D 86 (2012), 105034, 5 pages, arXiv:1107.5693.
  15. Freidel L., Livine E.R., 3D quantum gravity and effective noncommutative quantum field theory, Phys. Rev. Lett. 96 (2006), 221301, 4 pages, hep-th/0512113.
  16. Freidel L., Livine E.R., Ponzano-Regge model revisited. III. Feynman diagrams and effective field theory, Classical Quantum Gravity 23 (2006), 2021-2061, hep-th/0502106.
  17. Freidel L., Majid S., Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity, Classical Quantum Gravity 25 (2008), 045006, 37 pages, hep-th/0601004.
  18. Gitman D.M., Shelepin A.L., Poincaré group and relativistic wave equations in 2+1 dimensions, J. Phys. A: Math. Gen. 30 (1997), 6093-6121.
  19. Grigore D.R., The projective unitary irreducible representations of the Poincaré group in 1+2 dimensions, J. Math. Phys. 34 (1993), 4172-4189, hep-th/9304142.
  20. Guedes C., Oriti D., Raasakka M., Quantization maps, algebra representation, and non-commutative Fourier transform for Lie groups, J. Math. Phys. 54 (2013), 083508, 31 pages, arXiv:1301.7750.
  21. Imai S., Sasakura N., Scalar field theories in a Lorentz-invariant three-dimensional noncommutative space-time, J. High Energy Phys. 2000 (2000), no. 9, 032, 23 pages, hep-th/0005178.
  22. Jackiw R., Nair V.P., Relativistic wave equation for anyons, Phys. Rev. D 43 (1991), 1933-1942.
  23. Joung E., Mourad J., Noui K., Three dimensional quantum geometry and deformed symmetry, J. Math. Phys. 50 (2009), 052503, 29 pages, arXiv:0806.4121.
  24. Kempf A., Majid S., Algebraic q-integration and Fourier theory on quantum and braided spaces, J. Math. Phys. 35 (1994), 6802-6837, hep-th/9402037.
  25. Knapp A.W., Representation theory of semisimple groups. An overview based on examples, Princeton Mathematical Series, Vol. 36, Princeton University Press, Princeton, NJ, 1986.
  26. Koornwinder T.H., Muller N.M., The quantum double of a (locally) compact group, J. Lie Theory 7 (1997), 101-120, q-alg/9712042.
  27. Lukierski J., Ruegg H., Nowicki A., Tolstoy V.N., q-deformation of Poincaré algebra, Phys. Lett. B 264 (1991), 331-338.
  28. Majid S., Noncommutative-geometric groups by a bicrossproduct construction: Hopf algebras at the Planck scale, Ph.D. Thesis, Harvard University, 1988.
  29. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  30. Majid S., Ruegg H., Bicrossproduct structure of κ-Poincaré group and non-commutative geometry, Phys. Lett. B 334 (1994), 348-354, hep-th/9405107.
  31. Majid S., Schroers B.J., q-deformation and semidualization in 3D quantum gravity, J. Phys. A: Math. Theor. 42 (2009), 425402, 40 pages, arXiv:0806.2587.
  32. Matschull H.J., Welling M., Quantum mechanics of a point particle in (2+1)-dimensional gravity, Classical Quantum Gravity 15 (1998), 2981-3030, gr-qc/9708054.
  33. Meusburger C., Schroers B.J., Poisson structure and symmetry in the Chern-Simons formulation of (2+1)-dimensional gravity, Classical Quantum Gravity 20 (2003), 2193-2233, gr-qc/0301108.
  34. Meusburger C., Schroers B.J., The quantisation of Poisson structures arising in Chern-Simons theory with gauge group G×g*, Adv. Theor. Math. Phys. 7 (2003), 1003-1043, hep-th/0310218.
  35. Meusburger C., Schroers B.J., Quaternionic and Poisson-Lie structures in three-dimensional gravity: the cosmological constant as deformation parameter, J. Math. Phys. 49 (2008), 083510, 27 pages, arXiv:0708.1507.
  36. Raasakka M., Group Fourier transform and the phase space path integral for finite dimensional Lie groups, arXiv:1111.6481.
  37. Sasai Y., Sasakura N., Domain wall solitons and Hopf algebraic translational symmetries in noncommutative field theories, Internat. J. Modern Phys. A 23 (2008), 2277-2278, arXiv:0711.3059.
  38. Sasai Y., Sasakura N., The Cutkosky rule of three dimensional noncommutative field theory in Lie algebraic noncommutative spacetime, J. High Energy Phys. 2009 (2009), no. 6, 013, 22 pages, arXiv:0902.3050.
  39. Sasai Y., Sasakura N., Massive particles coupled with 2+1 dimensional gravity and noncommutative field theory, arXiv:0902.3502.
  40. Schroers B.J., Combinatorial quantisation of Euclidean gravity in three dimensions, in Quantization of Singular Symplectic Quotients, Progress in Mathematics, Vol. 198, Editors N.P. Landsman, M. Pflaum, M. Schlichenmaier, Birkhäuser Verlag, Basel, 2001, 307-328, math.QA/0006228.
  41. Schroers B.J., Quantum gravity and non-commutative spacetimes in three dimensions: a unified approach, Acta Phys. Polon. B Proc. Suppl. 4 (2011), 379-402, arXiv:1105.3945.
  42. Snyder H.S., Quantized space-time, Phys. Rev. 71 (1947), 38-41.
  43. Sternberg S., Group theory and physics, Cambridge University Press, Cambridge, 1994.
  44. 't Hooft G., Quantization of point particles in (2+1)-dimensional gravity and spacetime discreteness, Classical Quantum Gravity 13 (1996), 1023-1039, gr-qc/9601014.
  45. Witten E., 2+1-dimensional gravity as an exactly soluble system, Nuclear Phys. B 311 (1988), 46-78.

Previous article  Next article   Contents of Volume 10 (2014)