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


SIGMA 9 (2013), 080, 19 pages      arXiv:1308.1929      http://dx.doi.org/10.3842/SIGMA.2013.080
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Dirac Operators on Noncommutative Curved Spacetimes

Alexander Schenkel a and Christoph F. Uhlemann b
a) Fachgruppe Mathematik, Bergische Universität Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany
b) Department of Physics, University of Washington, Seattle, WA 98195-1560, USA

Received August 09, 2013, in final form December 11, 2013; Published online December 15, 2013

Abstract
We study the notion of a Dirac operator in the framework of twist-deformed noncommutative geometry. We provide a number of well-motivated candidate constructions and propose a minimal set of axioms that a noncommutative Dirac operator should satisfy. These criteria turn out to be restrictive, but they do not fix a unique construction: two of our operators generally satisfy the axioms, and we provide an explicit example where they are inequivalent. For highly symmetric spacetimes with Drinfeld twists constructed from sufficiently many Killing vector fields, all of our operators coincide. For general noncommutative curved spacetimes we find that demanding formal self-adjointness as an additional condition singles out a preferred choice among our candidates. Based on this noncommutative Dirac operator we construct a quantum field theory of Dirac fields. In the last part we study noncommutative Dirac operators on deformed Minkowski and AdS spacetimes as explicit examples.

Key words: Dirac operators; Dirac fields; Drinfeld twists; deformation quantization; noncommutative quantum field theory; quantum field theory on curved spacetimes.

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References

  1. Aschieri P., Castellani L., Noncommutative D=4 gravity coupled to fermions, J. High Energy Phys. 2009 (2009), no. 6, 086, 18 pages, arXiv:0902.3817.
  2. Aschieri P., Castellani L., Noncommutative gravity solutions, J. Geom. Phys. 60 (2010), 375-393, arXiv:0906.2774.
  3. Aschieri P., Castellani L., Noncommutative gravity coupled to fermions: second order expansion via Seiberg-Witten map, J. High Energy Phys. 2012 (2012), no. 7, 184, 27 pages, arXiv:1111.4822.
  4. Aschieri P., Dimitrijević M., Meyer F., Wess J., Noncommutative geometry and gravity, Classical Quantum Gravity 23 (2006), 1883-1911, hep-th/0510059.
  5. Aschieri P., Lizzi F., Vitale P., Twisting all the way: from classical mechanics to quantum fields, Phys. Rev. D 77 (2008), 025037, 16 pages, arXiv:0708.3002.
  6. Aschieri P., Schenkel A., Noncommutative connections on bimodules and Drinfeld twist deformation, arXiv:1210.0241.
  7. Balachandran A.P., Pinzul A., Qureshi B.A., Twisted Poincaré invariant quantum field theories, Phys. Rev. D 77 (2008), 025021, 9 pages, arXiv:0708.1779.
  8. Bär C., Green-hyperbolic operators on globally hyperbolic spacetimes, arXiv:1310.0738.
  9. Bär C., Ginoux N., Classical and quantum fields on Lorentzian manifolds, in Global Differential Geometry, Springer Proceedings in Mathematics, Vol. 17, Editors C. Bär, J. Lohkamp, M. Schwarz, Springer, Berlin, 2012, 359-400, arXiv:1104.1158.
  10. Borowiec A., Pachol A., κ-Minkowski spacetime as the result of Jordanian twist deformation, Phys. Rev. D 79 (2009), 045012, 11 pages, arXiv:0812.0576.
  11. Breitenlohner P., Freedman D.Z., Stability in gauged extended supergravity, Ann. Physics 144 (1982), 249-281.
  12. Bu J.-G., Kim H.-C., Lee Y., Vac C.H., Yee J.H., κ-deformed spacetime from twist, Phys. Lett. B 665 (2008), 95-99, hep-th/0611175.
  13. Connes A., Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994.
  14. Connes A., Landi G., Noncommutative manifolds, the instanton algebra and isospectral deformations, Comm. Math. Phys. 221 (2001), 141-159, math.QA/0011194.
  15. D'Andrea F., Spectral geometry of κ-Minkowski space, J. Math. Phys. 47 (2006), 062105, 19 pages, hep-th/0503012.
  16. Dimitrijević M., Jonke L., A twisted look on kappa-Minkowski: U(1) gauge theory, J. High Energy Phys. 2011 (2011), no. 12, 080, 23 pages, arXiv:1107.3475.
  17. Doplicher S., Fredenhagen K., Roberts J.E., Spacetime quantization induced by classical gravity, Phys. Lett. B 331 (1994), 39-44.
  18. Doplicher S., Fredenhagen K., Roberts J.E., The quantum structure of spacetime at the Planck scale and quantum fields, Comm. Math. Phys. 172 (1995), 187-220, hep-th/0303037.
  19. Fiore G., Wess J., Full twisted Poincaré symmetry and quantum field theory on Moyal-Weyl spaces, Phys. Rev. D 75 (2007), 105022, 13 pages, hep-th/0701078.
  20. Govindarajan T.R., Gupta K.S., Harikumar E., Meljanac S., Meljanac D., Twisted statistics in κ-Minkowski spacetime, Phys. Rev. D 77 (2008), 105010, 6 pages, arXiv:0802.1576.
  21. Iochum B., Masson T., Schücker T., Sitarz A., Compact κ-deformation and spectral triples, Rep. Math. Phys. 68 (2011), 37-64, arXiv:1004.4190.
  22. Jurić T., Meljanac S., Strajn R., Differential forms and κ-Minkowski spacetime from extended twist, Eur. Phys. J. C 73 (2013), 2472-2480, arXiv:1211.6612.
  23. Kim H.-C., Lee Y., Rim C., Yee J.H., Differential structure on the κ-Minkowski spacetime from twist, Phys. Lett. B 671 (2009), 398-401, arXiv:0808.2866.
  24. Meljanac S., Samsarov A., Strajn R., κ-deformation of phase space; generalized Poincaré algebras and R-matrix, J. High Energy Phys. 2012 (2012), no. 8, 127, 16 pages, arXiv:1204.4324.
  25. Mühlhoff R., Cauchy problem and Green's functions for first order differential operators and algebraic quantization, J. Math. Phys. 52 (2011), 022303, 7 pages, arXiv:1001.4091.
  26. Ohl T., Schenkel A., Cosmological and black hole spacetimes in twisted noncommutative gravity, J. High Energy Phys. 2009 (2009), no. 10, 052, 12 pages, arXiv:0906.2730.
  27. Ohl T., Schenkel A., Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes, Gen. Relativity Gravitation 42 (2010), 2785-2798, arXiv:0912.2252.
  28. Ohl T., Schenkel A., Uhlemann C.F., Spacetime noncommutativity in models with warped extradimensions, J. High Energy Phys. 2010 (2010), no. 7, 029, 16 pages, arXiv:1002.2884.
  29. Schenkel A., Noncommutative gravity and quantum field theory on noncommutative curved spacetimes, Ph.D. thesis, University of Würzburg, 2011, available at http://opus.bibliothek.uni-wuerzburg.de/volltexte/2011/6582/, arXiv:1210.1115.
  30. Schenkel A., QFT on homothetic Killing twist deformed curved spacetimes, Gen. Relativity Gravitation 43 (2011), 2605-2630, arXiv:1009.1090.
  31. Schenkel A., Uhlemann C.F., Field theory on curved noncommutative spacetimes, SIGMA 6 (2010), 061, 19 pages, arXiv:1003.3190.
  32. Schenkel A., Uhlemann C.F., High energy improved scalar quantum field theory from noncommutative geometry without UV/IR-mixing, Phys. Lett. B 694 (2010), 258-260, arXiv:1002.4191.
  33. Schupp P., Solodukhin S., Exact black hole solutions in noncommutative gravity, arXiv:0906.2724.
  34. Taylor M.E., Partial differential equations. I. Basic theory, Applied Mathematical Sciences, Vol. 115, Springer-Verlag, New York, 1996.
  35. Zahn J., Remarks on twisted noncommutative quantum field theory, Phys. Rev. D 73 (2006), 105005, 6 pages, hep-th/0603231.
  36. Zuckerman G.J., Action principles and global geometry, in Mathematical Aspects of String Theory (San Diego, Calif., 1986), Adv. Ser. Math. Phys., Vol. 1, World Sci. Publishing, Singapore, 1987, 259-284.

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