Advertisement

General Relativity and Gravitation

, Volume 42, Issue 12, pp 2785–2798 | Cite as

Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes

  • Thorsten Ohl
  • Alexander Schenkel
Research Article

Abstract

In this article we study the quantization of a free real scalar field on a class of noncommutative manifolds, obtained via formal deformation quantization using triangular Drinfel’d twists. We construct deformed quadratic action functionals and compute the corresponding equation of motion operators. The Green’s operators and the fundamental solution of the deformed equation of motion are obtained in terms of formal power series. It is shown that, using the deformed fundamental solution, we can define deformed *-algebras of field observables, which in general depend on the spacetime deformation parameter. This dependence is absent in the special case of Killing deformations, which include in particular the Moyal-Weyl deformation of the Minkowski spacetime.

Keywords

Noncommutative geometry Noncommutative field theory Quantum field theory on curved spacetimes 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Douglas M.R., Nekrasov N.A.: . Rev. Mod. Phys. 73, 977 (2001) arXiv:hep-th/0106048CrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Szabo R.J.: . Phys. Rept. 378, 207 (2003) arXiv:hep-th/0109162CrossRefADSzbMATHGoogle Scholar
  3. 3.
    Doplicher S., Fredenhagen K., Roberts J.E.: . Commun. Math. Phys. 172, 187 (1995) arXiv:hep-th/0303037CrossRefMathSciNetADSzbMATHGoogle Scholar
  4. 4.
    Oeckl R.: . Nucl. Phys. B 581, 559 (2000) arXiv:hep-th/0003018CrossRefMathSciNetADSzbMATHGoogle Scholar
  5. 5.
    Chaichian, M., Mnatsakanova, M.N., Nishijima, K., Tureanu, A., Vernov, Yu.S.: arXiv:hep-th/0402212Google Scholar
  6. 6.
    Zahn J.: . Phys. Rev. D 73, 105005 (2006) arXiv:hep-th/0603231CrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Bu J.G., Kim H.C., Lee Y., Vac C.H., Yee J.H.: . Phys. Rev. D 73, 125001 (2006) arXiv:hep-th/0603251CrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Fiore G., Wess J.: . Phys. Rev. D 75, 105022 (2007) arXiv:hep-th/0701078CrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Grosse H., Lechner G.: JHEP 0711, 012 (2007) arXiv:0706.3992 [hep-th]CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Balachandran A.P., Pinzul A., Qureshi B.A.: . Phys. Rev. D 77, 025021 (2008) arXiv:0708.1779 [hep-th]CrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Aschieri P., Lizzi F., Vitale P.: . Phys. Rev. D 77, 025037 (2008) arXiv:0708.3002 [hep-th]CrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Grosse H., Lechner G.: JHEP 0809, 131 (2008) arXiv:0808.3459 [math-ph]CrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Aschieri, P.: arXiv:0903.2457 [math.QA]Google Scholar
  14. 14.
    Arzano M., Marciano A.: . Phys. Rev. D 76, 125005 (2007) arXiv:0707.1329 [hep-th]CrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Arzano M.: Phys. Rev. D 77, 025013 (2008) arXiv:0710.1083 [hep-th]CrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Daszkiewicz M., Lukierski J., Woronowicz M.: J. Phys. A 42, 355201 (2009) arXiv:0807.1992 [hep-th]CrossRefMathSciNetGoogle Scholar
  17. 17.
    Daszkiewicz M., Lukierski J., Woronowicz M.: Phys. Rev. D 77, 105007 (2008) arXiv:0708.1561 [hep-th]CrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Gayral, V., Jureit, J.H., Krajewski, T., Wulkenhaar, R.: arXiv:hep-th/0612048Google Scholar
  19. 19.
    Paschke M., Verch R. Class.: Quantum Gravity 21, 5299 (2004) arXiv:gr-qc/0405057CrossRefzbMATHGoogle Scholar
  20. 20.
    Wald, R.M.: Quantum Field Theory in Curved Space–Time and Black Hole Thermodynamics, p. 205. University of Chicago Press, Chicago (1994)Google Scholar
  21. 21.
    Bär, C., Ginoux, N., Pfäffle, F.: Wave equations on Lorentzian manifolds and quantization. In: ESI Lectures in Mathematics and Physics. European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007. arXiv:0806.1036 [math.DG]Google Scholar
  22. 22.
    Drinfel’d V.G.: Soviet Math. Dokl. 28, 667–671 (1983)zbMATHGoogle Scholar
  23. 23.
    Ohl T., Schenkel A.: JHEP 0901, 084 (2009) arXiv:0810.4885 [hep-th]CrossRefMathSciNetADSGoogle Scholar
  24. 24.
    Schupp, P., Solodukhin, S.: arXiv:0906.2724 [hep-th]Google Scholar
  25. 25.
    Ohl T., Schenkel A.: JHEP 0910, 052 (2009) arXiv:0906.2730 [hep-th]CrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Aschieri P., Castellani L.: J. Geom. Phys. 60, 375 (2010) arXiv:0906.2774 [hep-th]CrossRefMathSciNetADSzbMATHGoogle Scholar
  27. 27.
    Aschieri P., Blohmann C., Dimitrijevic M., Meyer F., Schupp P., Wess J.: Class. Quantum Gravity 22, 351 (2005) arXiv:hep-th/0504183CrossRefMathSciNetGoogle Scholar
  28. 28.
    Aschieri P., Dimitrijevic M., Meyer F., Wess J. Class.: Quantum Gravity 23, 1883 (2006) arXiv:hep-th/0510059CrossRefzbMATHGoogle Scholar
  29. 29.
    Aschieri P., Castellani L.: JHEP 0906, 086 (2009) arXiv:0902.3817 [hep-th]CrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Reshetikhin N.: Lett. Math. Phys. 20, 331 (1990)CrossRefMathSciNetADSzbMATHGoogle Scholar
  31. 31.
    Jambor, C., Sykora, A.: arXiv:hep-th/0405268Google Scholar
  32. 32.
    Bursztyn H., Waldmann S.: Lett. Math. Phys. 53, 349–365 (2000) arXiv:math/0009170 [math.QA]CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Waldmann S.: Prog. Theor. Phys. Suppl. 144, 167 (2001)CrossRefMathSciNetADSGoogle Scholar
  34. 34.
    Waldmann S.: Lect. Notes Phys. 662, 143 (2005) arXiv:math/0304011CrossRefMathSciNetADSGoogle Scholar
  35. 35.
    Waldmann, S.: arXiv:0710.2140 [math.QA]Google Scholar
  36. 36.
    Aschieri, P., Schenkel, A.: in preparationGoogle Scholar
  37. 37.
    Duetsch M., Fredenhagen K.: Commun. Math. Phys. 203, 71 (1999) arXiv:hep-th/9807078CrossRefADSzbMATHGoogle Scholar
  38. 38.
    Waldmann S.: Rev. Math. Phys. 17, 15–75 (2005) arXiv:math/0408217 [math.QA]CrossRefMathSciNetzbMATHGoogle Scholar
  39. 39.
    Brunetti R., Fredenhagen K., Verch R.: Commun. Math. Phys. 237, 31 (2003) arXiv:math-ph/0112041MathSciNetADSzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institut für Theoretische Physik und AstrophysikUniversität WürzburgWürzburgGermany

Personalised recommendations