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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Douglas M.R., Nekrasov N.A.: . Rev. Mod. Phys. 73, 977 (2001) arXiv:hep-th/0106048
Szabo R.J.: . Phys. Rept. 378, 207 (2003) arXiv:hep-th/0109162
Doplicher S., Fredenhagen K., Roberts J.E.: . Commun. Math. Phys. 172, 187 (1995) arXiv:hep-th/0303037
Oeckl R.: . Nucl. Phys. B 581, 559 (2000) arXiv:hep-th/0003018
Chaichian, M., Mnatsakanova, M.N., Nishijima, K., Tureanu, A., Vernov, Yu.S.: arXiv:hep-th/0402212
Zahn J.: . Phys. Rev. D 73, 105005 (2006) arXiv:hep-th/0603231
Bu J.G., Kim H.C., Lee Y., Vac C.H., Yee J.H.: . Phys. Rev. D 73, 125001 (2006) arXiv:hep-th/0603251
Fiore G., Wess J.: . Phys. Rev. D 75, 105022 (2007) arXiv:hep-th/0701078
Grosse H., Lechner G.: JHEP 0711, 012 (2007) arXiv:0706.3992 [hep-th]
Balachandran A.P., Pinzul A., Qureshi B.A.: . Phys. Rev. D 77, 025021 (2008) arXiv:0708.1779 [hep-th]
Aschieri P., Lizzi F., Vitale P.: . Phys. Rev. D 77, 025037 (2008) arXiv:0708.3002 [hep-th]
Grosse H., Lechner G.: JHEP 0809, 131 (2008) arXiv:0808.3459 [math-ph]
Aschieri, P.: arXiv:0903.2457 [math.QA]
Arzano M., Marciano A.: . Phys. Rev. D 76, 125005 (2007) arXiv:0707.1329 [hep-th]
Arzano M.: Phys. Rev. D 77, 025013 (2008) arXiv:0710.1083 [hep-th]
Daszkiewicz M., Lukierski J., Woronowicz M.: J. Phys. A 42, 355201 (2009) arXiv:0807.1992 [hep-th]
Daszkiewicz M., Lukierski J., Woronowicz M.: Phys. Rev. D 77, 105007 (2008) arXiv:0708.1561 [hep-th]
Gayral, V., Jureit, J.H., Krajewski, T., Wulkenhaar, R.: arXiv:hep-th/0612048
Paschke M., Verch R. Class.: Quantum Gravity 21, 5299 (2004) arXiv:gr-qc/0405057
Wald, R.M.: Quantum Field Theory in Curved Space–Time and Black Hole Thermodynamics, p. 205. University of Chicago Press, Chicago (1994)
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]
Drinfel’d V.G.: Soviet Math. Dokl. 28, 667–671 (1983)
Ohl T., Schenkel A.: JHEP 0901, 084 (2009) arXiv:0810.4885 [hep-th]
Schupp, P., Solodukhin, S.: arXiv:0906.2724 [hep-th]
Ohl T., Schenkel A.: JHEP 0910, 052 (2009) arXiv:0906.2730 [hep-th]
Aschieri P., Castellani L.: J. Geom. Phys. 60, 375 (2010) arXiv:0906.2774 [hep-th]
Aschieri P., Blohmann C., Dimitrijevic M., Meyer F., Schupp P., Wess J.: Class. Quantum Gravity 22, 351 (2005) arXiv:hep-th/0504183
Aschieri P., Dimitrijevic M., Meyer F., Wess J. Class.: Quantum Gravity 23, 1883 (2006) arXiv:hep-th/0510059
Aschieri P., Castellani L.: JHEP 0906, 086 (2009) arXiv:0902.3817 [hep-th]
Reshetikhin N.: Lett. Math. Phys. 20, 331 (1990)
Jambor, C., Sykora, A.: arXiv:hep-th/0405268
Bursztyn H., Waldmann S.: Lett. Math. Phys. 53, 349–365 (2000) arXiv:math/0009170 [math.QA]
Waldmann S.: Prog. Theor. Phys. Suppl. 144, 167 (2001)
Waldmann S.: Lect. Notes Phys. 662, 143 (2005) arXiv:math/0304011
Waldmann, S.: arXiv:0710.2140 [math.QA]
Aschieri, P., Schenkel, A.: in preparation
Duetsch M., Fredenhagen K.: Commun. Math. Phys. 203, 71 (1999) arXiv:hep-th/9807078
Waldmann S.: Rev. Math. Phys. 17, 15–75 (2005) arXiv:math/0408217 [math.QA]
Brunetti R., Fredenhagen K., Verch R.: Commun. Math. Phys. 237, 31 (2003) arXiv:math-ph/0112041
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ohl, T., Schenkel, A. Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes. Gen Relativ Gravit 42, 2785–2798 (2010). https://doi.org/10.1007/s10714-010-1016-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10714-010-1016-2