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


SIGMA 6 (2010), 061, 19 pages      arXiv:1003.3190      https://doi.org/10.3842/SIGMA.2010.061
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Field Theory on Curved Noncommutative Spacetimes

Alexander Schenkel and Christoph F. Uhlemann
Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany

Received March 17, 2010, in final form July 14, 2010; Published online August 03, 2010

Abstract
We study classical scalar field theories on noncommutative curved spacetimes. Following the approach of Wess et al. [Classical Quantum Gravity 22 (2005), 3511 and Classical Quantum Gravity 23 (2006), 1883], we describe noncommutative spacetimes by using (Abelian) Drinfel'd twists and the associated *-products and *-differential geometry. In particular, we allow for position dependent noncommutativity and do not restrict ourselves to the Moyal-Weyl deformation. We construct action functionals for real scalar fields on noncommutative curved spacetimes, and derive the corresponding deformed wave equations. We provide explicit examples of deformed Klein-Gordon operators for noncommutative Minkowski, de Sitter, Schwarzschild and Randall-Sundrum spacetimes, which solve the noncommutative Einstein equations. We study the construction of deformed Green's functions and provide a diagrammatic approach for their perturbative calculation. The leading noncommutative corrections to the Green's functions for our examples are derived.

Key words: noncommutative field theory; Drinfel'd twists; deformation quantization; field theory on curved spacetimes.

pdf (411 kb)   ps (245 kb)   tex (52 kb)

References

  1. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
  2. Landi G., An introduction to noncommutative spaces and their geometry, hep-th/9701078.
  3. Doplicher S., Fredenhagen K., Roberts J.E., Spacetime quantization induced by classical gravity, Phys. Lett. B 331 (1994), 39-44.
    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.
  4. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 93 pages, hep-th/9908142.
  5. Amelino-Camelia G., Smolin L., Starodubtsev A., Quantum symmetry, the cosmological constant and Planck-scale phenomenology, Classical Quantum Gravity 21 (2004), 3095-3110, hep-th/0306134.
    Freidel L., Kowalski-Glikman J., Smolin L., 2+1 gravity and doubly special relativity, Phys. Rev. D 69 (2004), 044001, 7 pages, hep-th/0307085.
  6. Aschieri P., Blohmann C., Dimitrijevic M., Meyer F., Schupp P., Wess J., A gravity theory on noncommutative spaces, Classical Quantum Gravity 22 (2005), 3511-3532, hep-th/0504183.
  7. Aschieri P., Dimitrijevic M., Meyer F., Wess J., Noncommutative geometry and gravity, Classical Quantum Gravity 23 (2006), 1883-1911, hep-th/0510059.
  8. Kürkçüoglu S., Sämann C., Drinfeld twist and general relativity with fuzzy spaces, Classical Quantum Gravity 24 (2007), 291-311, hep-th/0606197.
  9. Chamseddine A.H., Felder G., Fröhlich J., Gravity in noncommutative geometry, Comm. Math. Phys. 155 (1993), 205-217, hep-th/9209044.
  10. Chamseddine A.H., Complexified gravity in noncommutative spaces, Comm. Math. Phys. 218 (2001), 283-292, hep-th/0005222.
  11. Chamseddine A.H., Deforming Einstein's gravity, Phys. Lett. B 504 (2001), 33-37, hep-th/0009153.
  12. Cardella M.A., Zanon D., Noncommutative deformation of four-dimensional Einstein gravity, Classical Quantum Gravity 20 (2003), L95-L103, hep-th/0212071.
  13. Chamseddine A.H., SL(2,C) gravity with complex vierbein and its noncommutative extension, Phys. Rev. D 69 (2004), 024015, 8 pages, hep-th/0309166.
  14. Banerjee R., Mukherjee P., Samanta S., Lie algebraic noncommutative gravity, Phys. Rev. D 75 (2007), 125020, 7 pages, hep-th/0703128.
  15. 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.
    Aschieri P., Castellani L., Noncommutative supergravity in D=3 and D=4, J. High Energy Phys. 2009 (2009), no. 6, 087, 22 pages, arXiv:0902.3823.
  16. Madore J., Mourad J., Quantum space-time and classical gravity, J. Math. Phys. 39 (1998), 423-442, gr-qc/9607060.
    Madore J., An introduction to noncommutative differential geometry and its physical applications, 2nd ed., London Mathematical Society Lecture Note Series, Vol. 257, Cambridge University Press, Cambridge, 1999.
    Langmann E., Szabo R.J., Teleparallel gravity and dimensional reductions of noncommutative gauge theory, Phys. Rev. D 64 (2001), 104019, 15 pages, hep-th/0105094.
    Chamseddine A.H., An invariant action for noncommutative gravity in four dimensions, J. Math. Phys. 44 (2003), 2534-2541, hep-th/0202137.
    García-Compeán H., Obregón O., Ramírez C., Sabido M., Noncommutative self-dual gravity, Phys. Rev. D 68 (2003), 044015, 8 pages, hep-th/0302180.
    Vassilevich D.V., Quantum noncommutative gravity in two dimensions, Nuclear Phys. B 715 (2005), 695-712, hep-th/0406163.
    Calmet X., Kobakhidze A., Noncommutative general relativity, Phys. Rev. D 72 (2005), 045010, 5 pages, hep-th/0506157.
    Buric M., Grammatikopoulos T., Madore J., Zoupanos G., Gravity and the structure of noncommutative algebras, J. High Energy Phys. 2006 (2006), no. 4, 054, 16 pages, hep-th/0603044.
    Majid S., Algebraic approach to quantum gravity. II. Noncommutative spacetime, hep-th/0604130.
    Majid S., Algebraic approach to quantum gravity. III. Noncommmutative Riemannian geometry, hep-th/0604132.
    Szabo R.J., Symmetry, gravity and noncommutativity, Classical Quantum Gravity 23 (2006), R199-R242, hep-th/0606233.
    Balachandran A.P., Pinzul A., Qureshi B.A., Vaidya S., Twisted gauge and gravity theories on the Groenewold-Moyal plane, Phys. Rev. D 76 (2007), 105025, 10 pages, arXiv:0708.0069.
    Müller-Hoissen F., Noncommutative geometries and gravity, in Recent Developments in Gravitation and Cosmology, AIP Conf. Proc., Vol. 977, Amer. Inst. Phys., Melville, NY, 2008, 12-29, arXiv:0710.4418.
    Vassilevich D.V., Diffeomorphism covariant star products and noncommutative gravity, Classical Quantum Gravity 26 (2009), 145010, 8 pages, arXiv:0904.3079.
    Vassilevich D.V., Tensor calculus on noncommutative spaces, Classical Quantum Gravity 27 (2010), 095020, 16 pages, arXiv:1001.0766.
  17. Rivelles V.O., Noncommutative field theories and gravity, Phys. Lett. B 558 (2003), 191-196, hep-th/0212262.
  18. Yang H.S., Emergent gravity from noncommutative space-time, Internat. J. Modern Phys. A 24 (2009), 4473-4517, hep-th/0611174.
    Yang H.S., On the correspondence between noncommuative field theory and gravity, Modern Phys. Lett. A 22 (2007), 1119-1132, hep-th/0612231.
  19. Steinacker H., Emergent gravity from noncommutative gauge theory, J. High Energy Phys. 2007 (2007), no. 12, 049, 36 pages, arXiv:0708.2426.
    Steinacker H., Emergent gravity and noncommutative branes from Yang-Mills matrix models, Nuclear Phys. B 810 (2009), 1-39, arXiv:0806.2032.
  20. Dito G., Sternheimer D., Deformation quantization: genesis, developments and metamorphoses, in Deformation Quantization (Strasbourg, 2001), IRMA Lect. Math. Theor. Phys., Vol. 1, de Gruyter, Berlin, 2002, 9-54, math.QA/0201168.
  21. Drinfel'd V.G., On constant quasiclassical solutions of the Yang-Baxter equations, Soviet Math. Dokl. 28 (1983), 667-671.
  22. Ohl T., Schenkel A., Symmetry reduction in twisted noncommutative gravity with applications to cosmology and black holes, J. High Energy Phys. 2009 (2009), no. 1, 084, 22 pages, arXiv:0810.4885.
  23. Schupp P., Solodukhin S., Exact black hole solutions in noncommutative gravity, arXiv:0906.2724.
  24. 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.
  25. Aschieri P, Castellani L., Noncommutative gravity solutions, J. Geom. Phys. 60 (2010), 375-393, arXiv:0906.2774.
  26. Asakawa T., Kobayashi S., Noncommutative solitons of gravity, Classical Quantum Gravity 27 (2010), 105014, 20 pages, arXiv:0911.2136.
  27. Stern A., Emergent Abelian gauge fields from noncommutative gravity, SIGMA 6 (2010), 019, 15 pages, arXiv:0912.3021.
  28. Chu C.S., Greene B.R., Shiu G., Remarks on inflation and noncommutative geometry, Modern Phys. Lett. A 16 (2001), 2231-2240, hep-th/0011241.
    Lizzi F., Mangano G., Miele G., Peloso M., Cosmological perturbations and short distance physics from noncommutative geometry, J. High Energy Phys. 2002 (2002), no. 6, 049, 16 pages, hep-th/0203099.
    Brandenberger R., Ho P.-M., Noncommutative spacetime, stringy spacetime uncertainty principle, and density fluctuations, Phys. Rev. D 66 (2002), 023517, 10 pages, hep-th/0203119.
    Huang Q.-G., Li M., CMB power spectrum from noncommutative spacetime, J. High Energy Phys. 2003 (2003), no. 6, 014, 7 pages, hep-th/0304203.
    Huang Q.G., Li M., Noncommutative inflation and the CMB multipoles, J. Cosmol. Astropart. Phys. 0311 (2003), 001, 9 pages, astro-ph/0308458.
    Tsujikawa S., Maartens R., Brandenberger R., Non-commutative inflation and the CMB, Phys. Lett. B 574 (2003), 141-148, astro-ph/0308169.
    Kim H.-C., Yee J.H., Rim C., Density fluctuations in kappa-deformed inflationary universe, Phys. Rev. D 72 (2005), 103523, 13 pages, gr-qc/0506122.
    Fatollahi A.H., Hajirahimi M., Noncommutative black-body radiation: implications on cosmic microwave background, Europhys. Lett. 75 (2006), 542-547, astro-ph/0607257.
    Akofor E., Balachandran A.P., Jo S.G., Joseph A., Qureshi B.A., Direction-dependent CMB power spectrum and statistical anisotropy from noncommutative geometry, J. High Energy Phys. 2008 (2008), no. 5, 092, 21 pages, arXiv:0710.5897.
    Akofor E., Balachandran A.P., Joseph A., Pekowsky L., Qureshi B.A., Constraints from CMB on spacetime noncommutativity and causality violation, Phys. Rev. D 79 (2009), 063004, 5 pages, arXiv:0806.2458.
    Fabi S., Harms B., Stern A., Noncommutative corrections to the Robertson-Walker metric, Phys. Rev. D 78 (2008), 065037, 7 pages, arXiv:0808.0943.
  29. Dolan B.P., Gupta K.S., Stern A., Noncommutative BTZ black hole and discrete time, Classical Quantum Gravity 24 (2007), 1647-1655, hep-th/0611233.
    Chaichian M., Tureanu A., Zet G., Corrections to Schwarzschild solution in noncommutative gauge theory of gravity, Phys. Lett. B 660 (2008), 573-578, arXiv:0710.2075.
    Mukherjee P., Saha A., Reissner-Nordstrom solutions in noncommutative gravity, Phys. Rev. D 77 (2008), 064014, 7 pages, arXiv:0710.5847.
    Buric M., Madore J., Spherically symmetric non-commutative space: d = 4, Eur. Phys. J. C 58 (2008), 347-353, arXiv:0807.0960.
    Nicolini P., Noncommutative black holes, the final appeal to quantum gravity: a review, Internat. J. Modern Phys. A 24 (2009), 1229-1308, arXiv:0807.1939.
    Wang D., Zhang R.B., Zhang X., Quantum deformations of Schwarzschild and Schwarzschild-de Sitter spacetimes, Classical Quantum Gravity 26 (2009), 085014, 14 pages, arXiv:0809.0614.
    Di Grezia E., Esposito G., Non-commutative Kerr black hole, arXiv:0906.2303.
    Banerjee R., Chakraborty B., Ghosh S., Mukherjee P., Samanta S., Topics in noncommutative geometry inspired physics, Found. Phys. 39 (2009), 1297-1345, arXiv:0909.1000.
  30. Oeckl R., Untwisting noncommutative Rd and the equivalence of quantum field theories, Nuclear Phys. B 581 (2000), 559-574, hep-th/0003018.
    Bahns D., Doplicher S., Fredenhagen K., Piacitelli G., Ultraviolet finite quantum field theory on quantum spacetime, Comm. Math. Phys. 237 (2003), 221-241, hep-th/0301100.
    Chaichian M., Mnatsakanova M.N., Nishijima K., Tureanu A., Vernov Yu.S., Towards an axiomatic formulation of noncommutative quantum field theory, hep-th/0402212.
    Paschke M., Verch R., Local covariant quantum field theory over spectral geometries, Classical Quantum Gravity 21 (2004), 5299-5316, gr-qc/0405057.
    Zahn J., Remarks on twisted noncommutative quantum field theory, Phys. Rev. D 73 (2006), 105005, 6 pages, hep-th/0603231.
    Bu J.-G., Kim H.-C., Lee Y., Vac C.H., Yee J.H., Noncommutative field theory from twisted Fock space, Phys. Rev. D 73 (2006), 125001, 10 pages, hep-th/0603251.
    Gayral V., Jureit J.H., Krajewski T., Wulkenhaar R., Quantum field theory on projective modules, hep-th/0612048.
    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.
    Freidel L., Kowalski-Glikman J., Nowak S., Field theory on κ-Minkowski space revisited: Noether charges and breaking of Lorentz symmetry, Internat. J. Modern Phys. A 23 (2008), 2687-2718, arXiv:0706.3658.
    Grosse H., Lechner G., Wedge-local quantum fields and noncommutative Minkowski space, J. High Energy Phys. 2007 (2007), no. 11, 012, 26 pages, arXiv:0706.3992.
    Arzano M., Marciano A., Fock space, quantum fields and kappa-Poincaré symmetries, Phys. Rev. D 76 (2007), 125005, 14 pages, arXiv:0707.1329.
    Daszkiewicz M., Lukierski J., Woronowicz M., Towards quantum noncommutative κ-deformed field theory, Phys. Rev. D 77 (2008), 105007, 10 pages, arXiv:0708.1561.
    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.
    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.
    Arzano M., Quantum fields, non-locality and quantum group symmetries, Phys. Rev. D 77 (2008), 025013, 5 pages, arXiv:0710.1083.
    Daszkiewicz M., Lukierski J., Woronowicz M., κ-deformed oscillators, the choice of star product and free κ-deformed quantum fields, J. Phys. A: Math. Theor. 42 (2009), 355201, 18 pages, arXiv:0807.1992.
    Grosse H., Lechner G., Noncommutative deformations of Wightman quantum field theories, J. High Energy Phys. 2008 (2008), no. 9, 131, 29 pages, arXiv:0808.3459.
    Fiore G., On second quantization on noncommutative spaces with twisted symmetries, J. Phys. A: Math. Theor. 43 (2010), 155401, 39 pages, arXiv:0811.0773.
    Borris M., Verch R., Dirac field on Moyal-Minkowski spacetime and non-commutative potential scattering, Comm. Math. Phys. 293 (2010), 399-448, arXiv:0812.0786.
    Aschieri P., Star product geometries, Russ. J. Math. Phys. 16 (2009), 371-383, arXiv:0903.2457.
  31. Ohl T., Schenkel A., Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes, arXiv:0912.2252.
  32. Wald R.M., Quantum field theory in curved spacetime and black hole thermodynamics, Chicago Lectures in Physics, University of Chicago Press, Chicago, IL, 1994.
  33. Bär C., Ginoux N., Pfäffle F., Wave equations on Lorentzian manifolds and quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2007, arXiv:0806.1036.
  34. Reshetikhin N., Multiparameter quantum groups and twisted quasitriangular Hopf algebras, Lett. Math. Phys. 20 (1990), 331-335.
  35. Jambor C., Sykora A., Realization of algebras with the help of *-products, hep-th/0405268.
  36. 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.
    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.
  37. Dimakis A., Madore J., Differential calculi and linear connections, J. Math. Phys. 37 (1996), 4647-4661, q-alg/9601023.
    Cerchiai L., Fiore G., Madore J., Frame formalism for the N-dimensional quantum Euclidean spaces, Internat. J. Modern Phys. B 14 (2000), 2305-2314, math.QA/0007044.
  38. 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.
  39. Schenkel A., Uhlemann C.F., High energy improved scalar quantum field theory from noncommutative geometry without UV/IR-mixing, arXiv:1002.4191.
  40. Horava P., Quantum gravity at a Lifshitz point, Phys. Rev. D 79 (2009), 084008, 15 pages, arXiv:0901.3775.
  41. Dappiaggi C., Moretti V., Pinamonti N., Cosmological horizons and reconstruction of quantum field theories, Comm. Math. Phys. 285 (2009), 1129-1163, arXiv:0712.1770.
    Dappiaggi C., Moretti V., Pinamonti N., Distinguished quantum states in a class of cosmological spacetimes and their Hadamard property, J. Math. Phys. 50 (2009), 062304, 38 pages, arXiv:0812.4033.
    Dappiaggi C., Moretti V., Pinamonti N., Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime, arXiv:0907.1034.


Previous article   Next article   Contents of Volume 6 (2010)