Noncommutative spacetime symmetries: Twist versus covariance

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We prove that the Moyal product is covariant under linear affine spacetime transformations. From the covariance law, by introducing an (x,Theta)-space where the spacetime coordinates and the noncommutativity matrix components are on the same footing, we obtain a noncommutative representation of the affine algebra, its generators being differential operators in (x,Theta)-space. As a particular case, the Weyl Lie algebra is studied and known results for Weyl invariant noncommutative field theories are rederived in a nutshell. We also show that this covariance cannot be extended to spacetime transformations generated by differential operators whose coefficients are polynomials of order larger than 1. We compare our approach with the twist deformed enveloping algebra description of spacetime transformations.
© 2006 The American Physical Society. The authors are grateful to V. Gayral, G. Marmo and C. Moreno for discussions. JMG-B acknowledges support from MEC, Spain, through a ‘Ramón y Cajal’ contract. Partial support from CICyT and UCM-CAM, Spain, through grants No. FIS2005-02309, 910770, and from the ‘‘Progetto di Ricerca di Interesse Nazionale‘‘, Italy, is also acknowledged.
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[1] D. Colladay and V. A. Kostelecky´, Phys. Rev. D 58, 116002 (1998). [2] A. Iorio and T. Sy´kora, Int. J. Mod. Phys. A 17, 2369 (2002); R. Jackiw and S.-Y. Pi, Phys. Rev. Lett. 88, 111603 (2002); L. Alvarez-Gaume´ and M. A. Va´zquez- Mozo, Nucl. Phys. B668, 293 (2003). [3] S. M. Carroll, J. A. Harvey, V. A. Kostelecky´, C.D. Lane, and T. Okamoto, Phys. Rev. Lett. 87, 141601 (2001). [4] A. A. Bichl, J. M. Grimstrup, H. Grosse, E. Kraus, L. Popp, M. Schweda, and R. Wulkenhaar, Eur. Phys. J. C 24, 165 (2002). [5] C. Gonera, P. Kosinski, P. Maslanka, and S. Giller, Phys. Rev. D 72, 067702 (2005); Phys. Lett. B 622, 192 (2005). [6] R. Oeckl, Nucl. Phys. B581, 559 (2000). [7] M. Chaichian, P. P. Kulish, K. Nishijima, and A. Tureanu, Phys. Lett. B 604, 98 (2004). [8] J. Wess, hep-th/0408080. [9] P. Matlock, Phys. Rev. D 71, 126007 (2005). [10] F. Lizzi, S. Vaidya, and P. Vitale, Phys. Rev. D 73, 125020 (2006). [11] P. Aschieri, C. Blohmann, M. Dimitrijevic, F. Meyer, P. Schupp, and J. Wess, Class. Quant. Grav. 22, 3511 (2005). [12] D.V. Vassilevich, Mod. Phys. Lett. A 21, 1279 (2006); P. Aschieri, M. Dimitrijevic, F. Meyer, S. Schraml, and J. Wess, hep-th/0603024; J. Zahn, Phys. Rev. D 73, 105005 (2006); M. Chaichian and A. Tureanu, Phys. Lett. B 637, 199 2006). [13] M. A. Rieffel, Deformation Quantization for Actions of Rd, Memoirs Amer. Math. Soc. Vol. 506 (Providence, RI, USA, 1993). [14] R. Estrada, J. M. Gracia-Bondı´a, and J. C. Várilly, J. Math. Phys. (N.Y.) 30, 2789 (1989). [15] S. Gutt and J. Rawnsley, J. Geom. Phys. 29, 347 (1999). [16] F.W. Hehl, J. D. McCrea, E.W. Mielke, and Y. Ne’eman, Phys. Rep. 258, 1 (1995). [17] S. Majid, Foundations of Quantum Group Theor (Cambridge University Press, Cambridge, 1995). [18] Expressions for the ___-derivative terms in these generators have been searched for in the past. See M. Chaichian, K. Nishijima, and A. Tureanu, Phys. Lett. B 633, 129 (2006) for an approach to find what we now understand to be M_Θμγ M__μγ. [19] F. Lizzi, R. Szabo, and A. Zampini, J. High Energy Phys. 08 (2001) 032. [20] A. Stern and A. Pinzul, Int. J. Mod. Phys. A 20, 5871 (2005). [21] N. Seiberg and E. Witten, J. High Energy Phys. 09 (1999) 032. [22] N. Seiberg, J. High Energy Phys. 06 (2003) 010. [23] For the twist approach, this was first considered in M. Ihl and C. Saemann, J. High Energy Phys. 01 (2006) 065; See also Banerjee, C. Lee and S. Siwach, hep-th/0511205. [24] M. Dubois-Violette, A. Kriegl, Y. Maeda, and P.W. Michor, Prog. Theor. Phys. Suppl. 144, 54 (2001). GRACIA BONDI´A et al. PHYSICAL REVIEW D 74, 025014 (2006)