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Moyal deformations of Clifford gauge theories of gravity

Carlos Castro

Carlos Castro

Universidad Tecnica Particular de Loja, San Cayetano Alto, Loja 1101608, Ecuador

Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, GA 30314, USA

E-mail Address: perelmanc@hotmail.com

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https://doi.org/10.1142/S0219887816500183Cited by:0 (Source: Crossref)

Abstract

A Moyal deformation of a Clifford $Cl (3, 1)$ Gauge Theory of (Conformal) Gravity is performed for canonical noncommutativity (constant $Θ^{μ ν}$ parameters). In the very special case when one imposes certain constraints on the fields, there are $n o$ first-order contributions in the $Θ^{μ ν}$ parameters to the Moyal deformations of Clifford gauge theories of gravity. However, when one does $n o t$ impose constraints on the fields, there are first-order contributions in $Θ^{μ ν}$ to the Moyal deformations in variance with the previous results obtained by other authors and based on different gauge groups. Despite that the generators of $U (2, 2), SO (4, 2), SO (2, 3)$ can be expressed in terms of the Clifford algebra generators this does $n o t$ imply that these algebras are isomorphic to the Clifford algebra. Therefore one should not expect identical results to those obtained by other authors. In particular, there are Moyal deformations of the Einstein–Hilbert gravitational action with a cosmological constant to first-order in $Θ^{μ ν}$ . Finally, we provide a mechanism which furnishes a plausible cancellation of the huge vacuum energy density.

Keywords:

PACS: 04.50.Kd, 04.60.-m, 02.20.-a, 02.40.Gh

References

1. I. R. Porteous, Clifford Algebras and Classical Groups (Cambridge University Press, 1995). Crossref, Google Scholar
2. C. H. Tze and F. Gursey, On the Role of Divison, Jordan and Related Algebras in Particle Physics (World Scientific, Singapore, 1996); S. Okubo, Introduction to Octonion and Other Nonassociative Algebras in Physics (Cambridge University Press, 2005); R. Schafer, An introduction to Nonassociative Algebras (Academic Press, New York 1966); G. Dixon, Division Algebras, Octonions, Quaternions, Complex Numbers, and the Algebraic Design of Physics (Kluwer, Dordrecht, 1994); G. Dixon, J. Math. Phys. 45(10) (2004) 3678; P. Ramond, Exceptional groups and physics, preprint (2003), arXiv: hep-th/0301050; J. Baez, Bull. Amer. Math. Soc. 39(2) (2002) 145. Google Scholar
3. P. Jordan, J. von Neumann and E. Wigner, Ann. Math. 35 (1934) 2964; K. MacCrimmon, A Taste of Jordan Algebras (Springer-Verlag, New York, 2003); H. Freudenthal, Nederl. Akad. Wetensch. Proc. Ser. A 57 (1954) 218; J. Tits, Nederl. Akad. Wetensch. Proc. Ser. A 65 (1962) 530; T. Springer, Nederl. Akad. Wetensch. Proc. Ser. A 65 (1962) 259. Google Scholar
4. C. Castro and M. Pavsic, Prog. Phys. 1 (2005) 31; Phys. Lett. B 559 (2003) 74; Int. J. Theor. Phys. 42 (2003) 1693. Google Scholar
5. M. Pavsic, The Landscape of Theoretical Physics: A Global View, From Point Particles to the Brane World and Beyond, in Search of a Unifying Principle (Kluwer Academic Publishers, Dordrecht, 2001). Google Scholar
6. S. W. MacDowell and F. Mansouri, Phys. Rev. Lett 38 (1977) 739; F. Mansouri, Phys. Rev. D 16 (1977) 2456. Google Scholar
7. A. Chamseddine and P. West, Nuc. Phys. B 129 (1977) 39; D. Wise, MacDowell–Mansouri gravity and Cartan geometry, preprint (2006), arXiv: gr-qc/0611154. Google Scholar
8. E. Fradkin and A. Tseytlin, Phys. Rep. 119(4–5) (1985) 233–362. Crossref, Web of Science, Google Scholar
9. S. Capozziello and M. De Laurentis, Extended theories of gravity, Phys. Rep. 509 (2011) 167, arXiv: 1108.6266; S. Nojiri and S. Odintsov, Unified cosmic history in modified gravity: From $F (R)$ theory to Lorentz non-invariant models, Phys. Rep. 505 (2011) 59, arXiv: 1011.0544. Google Scholar
10. I. Obata, T. Miura and J. Soda, Dynamics of electroweak gauge fields during and after Higgs inflation, preprint (2014), arXiv: 1405.3091; C. J. Feng and Z. Z. Li, Towards a realistic solution of the cosmological constant fine-tuning problem by Higgs inflation, preprint (2014), arXiv: 1405.3056. Google Scholar
11. S. Capozziello and V. Faraoni, Beyond Einstein Gravity. A Survey of Gravitational Theories for Cosmology and Astrophysics, Fundamental Theories of Physics, Vol. 170 (Springer, New York, 2011); V. Faraoni, Cosmology in Scalar–Tesnor Gravity, Fundamental Theories of Physics, Vol. 139 (Kluwer Academic Publishers, Dordrecht, 2004). Google Scholar
12. C. Castro, A Clifford-gravity based cosmology, dark matter and dark energy, Int. J. Geom. Meth. Mod. Phys. 12(3) (2015) 1550037. Link, Web of Science, Google Scholar
13. F. Hehl, J. McCrea, E. Mielke and Y. Ne’eman, Phys. Rep. 258 (1995) 1; C. Y Lee, Class. Quantum Grav. 9 (1992) 2001; C. Y. Lee and Y. Ne’eman, Phys. Lett. B 242 (1990) 59. Google Scholar
14. R. Gilmore, Lie Groups, Lie Algebras and Some of Their Applications (John Wiley and Sons, New York, 1974). Crossref, Google Scholar
15. A. Chamseddine, An invariant action for noncommutative gravity in four dimensions, preprint (2002), arXiv: hep-th/0202137; Comm. Math. Phys. 218 (2001) 283; A. Chamseddine, Gravity in complex hermitian spacetime, preprint (2006), arXiv: hep-th/0610099. Google Scholar
16. N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High. Energy Phys. 9909 (1999) 032, arXiv: hep-th/9908142. Crossref, Web of Science, Google Scholar
17. A. Agostini, G. Amelino-Camelia, M. Arzano and F. D’Andrea, Action functional for kappa-Minkowski noncommutative spacetime, preprint (2004), arXiv: hep-th/0407227. Google Scholar
18. J. Wess, J. Phys. Conf. Ser. 53 (2006) 752, arXiv: hep-th/0608135; P. Aschieri, M. Dimitrijević, F. Meyer, S. Schraml and J. Wess, Lett. Math. Phys. 78 (2006) 61, arXiv: hep-th/0603024. Google Scholar
19. R. Banerjee, P. Mukherjee and S. Samanta, Phys. Rev. D 75 (2007) 125020, arXiv: hep-th/0703207. Crossref, Web of Science, Google Scholar
20. M. Chaichian, M. Oksanen, A. Tureanu and G. Zet, Noncommutative gauge theory using covariant star product defined between Lie valued differential forms, preprint (2010), arXiv: 1001.0508; M. Chaichian, M. Oksanen, A. Tureanu and G. Zet, Covariant star product on symplectic and Poisson spacetime manifolds, preprint (2010), arXiv: 1001.0503; A. Schenkel and C. Uhlemann, Field theory on curved noncommutative spacetimes, preprint (2010), arXiv: 1003.3190. Google Scholar
21. J. Madore, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C 16 (2000) 161; B. Jurco, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C 17 (2000) 521; B. Jurco, L. Moller, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C 21 (2001) 383; P. Aschieri, M. Dimitrijević, F. Meyer and J. Wess, Class. Quantum Grav. 23 (2006) 1883. Google Scholar
22. M. Dimitrijević and V. Radovanonic, Noncommutative $SO (2, 3)$ gauge theory and noncommutative gravity, preprint (2014), arXiv: 1404.4213. Google Scholar
23. Y. G Miao, Z. Xue and S. J. Zhang, $U (2, 2)$ gravity on noncommutative space with a symplectic structure, preprint (2010), arXiv: 1006.4074. Google Scholar
24. P. Aschieri, L. Castellani and M. Dimitrijević, Noncommutative gravity at second order via Seiberg–Witten map, Phys. Rev. D 87 (2013) 024017. Crossref, Web of Science, Google Scholar
25. M. Dimitrijević, L. Jonke and L. Moeller, J. High Energy Phys. 0509 (2005) 068, arXiv: hep-th/0504129; M. Dimitrijević, F. Meyer, L. Moeller and J. Wess, Eur. Phys. J. C 36 (2004) 117, arXiv: hep-th/0310116; M. Dimitrijević, L. Jonke, L. Moeller, E. Tsouchnika, J. Wess and M. Wohlgenannt, Eur. Phys. J. C 31 (2003) 129, arXiv: hep-th/0307149; L. Moeller, A symmetry invariant integral on $κ$ -deformed spacetime, arXiv: hep-th/0409128. Google Scholar
26. G. Felder and B. Shoihket, Lett. Math. Phys. 53 (2000) 75. Crossref, Web of Science, Google Scholar
27. C. Chryssomalakos and E. Okon, Star product and invariant integration for Lie type noncommutative spacetimes, preprint (2007), arXiv: 0705.3780. Google Scholar
28. W. Behr and A. Sykora, Construction of gauge theories on curved noncommutative spacetime, Nucl. Phys. B 698 (2004) 473; W. Behr, Noncommutative gauge theory beyond the canonical case, preprint (2005), arXiv: hep-th/0511119; A. Sykora, The application of star products to noncommutative geometry and gauge theory, preprint (2004), arXiv: hep-th/0412012. Google Scholar
29. R. Ablamowicz and B. Fauser, Clifford and Grassmann Hopf algebras via the BIGEBRA package for Maple, Comput. Phys. Commun. 170 (2005) 115; B. Fauser, A treatise on quantum Clifford algebras, preprint (2002), arXiv: math/0202059. Google Scholar
30. K. Becker, M. Becker and J. Schwarz, String Theory and M-Theory: An Introduction (Cambridge University Press, 2007), pp. 543–545. Google Scholar
31. S. Capozziello and C. Stornaiolo, Spacetime deformations as extended conformal transformations, Int. J. Geom. Meth. Mod. Phys. 5 (2008) 185–195. Link, Web of Science, Google Scholar
32. S. Capozziello and S. Vignolo, The Cauchy problem for $f (R)$ gravity: An overview, Int. J. Geom. Meth. Mod. Phys. 09 (2012) 1250006. Link, Web of Science, Google Scholar