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κ-Minkowski spacetime and the star product realizations

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Abstract

We investigate a Lie algebra-type κ-deformed Minkowski spacetime with undeformed Lorentz algebra and mutually commutative vector-like Dirac derivatives. There are infinitely many realizations of κ-Minkowski space. The coproduct and the star product corresponding to each of them are found. An explicit connection between realizations and orderings is established and the relation between the coproduct and the star product, provided through an exponential map, is proved. Utilizing the properties of the natural realization, we construct a scalar field theory on κ-deformed Minkowski space and show that it is equivalent to the scalar, nonlocal, relativistically invariant field theory on the ordinary Minkowski space. This result is universal and does not depend on the realizations, i.e. the orderings, used.

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References

  1. S. Doplicher, K. Fredenhagen, J.E. Roberts, Phys. Lett. B 331, 39 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  2. S. Doplicher, K. Fredenhagen, J.E. Roberts, Commun. Math. Phys. 172, 187 (1995) [hep-th/0303037]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. N. Seiberg, E. Witten, JHEP 09, 032 (1999) [hep-th/9908142]

    Article  ADS  MathSciNet  Google Scholar 

  4. J. de Boer, P.A. Grassi, P. van Nieuwenhuizen, Phys. Lett. B 574, 98 (2003) [hep-th/0302078]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. M.R. Douglas, N.A. Nekrasov, Rev. Mod. Phys. 73, 977 (2001) [hep-th/0106048]

    Article  ADS  MathSciNet  Google Scholar 

  6. A.P. Balachandran, T.R. Govindarajan, C. Molina, P. Teotonio-Sobrinho, JHEP 0410, 72 (2004)

    Article  ADS  Google Scholar 

  7. A.P. Balachandran, T.R. Govindarajan, A.G. Martins, P. Teotonio-Sobrinho, JHEP 0411, 68 (2004)

    Article  ADS  Google Scholar 

  8. R.J. Szabo, Phys. Rep. 378, 207 (2003) [hep-th/0109162]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. R.J. Szabo, Class. Quantum Grav. 23, R199 (2006) [hep-th/0606233]

    Article  MATH  ADS  Google Scholar 

  10. P. Aschieri, B. Jurco, P. Schupp, J. Wess, Nucl. Phys. B 651, 45 (2003) [hep-th/0205214]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. P. Aschieri, C. Blohmann, M. Dimitrijević, F. Meyer, P. Schupp, J. Wess, Class. Quantum Grav. 22, 3511 (2005) [hep-th/0504183]

    Article  MATH  ADS  Google Scholar 

  12. A.P. Balachandran, T.R. Govindarajan, Kumar S. Gupta, S. Kurkcuoglu, Class. Quantum Grav. 23, 5799 (2006)

    Article  MATH  ADS  Google Scholar 

  13. X. Calmet, A. Kobakhidze, Phys. Rev. D 72, 045010 (2005) [hep-th/0506157]

    Article  ADS  MathSciNet  Google Scholar 

  14. J. Lukierski, A. Nowicki, H. Ruegg, V.N. Tolstoy, Phys. Lett. B 264, 331 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  15. J. Lukierski, A. Nowicki, H. Ruegg, Phys. Lett. B 293, 344 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  16. J. Lukierski, H. Ruegg, Phys. Lett. B 329, 189 (1994) [hep-th/9310117]

    Article  ADS  MathSciNet  Google Scholar 

  17. S. Majid, H. Ruegg, Phys. Lett. B 334, 348 (1994) [hep-th/9404107]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. P. Kosiński, P. Maślanka, hep-th/9411033

  19. A. Sitarz, Phys. Lett. B 349, 42 (1995) [hep-th/9409014]

    Article  ADS  MathSciNet  Google Scholar 

  20. K. Kosiński, J. Lukierski, P. Maślanka, Phys. Rev. D 62, 025004 (2000) [hep-th/9902037]

    Article  ADS  MathSciNet  Google Scholar 

  21. K. Kosiński, J. Lukierski, P. Maślanka, Czech. J. Phys. 50, 1283 (2000) [hep-th/0009120]

    Article  ADS  Google Scholar 

  22. P. Kosiński, J. Lukierski, P. Maślanka, A. Sitarz, hep-th/0307038

  23. G. Amelino-Camelia, Phys. Lett. B 510, 255 (2001) [hep-th/0012238]

    Article  MATH  ADS  Google Scholar 

  24. G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 35 (2002) [gr-qc/0012051]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. N.R. Bruno, G. Amelino-Camelia, J. Kowalski-Glikman, Phys. Lett. B 522, 133 (2001) [hep-th/0107039]

    Article  MATH  ADS  Google Scholar 

  26. J. Kowalski-Glikman, S. Nowak, Phys. Lett. B 539, 126 (2002) [hep-th/0203040]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. G. Amelino-Camelia, M. Arzano, Phys. Rev. D 65, 084044 (2002) [hep-th/0105120]

    Article  ADS  MathSciNet  Google Scholar 

  28. G. Amelino-Camelia, F. D’Andrea, G. Mandanici, JCAP 0309, 006 (2003) [hep-th/0211022]

    ADS  Google Scholar 

  29. M. Dimitrijević, L. Jonke, L. Möller, E. Tsouchnika, J. Wess, M. Wohlgenannt, Eur. Phys. J. C 31, 129 (2003) [hep-th/0307149]

    Article  ADS  Google Scholar 

  30. M. Dimitrijević, F. Meyer, L. Möller, J. Wess, Eur. Phys. J. C 36, 117 (2004) [hep-th/0310116]

    ADS  Google Scholar 

  31. M. Dimitrijević, L. Möller, E. Tsouchnika, J. Phys. A 37, 9749 (2004) [hep-th/0404224]

    Article  ADS  MathSciNet  Google Scholar 

  32. C. Jambor, A. Sykora, hep-th/0405268

  33. A. Sykora, hep-th/0412012

  34. S. Meljanac, M. Stojic, Eur. Phys. J. C 47, 531 (2006) [hep-th/0605133]

    Article  ADS  MathSciNet  Google Scholar 

  35. S. Meljanac, S. Kresic-Juric, M. Stojic, Eur. Phys. J. C 51, 229 (2007) [hep-th/0702215]

    Article  ADS  MathSciNet  Google Scholar 

  36. L. Moller, JHEP 0512, 029 (2005) [hep-th/0409128]

    Article  ADS  MathSciNet  Google Scholar 

  37. H. Grosse, M. Wohlgenannt, Nucl. Phys. B 748, 473 (2006) [hep-th/0507030]

    Article  MATH  ADS  Google Scholar 

  38. A. Agostini, G. Amelino-Camelia, M. Arzano, F. D’Andrea, hep-th/0407227

  39. A. Agostini, G. Amelino-Camelia, M. Arzano, A. Marciano, R.A. Tacchi, hep-th/0607221

  40. M. Arzano, A. Marciano, Phys. Rev. D 75, 081701 (2007) [hep-th/0701268]

    Article  ADS  MathSciNet  Google Scholar 

  41. A. Agostini, F. Lizzi, A. Zampini, Mod. Phys. Lett. A 17, 2105 (2002) [hep-th/0209174]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  42. L. Freidel, J. Kowalski-Glikman, S. Nowak, hep-th/0612170

  43. S. Ghosh, Phys. Lett. B 623, 251 (2005) [hep-th/0506084]

    Article  ADS  MathSciNet  Google Scholar 

  44. S. Ghosh, P. Pal, Phys. Lett. B 618, 243 (2005) [hep-th/0502192]

    Article  ADS  MathSciNet  Google Scholar 

  45. S. Ghosh, Phys. Rev. D 74, 084019 (2006) [hep-th/0608206]

    Article  ADS  MathSciNet  Google Scholar 

  46. S. Ghosh, P. Pal, hep-th/0702159

  47. N. Durov, S. Meljanac, A. Samsarov, Z. Škoda, J. Algebra 309, 318 (2007) [math.RT/0604096]

    Article  MATH  MathSciNet  Google Scholar 

  48. D. Bonatsos, C. Daskaloyannis, Phys. Lett. B 307, 100 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  49. S. Meljanac, M. Mileković, S. Pallua, Phys. Lett. B 328, 55 (1994) [hep-th/9404039]

    Article  ADS  MathSciNet  Google Scholar 

  50. S. Meljanac, M. Mileković, Int. J. Mod. Phys. A 11, 1391 (1996)

    Article  MATH  ADS  Google Scholar 

  51. S. Meljanac, A. Perica, J. Phys. A 27, 4737 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  52. S. Meljanac, A. Perica, Mod. Phys. Lett. A 9, 3293 (1994) [hep-th/9409180]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  53. S. Meljanac, A. Perica, D. Svrtan, J. Phys. A 36, 6337 (2003) [math-ph/0304038]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  54. V. Bardek, S. Meljanac, Eur. Phys. J. C 17, 539 (2000) [hep-th/0009099]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  55. V. Bardek, L. Jonke, S. Meljanac, M. Mileković, Phys. Lett. B 531, 311 (2002) [hep-th/0107053]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  56. S. Meljanac, M. Mileković, M. Stojić, Eur. Phys. J. C 24, 331 (2002) [math-ph/0201061]

    Article  MATH  ADS  Google Scholar 

  57. L. Jonke, S. Meljanac, Phys. Lett. B 526, 149 (2002) [hep-th/0106135]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  58. V.P. Nair, A.P. Polychronakos, Phys. Lett. B 505, 267 (2001) [hep-th/0011172]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  59. L. Jonke, S. Meljanac, Eur. Phys. J. C 29, 433 (2003) [hep-th/0210042]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  60. I. Dadić, L. Jonke, S. Meljanac, Acta Phys. Slovaca 55, 149 (2005) [hep-th/0301066]

    Google Scholar 

  61. J. Wess, Deformed coordinates spaces, derivatives. Lecture given at the Balkan workshop BW2003, August 2003, Vrnjačka Banja, Serbia [hep-th/0408080]

  62. M. Chaichian, P.P. Kulish, K. Nishijima, A. Tureanu, Phys. Lett. B 604, 98 (2004) [hep-th/0408069]

    Article  ADS  MathSciNet  Google Scholar 

  63. M. Chaichian, P. Presnajder, A. Tureanu, Phys. Rev. Lett. 94, 151602 (2005) [hep-th/0409096]

    Article  ADS  Google Scholar 

  64. T.R. Govindarajan, private communication

  65. L. Freidel, E.R. Livine, Class. Quantum Grav. 23, 2021 (2006) [hep-th/0502106]

    Article  MATH  ADS  MathSciNet  Google Scholar 

  66. L. Freidel, E.R. Livine, Phys. Rev. Lett. 96, 221301 (2006) [hep-th/0512113]

    Article  ADS  MathSciNet  Google Scholar 

  67. L. Freidel, S. Majid, hep-th/0601004

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Meljanac, S., Samsarov, A., Stojić, M. et al. κ-Minkowski spacetime and the star product realizations. Eur. Phys. J. C 53, 295–309 (2008). https://doi.org/10.1140/epjc/s10052-007-0450-0

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  • DOI: https://doi.org/10.1140/epjc/s10052-007-0450-0

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