Vol. 24 No. 1 (2014)
Papers

A New Solution to the Structure Equation in Noncommutative Spacetime

Nguyen Ai Viet
Information Technology Institute, Vietnam National Univeity

Published 12-03-2014

Keywords

  • non-commutative geometry,
  • gravity,
  • general relativity,
  • Kaluza-Klein theory,
  • Cartan formalism

How to Cite

Viet, N. A. (2014). A New Solution to the Structure Equation in Noncommutative Spacetime. Communications in Physics, 24(1), 21. https://doi.org/10.15625/0868-3166/24/1/3606

Abstract

In this paper, starting from the common foundation of Connes' noncommutative geometry ( NCG)\cite{Connes1, Connes2, CoLo, Connes3}, various possible alternatives in the formulation of atheory of gravity in noncommutative spacetime are discussed indetails. The diversity in the final physical content of the theory is shown to be the consequence of the arbitrary choices in each construction steps. As an alternative in the last step, when the structure equations are to be solved, a minimal set of constraints on the torsion and connection is found to determine all the geometric notions in terms of the metric. In the Connes-Lott model of noncommutative spacetime, in order to keep the full spectrum of the discretized Kaluza-Klein theory \cite{VW2}, it is necessary to include the torsion in the generalized Einstein-Hilbert-Cartan action.

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

References

  1. bibitem{Connes1} A.Connes, {it Noncommutative Geometry}, (Academic Press,
  2. ; A.Connes, {it Noncommutative Differential Geometry}, Publ.
  3. I.H.E.S. {bf 62} (1986), 257; A.Connes, { it Noncommutative
  4. Geometry and Physics} in {it Gravitation and Quantizations}, Les
  5. Houches, Session LVII, (Elsevier Science B.V. 1995).
  6. bibitem{Connes2} A.Connes, {it Essay on physics and noncommutative geometry},
  7. in {it The interface of mathematics and particle physics}, Oxford
  8. Univ.Press ( 1990), 9.
  9. bibitem{CoLo} A.Connes and J.Lott Nucl.Phys.{bf B18} Suppl.(1990),
  10. ; A.Connes and J.Lott, {it The metric Aspect on Nonncommutative
  11. Geometry}, in {it Proceedings of the 1991 Carges Summer School},
  12. ed.J.Fr"ohlich et al.( Plenum, 1992).
  13. bibitem{Connes3} A.Connes, {it Gravity coupled with matter and
  14. the foundation of noncommutative geometry}, hep-th/8603053 (1996).
  15. bibitem{Kast} D.Kastler, Commun.Math.Phys. {bf 166} (1995), 633.
  16. bibitem{COQUE} R.Coquereaux, J.Geom.Phys. {bf 6} (1989),425.
  17. bibitem{Madore} J.Madore, {it An Introduction to Noncommutative Geometry and its
  18. Physical Applications}, ( LMS Lecture Notes 206, 1995).
  19. bibitem{VW2} N.A.Viet and K.C.Wali, Intl. J. Modern Phys., {bf A11} (1996), 2403.
  20. bibitem{Landi} G.Landi {it An Introduction to Noncommutative
  21. Spaces and their Geometries},( Springer-Verlag 1997).
  22. bibitem{CHAM}
  23. A.H.Chamseddine, G.Felder and Fr"ohlich Comm.Math.Phys. {bf
  24. },(1993), 205; A.H.Chamseddine, J.Fr"ohlich, O.Grandjean,
  25. J.Math.Phys. {bf 36} (1995), 6255.
  26. bibitem{WODZICKI} W.Kalau and M.Walze, J.Geom.Phys. {bf 16} (1995), 327.
  27. bibitem{LVW} G.Landi, Nguyen Ai Viet, K.C.Wali Phys.Letters {bf B326} (1994), 32.
  28. bibitem{VW1} Nguyen Ai Viet, K.C.Wali, Intl. J. Modern Phys., {bf A11} (1996), 533.
  29. bibitem{VW3} Nguyen Ai Viet, K.C.Wali, {it Matter Fields in
  30. Curved Space-Time} in {it Theoretical High-Energy Physics
  31. MRST'2000}, ed. C.R.Hagen (2000), 27.
  32. bibitem{CHIRAL} Nguyen Ai Viet and K.C.Wali {it Chiral spinors and Gauge Fields in curved
  33. noncommutative space-time}, hep-th/0212064 ( to be published).
  34. bibitem{V1} Nguyen Ai Viet {it Predictions of Noncommutative
  35. space-time} in MRST'94 What Next? Exploring the Future of
  36. High-Energy Physics, ed.K.R.Cudel et al,(World Scientific, 1994).
  37. bibitem{V2} Nguyen Ai Viet, (To memory of E.Wigner) Heavy-Ion Phys.
  38. {bf 1} (1995) 263.
  39. bibitem{wheeler} Ch.W.Misner, K.S.Thorne and J.A.Wheeler {it Gravitation},
  40. (W.H.Freeman and Company, New York, 1973).
  41. bibitem{wald} R.M.Wald {it General Relativity}, (The University of Chicago Press,
  42. Chicago and London, 1984).
  43. bibitem{Naka} M.Nakahara, {it Geometry, Topology and Physics}, (
  44. Institute of Physics Press, 1992).
  45. bibitem{Eguchi} T.Eguchi, P.B.Gilkey and A.J.Hanson, {it
  46. Gravitation, Gauge theories and Differential Geometry} Physics
  47. Reports {bf 66} No 6 (1990).
  48. bibitem{Dubois} M.Dubois-Viollete {it Lectures on graded
  49. Differential algebras and Noncommutative Geometry} LPT-ORSAY
  50. /100, qa/9912017 ( 1999).
  51. bibitem{SITARZ} A.Sitarz Class.Quant.Grav. {bf 11} (1994)
  52. bibitem{Klim} C.Klimcik, A.Pompos, V.Soucek, Lett.Math.Phys. {bf
  53. } (1994), 259.
  54. bibitem{CHINA} Bin Chen, Takesi Saito, Ke Wu,
  55. Prog.Theor.Phys. {bf 92}, (1994), 881; G. Konisi, Takesi Saito,
  56. Ke Wu, Prog.Theor.Phys. {bf 93}, (1995), 621.
  57. bibitem{LiConKK} M.Dubois-Violette, R.Kerner, J.Madore,
  58. J.Math.Phys. {bf 31} (1990), 316; J.Madore, Phys.Rev. {D41}
  59. (1990), 3790; M.Dubois-Violette, J.Madore, T.Masson, J.Mourad,
  60. J.Mourad, J.Math.Phys. {bf 37} (1996), 4089; J.Madore,
  61. Class.Quant.Grav. {bf 13} (1996), 2109; J.Mourad,
  62. Class.Quant.Grav. {bf 12} (1995), 965.
  63. bibitem{KK} Th.Kaluza, Sitzuuza, Sitzungsber. Preuss. Akad. Wiss. Phys.
  64. Math. Klasse 966 (1921);O.Klein, Z.F. Physik {bf 37} (1926) 895;
  65. Y.Thirry, Comptes Rendus (Paris) {bf 226} (1948) 216.