A New Solution to the Structure Equation in Noncommutative Spacetime

Nguyen Ai Viet
Author affiliations

Authors

  • Nguyen Ai Viet Information Technology Institute, Vietnam National Univeity

DOI:

https://doi.org/10.15625/0868-3166/24/1/3606

Keywords:

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

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

bibitem{Connes1} A.Connes, {it Noncommutative Geometry}, (Academic Press,

; A.Connes, {it Noncommutative Differential Geometry}, Publ.

I.H.E.S. {bf 62} (1986), 257; A.Connes, { it Noncommutative

Geometry and Physics} in {it Gravitation and Quantizations}, Les

Houches, Session LVII, (Elsevier Science B.V. 1995).

bibitem{Connes2} A.Connes, {it Essay on physics and noncommutative geometry},

in {it The interface of mathematics and particle physics}, Oxford

Univ.Press ( 1990), 9.

bibitem{CoLo} A.Connes and J.Lott Nucl.Phys.{bf B18} Suppl.(1990),

; A.Connes and J.Lott, {it The metric Aspect on Nonncommutative

Geometry}, in {it Proceedings of the 1991 Carges Summer School},

ed.J.Fr"ohlich et al.( Plenum, 1992).

bibitem{Connes3} A.Connes, {it Gravity coupled with matter and

the foundation of noncommutative geometry}, hep-th/8603053 (1996).

bibitem{Kast} D.Kastler, Commun.Math.Phys. {bf 166} (1995), 633. DOI: https://doi.org/10.1007/BF02099890

bibitem{COQUE} R.Coquereaux, J.Geom.Phys. {bf 6} (1989),425. DOI: https://doi.org/10.1016/0393-0440(89)90013-2

bibitem{Madore} J.Madore, {it An Introduction to Noncommutative Geometry and its

Physical Applications}, ( LMS Lecture Notes 206, 1995).

bibitem{VW2} N.A.Viet and K.C.Wali, Intl. J. Modern Phys., {bf A11} (1996), 2403. DOI: https://doi.org/10.1142/S0217751X96001206

bibitem{Landi} G.Landi {it An Introduction to Noncommutative

Spaces and their Geometries},( Springer-Verlag 1997).

bibitem{CHAM}

A.H.Chamseddine, G.Felder and Fr"ohlich Comm.Math.Phys. {bf

},(1993), 205; A.H.Chamseddine, J.Fr"ohlich, O.Grandjean,

J.Math.Phys. {bf 36} (1995), 6255.

bibitem{WODZICKI} W.Kalau and M.Walze, J.Geom.Phys. {bf 16} (1995), 327. DOI: https://doi.org/10.1016/0393-0440(94)00032-Y

bibitem{LVW} G.Landi, Nguyen Ai Viet, K.C.Wali Phys.Letters {bf B326} (1994), 32.

bibitem{VW1} Nguyen Ai Viet, K.C.Wali, Intl. J. Modern Phys., {bf A11} (1996), 533. DOI: https://doi.org/10.1142/S0217751X96000249

bibitem{VW3} Nguyen Ai Viet, K.C.Wali, {it Matter Fields in

Curved Space-Time} in {it Theoretical High-Energy Physics

MRST'2000}, ed. C.R.Hagen (2000), 27.

bibitem{CHIRAL} Nguyen Ai Viet and K.C.Wali {it Chiral spinors and Gauge Fields in curved

noncommutative space-time}, hep-th/0212064 ( to be published).

bibitem{V1} Nguyen Ai Viet {it Predictions of Noncommutative

space-time} in MRST'94 What Next? Exploring the Future of

High-Energy Physics, ed.K.R.Cudel et al,(World Scientific, 1994).

bibitem{V2} Nguyen Ai Viet, (To memory of E.Wigner) Heavy-Ion Phys.

{bf 1} (1995) 263.

bibitem{wheeler} Ch.W.Misner, K.S.Thorne and J.A.Wheeler {it Gravitation},

(W.H.Freeman and Company, New York, 1973).

bibitem{wald} R.M.Wald {it General Relativity}, (The University of Chicago Press,

Chicago and London, 1984).

bibitem{Naka} M.Nakahara, {it Geometry, Topology and Physics}, (

Institute of Physics Press, 1992). DOI: https://doi.org/10.1049/ee.1992.0057

bibitem{Eguchi} T.Eguchi, P.B.Gilkey and A.J.Hanson, {it

Gravitation, Gauge theories and Differential Geometry} Physics

Reports {bf 66} No 6 (1990).

bibitem{Dubois} M.Dubois-Viollete {it Lectures on graded

Differential algebras and Noncommutative Geometry} LPT-ORSAY

/100, qa/9912017 ( 1999).

bibitem{SITARZ} A.Sitarz Class.Quant.Grav. {bf 11} (1994) DOI: https://doi.org/10.1088/0264-9381/11/8/017

bibitem{Klim} C.Klimcik, A.Pompos, V.Soucek, Lett.Math.Phys. {bf

} (1994), 259.

bibitem{CHINA} Bin Chen, Takesi Saito, Ke Wu,

Prog.Theor.Phys. {bf 92}, (1994), 881; G. Konisi, Takesi Saito,

Ke Wu, Prog.Theor.Phys. {bf 93}, (1995), 621.

bibitem{LiConKK} M.Dubois-Violette, R.Kerner, J.Madore,

J.Math.Phys. {bf 31} (1990), 316; J.Madore, Phys.Rev. {D41} DOI: https://doi.org/10.1063/1.528916

(1990), 3790; M.Dubois-Violette, J.Madore, T.Masson, J.Mourad,

J.Mourad, J.Math.Phys. {bf 37} (1996), 4089; J.Madore, DOI: https://doi.org/10.1063/1.531618

Class.Quant.Grav. {bf 13} (1996), 2109; J.Mourad, DOI: https://doi.org/10.1088/0264-9381/13/8/008

Class.Quant.Grav. {bf 12} (1995), 965. DOI: https://doi.org/10.1088/0264-9381/12/4/007

bibitem{KK} Th.Kaluza, Sitzuuza, Sitzungsber. Preuss. Akad. Wiss. Phys.

Math. Klasse 966 (1921);O.Klein, Z.F. Physik {bf 37} (1926) 895; DOI: https://doi.org/10.1007/BF01397481

Y.Thirry, Comptes Rendus (Paris) {bf 226} (1948) 216.

Downloads

Published

12-03-2014

How to Cite

[1]
N. A. Viet, “A New Solution to the Structure Equation in Noncommutative Spacetime”, Comm. Phys., vol. 24, no. 1, p. 21, Mar. 2014.

Issue

Section

Papers
Received 30-01-2014
Published 12-03-2014

Most read articles by the same author(s)

1 2 > >>