International Journal of Theoretical Physics

, Volume 56, Issue 1, pp 259–269 | Cite as

The Geometry of Noncommutative Space-Time

  • R. Vilela MendesEmail author


Stabilization, by deformation, of the Poincaré-Heisenberg algebra requires both the introduction of a fundamental lentgh and the noncommutativity of translations which is associated to the gravitational field. The noncommutative geometry structure that follows from the deformed algebra is studied both for the non-commutative tangent space and the full space with gravity. The contact points of this approach with the work of David Finkelstein are emphasized.


Spacetime Noncommutative geometry Gravity 


  1. 1.
    Finkelstein, D.R.: General quantization. Int. J. Theor. Phys. 45, 1399–1427 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Segal, I.E.: A class of operator algebras which are determined by groups. Duke Math. J. 18, 221–265 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Finkelstein, D.R., Shiri-Garakani, M.: Finite quantum dynamics. Int. J. Theor. Phys. 50, 1731–1751 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Vilela Mendes, R.: The stability of physical theories principle. In: Licata, I. (ed.) Beyond Peaceful Coexistence, The Emergence of Space, Time and Quantum, pp. 153–200. Imperial College Press, London (2016)Google Scholar
  5. 5.
    Vilela Mendes, R.: Deformations, stable theories and fundamental constants. J. Phys. A: Math. Gen. 27, 8091–8104 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Vilela Mendes, R.: Quantum mechanics and non-commutative space-time. Phys. Lett. A 210, 232–240 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Vilela Mendes, R.: Geometry, stochastic calculus and quantum fields in a non-commutative space-time. J. Math. Phys. 41, 156–186 (2000)ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Carlen, E., Vilela Mendes, R.: Non-commutative space-time and the uncertainty principle. Phys. Lett. A 290, 109–114 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Vilela Mendes, R.: Stochastic calculus and processes in non-commutative space-time. Prog. Probab. 52, 205–217 (2002)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Vilela Mendes, R.: Some consequences of a noncommutative space-time structure. Eur. Phys. J. C 42, 445–452 (2005)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Vilela Mendes, R.: The deformation-stability fundamental length and deviations from c. Phys. Lett. A 376, 1823–1826 (2012)ADSCrossRefGoogle Scholar
  12. 12.
    Vilela Mendes, R.: A laboratory scale fundamental time? Eur. Phys. J. C 72, 2239 (2012)ADSCrossRefGoogle Scholar
  13. 13.
    Finkelstein, D.R.: Quantum set theory and Clifford algebra. Int. J. Theor. Phys. 21, 489–503 (1982)CrossRefzbMATHGoogle Scholar
  14. 14.
    Baugh, J., Finkelstein, D.R., Galiautdinov, A., Saller, H.: Clifford algebra as quantum language. J. Math. Phys. 42, 1489–1500 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Galiautdinov, A.A., Finkelstein, D.R.: Chronon corrections to the Dirac equation. J. Math. Phys. 43, 4741–4752 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Baugh, J., Finkelstein, D.R., Galiautdinov, A., Shiri-Garakani, M.: Transquantum dynamics. Found. Phys. 33, 12671275 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Dubois-Violette, M., Kerner, R., Madore, J.: Noncommutative differential geometry of matrix algebras. J. Math. Phys. 31, 316–322 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Connes, A.: Non-commutative Geometry. Academic, New York (1994)zbMATHGoogle Scholar
  19. 19.
    Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives. American Math. Society Colloquium Publications (2007)Google Scholar
  20. 20.
    Vilela Mendes, R.: An extended Dirac equation in noncommutative space-time. Mod. Phys. Lett. A 31, 1650089 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Finkelstein, D.R.: Homotopy approach to quantum gravity. Int. J. Theor. Phys. 47, 534–552 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Finkelstein, D.R.: arXiv:1108.1495, arXiv:1403.3725, arXiv:1403.3726
  23. 23.
    Vilenkin, N. Ja., Klimik, A.U.: Representation of Lie Groups and Special Functions, vol. 2. Kluwer, Dordrecht (1993)Google Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Centro de Matemática e Aplicações FundamentaisUniversity of LisbonLisboaPortugal

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