Noncommutative geometry and structure of space–time

  • Ali H. ChamseddineEmail author


I give a summary review of the research program using noncommutative geometry as a framework to determine the structure of space–time. Classification of finite noncommutative spaces under few assumptions reveals why nature chose the Standard Model and the reasons behind the particular form of gauge fields, Higgs fields and fermions as well as the origin of symmetry breaking. It also points that at high energies the Standard Model is a truncation of Pati–Salam unified model of leptons and quarks. The same conclusions are arrived at uniquely without making any assumptions except for an axiom which is a higher form of Heisenberg commutation relations quantizing the volume of space–time. We establish the existence of two kinds of quanta of geometry in the form of unit spheres of Planck length. We provide answers to many of the questions which are not answered by other approaches, however, more research is needed to answer the remaining challenging questions.


Noncommutative Geometry Unification 

Mathematics Subject Classification

58B34 81T75 83C65 



I would like to thank Alain Connes for a fruitful collaboration on the topic of noncommutative geometry since 1996. I would also like to thank Walter van Suijlekom and Slava Mukhanov for essential contributions to this program of research. This research is supported in part by the National Science Foundation under Grant No. Phys-1518371.


  1. 1.
    Connes, A.: Noncommutative Geometry. Academic Press, New York (1994)zbMATHGoogle Scholar
  2. 2.
    Connes, A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Connes, A.: A Short Survey of Noncommutative Geometry. arXiv:hep-th/0003006
  4. 4.
    Gracia-Bondia, J., Varilly, J., Figueroa, H.: Elements of noncommutative geometry. Birkhauser, Basel (2018)zbMATHGoogle Scholar
  5. 5.
    Kac, M.: Can one hear the shape of a drum? Am. Math. Month. 73, 1–23 (1996)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Milnor, J.: Eigenvalues of the laplac operator on certain manifolds. Proc. Natl. Acad. Sci. USA 51, 542 (1964)CrossRefGoogle Scholar
  7. 7.
    Chamseddine, A.H., Connes, A.: The spectral action principle. Comm. Math. Phys. 186, 731 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chamseddine, A.H., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991–1089 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chamseddine, A.H., Connes, A.: Noncommutative geometry as a framework for unification of all fundamental interactions including gravity. Fortsch. Phys. 58, 553 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chamseddine, A.H., Connes, A.: Space–time from the spectral point of view. In: Damour T et al (eds) Proceedings 12th Marcel Grossmann Meeting, Paris, July 12–18 (2009)Google Scholar
  11. 11.
    Chamseddine, A.H., Connes, A.: Why the standard model. J. Geom. Phys. 58, 38 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chamseddine, A.H., Connes, A., van Suijlekom, W.: Inner fluctuations in noncommutative geometry without the first order condition. J. Geom. Phys. 73, 222 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chamseddine, A.H., Connes, A.: Reselience of the spectral standard model. JHEP 09, 104 (2009)zbMATHGoogle Scholar
  14. 14.
    Karimi, H.: Implications of the complex singlet field for noncommutative geometry model. JHEP 1712, 040 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chamseddine, A.H., Connes, A., van Suijlekom, W.: Beyond the spectral standard model: emergence of Pati–Salam unification. JHEP 11, 132 (2013)CrossRefGoogle Scholar
  16. 16.
    Chamseddine, A.H., Connes, A., van Suijlekom, W.: Grand unification in the spectral Pati–Salam model. JHEP 1511, 011 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Chamseddine, A.H., Connes, A.: The Uncanny Precision of the spectral action. Comm. Math. Phys. 293, 867–897 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Greub, W., Halperin, S., Vanstone, R.: Connections, Curvature and Cohomology, volumes 1–3, and in particular pages 347–351 volume 2 (sphere maps). Academic Press, New York (1973)Google Scholar
  19. 19.
    Lawson, H., Michelson, M.: Spin Geometry. Princeton University Press, Princeton (1989)Google Scholar
  20. 20.
    Moser, J.: On the volume elements on a manifold. Trans. Am. Math. Soc. 120, 286–294 (1965)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Chamseddine, A.H., Connes, A., Mukhanov, Viatcheslav: Geometry and the quantum: basics. JHEP 12, 098 (2014). arXiv:1411.0977 MathSciNetCrossRefGoogle Scholar
  22. 22.
    Chamseddine, A.H., Connes, A., Mukhanov, V.: Quanta of geometry: noncommutative aspects. Phys. Rev. Lett. 114, 091302 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Physics DepartmentAmerican University of BeirutBeirutLebanon
  2. 2.I.H.E.S.Bures-sur-YvetteFrance

Personalised recommendations