Skip to main content
SpringerLink
Log in

Noncommutative geometry and structure of space–time

  • Published:
Afrika Matematika Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Connes, A.: Noncommutative Geometry. Academic Press, New York (1994)

    MATH  Google Scholar 

  2. Connes, A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194 (1995)

    Article  MathSciNet  Google Scholar 

  3. Connes, A.: A Short Survey of Noncommutative Geometry. arXiv:hep-th/0003006

  4. Gracia-Bondia, J., Varilly, J., Figueroa, H.: Elements of noncommutative geometry. Birkhauser, Basel (2018)

    MATH  Google Scholar 

  5. Kac, M.: Can one hear the shape of a drum? Am. Math. Month. 73, 1–23 (1996)

    Article  MathSciNet  Google Scholar 

  6. Milnor, J.: Eigenvalues of the laplac operator on certain manifolds. Proc. Natl. Acad. Sci. USA 51, 542 (1964)

    Article  Google Scholar 

  7. Chamseddine, A.H., Connes, A.: The spectral action principle. Comm. Math. Phys. 186, 731 (1997)

    Article  MathSciNet  Google Scholar 

  8. Chamseddine, A.H., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991–1089 (2007)

    Article  MathSciNet  Google Scholar 

  9. Chamseddine, A.H., Connes, A.: Noncommutative geometry as a framework for unification of all fundamental interactions including gravity. Fortsch. Phys. 58, 553 (2010)

    Article  MathSciNet  Google Scholar 

  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)

  11. Chamseddine, A.H., Connes, A.: Why the standard model. J. Geom. Phys. 58, 38 (2008)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  13. Chamseddine, A.H., Connes, A.: Reselience of the spectral standard model. JHEP 09, 104 (2009)

    MATH  Google Scholar 

  14. Karimi, H.: Implications of the complex singlet field for noncommutative geometry model. JHEP 1712, 040 (2017)

    Article  MathSciNet  Google Scholar 

  15. Chamseddine, A.H., Connes, A., van Suijlekom, W.: Beyond the spectral standard model: emergence of Pati–Salam unification. JHEP 11, 132 (2013)

    Article  Google Scholar 

  16. Chamseddine, A.H., Connes, A., van Suijlekom, W.: Grand unification in the spectral Pati–Salam model. JHEP 1511, 011 (2015)

    Article  MathSciNet  Google Scholar 

  17. Chamseddine, A.H., Connes, A.: The Uncanny Precision of the spectral action. Comm. Math. Phys. 293, 867–897 (2010)

    Article  MathSciNet  Google Scholar 

  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. Lawson, H., Michelson, M.: Spin Geometry. Princeton University Press, Princeton (1989)

    Google Scholar 

  20. Moser, J.: On the volume elements on a manifold. Trans. Am. Math. Soc. 120, 286–294 (1965)

    Article  MathSciNet  Google Scholar 

  21. Chamseddine, A.H., Connes, A., Mukhanov, Viatcheslav: Geometry and the quantum: basics. JHEP 12, 098 (2014). arXiv:1411.0977

    Article  MathSciNet  Google Scholar 

  22. Chamseddine, A.H., Connes, A., Mukhanov, V.: Quanta of geometry: noncommutative aspects. Phys. Rev. Lett. 114, 091302 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Physics Department, American University of Beirut, Beirut, Lebanon

    Ali H. Chamseddine

  2. I.H.E.S., 91440, Bures-sur-Yvette, France

    Ali H. Chamseddine

Authors
  1. Ali H. Chamseddine
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ali H. Chamseddine.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chamseddine, A.H. Noncommutative geometry and structure of space–time. Afr. Mat. 31, 15–27 (2020). https://doi.org/10.1007/s13370-018-0648-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-018-0648-3

Keywords

Mathematics Subject Classification

Search

Navigation