Letters in Mathematical Physics

, Volume 108, Issue 5, pp 1323–1340 | Cite as

The Standard Model in noncommutative geometry: fundamental fermions as internal forms

  • Ludwik Dąbrowski
  • Francesco D’AndreaEmail author
  • Andrzej Sitarz


Given the algebra, Hilbert space H, grading and real structure of the finite spectral triple of the Standard Model, we classify all possible Dirac operators such that H is a self-Morita equivalence bimodule for the associated Clifford algebra.


Spectral triples Morita equivalence Standard Model 

Mathematics Subject Classification

Primary 58B34 Secondary 46L87 81T13 



L.D. is grateful for his support at IMPAN provided by Simons-Foundation Grant 346300 and a Polish Government MNiSW 2015–2019 matching fund. A.S. acknowledges the support from the Grant NCN 2015/19/B/ST1/03098. This work is part of the project Quantum Dynamics sponsored by the EU-grant RISE 691246 and Polish Government grant 317281.


  1. 1.
    Boyle, L., Farnsworth, S.: A new algebraic structure in the standard model of particle physics. arXiv:1604.00847 [hep-th]
  2. 2.
    Brouder, Ch., Bizi, N., Besnard, F.: The Standard Model as an extension of the noncommutative algebra of forms. arXiv:1504.03890 [hep-th]
  3. 3.
    Chamseddine, A.H., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991–1089 (2007). arXiv:hep-th/0610241
  4. 4.
    Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives, Colloquium Publications, 55th edn. AMS, Providence (2008)zbMATHGoogle Scholar
  5. 5.
    Connes, A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194–6231 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Connes, A.: Gravity coupled with matter and foundation of non-commutative geometry. Commun. Math. Phys. 182, 155–176 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    D’Andrea, F., Dąbrowski, L.: The Standard Model in Noncommutative Geometry and Morita equivalence. J. Noncommut. Geom. 10, 551–578 (2016). arXiv:1501.00156 [math-ph]
  8. 8.
    Kurkov, M., Lizzi, F.: Clifford Structures in Noncommutative Geometry and Extended Bosonic Sector (in preparation) Google Scholar
  9. 9.
    Krajewski, T.: Classification of finite spectral triples. J. Geom. Phys. 28, 1–30 (1998). arXiv:9701081 [hep-th]
  10. 10.
    Farnsworth, S., Boyle, L.: Non-commutative geometry, non-associative geometry and the Standard Model of particle physics. N. J. Phys. 16, 123027 (2014). arXiv:1401.5083 [hep-th]
  11. 11.
    Plymen, R.J.: Strong Morita equivalence, spinors and symplectic spinors. J. Oper. Theory 16, 305–324 (1986)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Paschke, M., Sitarz, A.: Discrete spectral triples and their symmetries. J. Math. Phys. 39, 6191–6205 (1996). arXiv:q-alg/9612029
  13. 13.
    Paschke, M., Scheck, F., Sitarz, A.: Can (noncommutative) geometry accommodate leptoquarks? Phys. Rev. D 59, 035003 (1999). arXiv:hepth/9709009
  14. 14.
    van Suijlekom, W.D.: Noncommutative Geometry and Particle Physics. Springer, New York (2015)CrossRefzbMATHGoogle Scholar
  15. 15.
    Várilly, J.C.: An Introduction to Noncommutative Geometry. EMS Series of Lectures in Mathematics (2006)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Scuola Internazionale Superiore di Studi Avanzati (SISSA)TriesteItaly
  2. 2.Università di Napoli “Federico II”NaplesItaly
  3. 3.I.N.F.N. Sezione di NapoliNaplesItaly
  4. 4.Instytut Fizyki Uniwersytetu JagiellońskiegoKrakówPoland
  5. 5.IMPANWarszawaPoland

Personalised recommendations