A survey of spectral models of gravity coupled to matter

  • Ali Chamseddine
  • Walter D. van SuijlekomEmail author


This is a survey of the historical development of the spectral Standard Model and beyond, starting with the ground breaking paper of Alain Connes in 1988 where he observed that there is a link between Higgs fields and finite noncommutative spaces. We present the important contributions that helped in the search and identification of the noncommutative space that characterizes the fine structure of space-time. The nature and properties of the noncommutative space are arrived at by independent routes and show the uniqueness of the spectral Standard Model at low energies and the Pati–Salam unification model at high energies.



The work of A. H. C. is supported in part by the National Science Foundation Grant No. Phys-1518371. He also thanks the Radboud Excellence Initiative for hosting him at Radboud University where this research was carried out. We would like to thank Alain Connes for sharing with us his insights and ideas.


  1. 1.
    G. Aad et al. Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys. Lett. B716 (2012) 1–29.CrossRefGoogle Scholar
  2. 2.
    U. Aydemir. Clifford-based spectral action and renormalization group analysis of the gauge couplings. arXiv:1902.08090.Google Scholar
  3. 3.
    U. Aydemir, D. Minic, C. Sun, and T. Takeuchi. Pati–Salam unification from noncommutative geometry and the TeV-scale W R boson. Int. J. Mod. Phys. A31 (2016) 1550223.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    J. W. Barrett. A Lorentzian version of the non-commutative geometry of the standard model of particle physics. J. Math. Phys. 48 (2007) 012303.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    W. Beenakker, T. van den Broek, and W. D. van Suijlekom. Supersymmetry and noncommutative geometry, volume 9 of SpringerBriefs in Mathematical Physics. Springer, Cham, 2016.zbMATHCrossRefGoogle Scholar
  6. 6.
    F. Besnard. On the uniqueness of Barrett’s solution to the fermion doubling problem in noncommutative geometry. arXiv:1903.04769.Google Scholar
  7. 7.
    L. Boyle and S. Farnsworth. Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics, 1401.5083.Google Scholar
  8. 8.
    L. Boyle and S. Farnsworth. A new algebraic structure in the standard model of particle physics. JHEP 06 (2018) 071.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    T. van den Broek and W. D. van Suijlekom. Supersymmetric QCD and noncommutative geometry. Commun. Math. Phys. 303 (2011) 149–173.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    T. van den Broek and W. D. van Suijlekom. Supersymmetric QCD from noncommutative geometry. Phys. Lett. B699 (2011) 119–122.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    C. Brouder, N. Bizi, and F. Besnard. The Standard Model as an extension of the noncommutative algebra of forms. arXiv:1504.03890.Google Scholar
  12. 12.
    A. H. Chamseddine and A. Connes. Universal formula for noncommutative geometry actions: Unifications of gravity and the Standard Model. Phys. Rev. Lett. 77 (1996) 4868–4871.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    A. H. Chamseddine. Connection between space-time supersymmetry and noncommutative geometry. Phys. Lett. B332 (1994) 349–357.CrossRefGoogle Scholar
  14. 14.
    A. H. Chamseddine and A. Connes. Why the Standard Model. J. Geom. Phys. 58 (2008) 38–47.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    A. H. Chamseddine and A. Connes. Noncommutative geometry as a framework for unification of all fundamental interactions including gravity. Part I. Fortsch. Phys. 58 (2010) 553–600.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    A. H. Chamseddine and A. Connes. Resilience of the Spectral Standard Model. JHEP 1209 (2012) 104.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    A. H. Chamseddine, A. Connes, and M. Marcolli. Gravity and the Standard Model with neutrino mixing. Adv. Theor. Math. Phys. 11 (2007) 991–1089.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    A. H. Chamseddine, A. Connes, and V. Mukhanov. Geometry and the quantum: Basics. JHEP 1412 (2014) 098.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    A. H. Chamseddine, A. Connes, and V. Mukhanov. Quanta of geometry: Noncommutative aspects. Phys. Rev. Lett. 114 (2015) 091302.MathSciNetCrossRefGoogle Scholar
  20. 20.
    A. H. Chamseddine, A. Connes, and W. D. van Suijlekom. Entropy and the spectral action. Commun. Math. Phys. (online first) [arXiv:1809.02944].Google Scholar
  21. 21.
    A. H. Chamseddine, A. Connes, and W. D. van Suijlekom. Beyond the spectral Standard Model: Emergence of Pati-Salam unification. JHEP 1311 (2013) 132.zbMATHCrossRefGoogle Scholar
  22. 22.
    A. H. Chamseddine, A. Connes, and W. D. van Suijlekom. Grand unification in the spectral Pati-Salam model. JHEP 11 (2015) 011.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    A. H. Chamseddine, G. Felder, and J. Fröhlich. Unified gauge theories in noncommutative geometry. Phys. Lett. B. 296 (1992) 109.MathSciNetCrossRefGoogle Scholar
  24. 24.
    A. H. Chamseddine, G. Felder, and J. Fröhlich. Gravity in noncommutative geometry. Comm. Math. Phys. 155 (1993) 205–217.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    A. H. Chamseddine, J. Fröhlich, and O. Grandjean. The gravitational sector in the Connes-Lott formulation of the standard model. J. Math. Phys. 36 (1995) 6255–6275.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    S. Chatrchyan et al. Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Phys. Lett. B716 (2012) 30–61.CrossRefGoogle Scholar
  27. 27.
    A. Connes. Noncommutative differential geometry. Publ. Math. IHES 39 (1985) 257–360.MathSciNetGoogle Scholar
  28. 28.
    A. Connes. Essay on physics and noncommutative geometry. In The interface of mathematics and particle physics (Oxford, 1988), volume 24 of Inst. Math. Appl. Conf. Ser. New Ser., pages 9–48. Oxford Univ. Press, New York, 1990.Google Scholar
  29. 29.
    A. Connes. Noncommutative Geometry. Academic Press, San Diego, 1994.zbMATHGoogle Scholar
  30. 30.
    A. Connes. Noncommutative geometry and reality. J. Math. Phys. 36(11) (1995) 6194–6231.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    A. Connes. On the foundations of noncommutative geometry. In The unity of mathematics, volume 244 of Progr. Math., pages 173–204. Birkhäuser Boston, Boston, MA, 2006.Google Scholar
  32. 32.
    A. Connes. Noncommutative geometry year 2000 [ MR1826266 (2003g:58010)]. In Highlights of mathematical physics (London, 2000), pages 49–110. Amer. Math. Soc., Providence, RI, 2002.Google Scholar
  33. 33.
    A. Connes and J. Lott. Particle models and noncommutative geometry. Nucl. Phys. Proc. Suppl. 18B (1991) 29–47.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    A. Connes and H. Moscovici. Type III and spectral triples. In Traces in number theory, geometry and quantum fields, Aspects Math., E38, pages 57–71. Friedr. Vieweg, Wiesbaden, 2008.Google Scholar
  35. 35.
    L. Dabrowski. On noncommutative geometry of the Standard Model: fermion multiplet as internal forms. In 36th Workshop on Geometric Methods in Physics (XXXVI WGMP) Bialowieza, Poland, July 2–8, 2017, 2017.Google Scholar
  36. 36.
    L. Dabrowski, F. D’Andrea, and A. Sitarz. The Standard Model in noncommutative geometry: fundamental fermions as internal forms. Lett. Math. Phys. 108 (2018) 1323–1340.MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    L. Dabrowski and A. Sitarz. Fermion masses, mass-mixing and the almost commutative geometry of the Standard Model, 1806.07282.Google Scholar
  38. 38.
    A. Devastato, F. Lizzi, and P. Martinetti. Higgs mass in noncommutative geometry. Fortschr. Phys. 62 (2014) 863–868.MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    A. Devastato, S. Farnsworth, F. Lizzi, and P. Martinetti. Lorentz signature and twisted spectral triples. JHEP 03 (2018) 089.MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    A. Devastato, F. Lizzi, and P. Martinetti. Grand symmetry, spectral action, and the Higgs mass. JHEP 1401 (2014) 042.CrossRefGoogle Scholar
  41. 41.
    A. Devastato and P. Martinetti. Twisted spectral triple for the Standard Model and spontaneous breaking of the Grand Symmetry. Math. Phys. Anal. Geom. 20 (2017) 2.MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    R. Dong and M. Khalkhali. Second quantization and the spectral action. arXiv:1903.09624.Google Scholar
  43. 43.
    M. Dubois-Violette. Dérivations et calcul différentiel non commutatif. C. R. Acad. Sci. Paris Sér. I Math. 307 (1988) 403–408.MathSciNetzbMATHGoogle Scholar
  44. 44.
    M. Dubois-Violette, R. Kerner, and J. Madore. Classical bosons in a noncommutative geometry. Class. Quant. Grav. 6 (1989) 1709.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    M. Dubois-Violette, R. Kerner, and J. Madore. Gauge bosons in a noncommutative geometry. Phys. Lett. B217 (1989) 485–488.MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    M. Dubois-Violette, J. Madore, and R. Kerner. Noncommutative differential geometry and new models of gauge theory. J. Math. Phys. 31 (1990) 323.MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    K. van den Dungen and W. D. van Suijlekom. Electrodynamics from noncommutative geometry. J. Noncommut. Geom. 7, 433–456 (2013).MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    M. Eckstein and B. Iochum. Spectral Action in Noncommutative Geometry. Mathematical Physics Studies. Springer, 2018.zbMATHCrossRefGoogle Scholar
  49. 49.
    S. Farnsworth and L. Boyle. Non-associative geometry and the spectral action principle. JHEP (2015) 023, front matter+25.Google Scholar
  50. 50.
    H. Figueroa, J. M. Gracia-Bondía, F. Lizzi, and J. C. Várilly. A nonperturbative form of the spectral action principle in noncommutative geometry. J. Geom. Phys. 26 (1998) 329–339.MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    M. Gonderinger, Y. Li, H. Patel, and M. J. Ramsey-Musolf. Vacuum Stability, Perturbativity, and Scalar Singlet Dark Matter. JHEP 1001 (2010) 053.zbMATHCrossRefGoogle Scholar
  52. 52.
    J. M. Gracia-Bondía, B. Iochum, and T. Schücker. The standard model in noncommutative geometry and fermion doubling. Phys. Lett. B 416 (1998) 123–128.MathSciNetCrossRefGoogle Scholar
  53. 53.
    B. Iochum, T. Schucker, and C. Stephan. On a classification of irreducible almost commutative geometries. J. Math. Phys. 45 (2004) 5003–5041.MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    D. Kastler. A detailed account of Alain Connes’ version of the standard model in noncommutative geometry. I, II. Rev. Math. Phys. 5 (1993) 477–532.MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    D. Kastler. A detailed account of Alain Connes’ version of the standard model in non-commutative differential geometry. III. Rev. Math. Phys. 8 (1996) 103–165.MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    D. Kastler and T. Schücker. A detailed account of Alain Connes’ version of the standard model. IV. Rev. Math. Phys. 8 (1996) 205–228.MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    D. Kastler and T. Schücker. The standard model à la Connes-Lott. J. Geom. Phys. 24 (1997) 1–19.MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    T. Krajewski. Classification of finite spectral triples. J. Geom. Phys. 28 (1998) 1–30.MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    M.A. Kurkov and F. Lizzi. Clifford Structures in noncommutative geometry and the extended scalar sector. Phys.Rev. D97 (2018) 085024.MathSciNetGoogle Scholar
  60. 60.
    G. Landi. An Introduction to Noncommutative Spaces and their Geometry. Springer-Verlag, 1997.zbMATHGoogle Scholar
  61. 61.
    G. Landi and P. Martinetti. On twisting real spectral triples by algebra automorphisms. Lett. Math. Phys. 106 (2016) 1499–1530.MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    G. Landi and P. Martinetti. Gauge transformations for twisted spectral triples. Lett. Math. Phys. 108 (2018) 2589–2626.MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    F. Lizzi, G. Mangano, G. Miele, and G. Sparano. Fermion Hilbert space and fermion doubling in the noncommutative geometry approach to gauge theories. Phys. Rev. D55 (1997) 6357–6366.MathSciNetGoogle Scholar
  64. 64.
    F. Lizzi. Noncommutative Geometry and Particle Physics. PoS CORFU2017 (2018) 133.Google Scholar
  65. 65.
    J. Madore. An Introduction to Noncommutative Differential Geometry and its Physical Applications. Cambridge University Press, 1995.zbMATHGoogle Scholar
  66. 66.
    C. P. Martín, J. M. Gracia-Bondía, and J. C. Várilly. The standard model as a noncommutative geometry: the low-energy regime. Phys. Rep. 294 (1998) 363–406.MathSciNetCrossRefGoogle Scholar
  67. 67.
    R. Mohapatra. Unification and supersymmetry. The frontiers of quark-lepton physics. Springer, New York, third edition, 2003.Google Scholar
  68. 68.
    M. Paschke and A. Sitarz. Discrete spectral triples and their symmetries. J. Math. Phys. 39 (1998) 6191–6205.MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    F. Scheck, H. Upmeier, and W. Werner, editors. Noncommutative geometry and the standard model of elementary particle physics, volume 596 of Lecture Notes in Physics. Springer-Verlag, Berlin, 2002. Papers from the conference held in Hesselberg, March 14–19, 1999.Google Scholar
  70. 70.
    C. A. Stephan. Almost-commutative geometries beyond the Standard Model. J. Phys. A39 (2006) 9657.MathSciNetzbMATHGoogle Scholar
  71. 71.
    C. A. Stephan. Almost-commutative geometries beyond the Standard Model. ii. new colours. J. Phys. A40 (2007) 9941.MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    C. A. Stephan. Beyond the Standard Model: A Noncommutative Approach, 0905.0997.Google Scholar
  73. 73.
    C. A. Stephan. New scalar fields in noncommutative geometry. Phys. Rev. D79 (2009) 065013.MathSciNetGoogle Scholar
  74. 74.
    C. A. Stephan. A Dark Sector Extension of the Almost-Commutative Standard Model. Int. J. Mod. Phys. A29 (2014) 1450005.MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    W. D. van Suijlekom. Noncommutative Geometry and Particle Physics. Springer, 2015.zbMATHCrossRefGoogle Scholar
  76. 76.
    I. Todorov and M. Dubois-Violette. Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra. Int. J. Mod. Phys. A33 (2018) 1850118.MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    J. C. Várilly and J. M. Gracia-Bondía. Connes’ noncommutative differential geometry and the standard model. J. Geom. Phys. 12 (1993) 223–301.MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    E. P. Wigner. Normal form of antiunitary operators. J. Mathematical Phys. 1 (1960) 409–413.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Physics, Institute for MathematicsAstrophysics and Particle PhysicsNijmegenThe Netherlands
  2. 2.American University of BeirutBeirutLebanon
  3. 3.Radboud UniversityNijmegenThe Netherlands
  4. 4.Institute for Mathematics, Astrophysics and Particle PhysicsRadboud University NijmegenNijmegenThe Netherlands

Personalised recommendations