Noncommutative geometry and fundamental physical interactions: The Lagrangian level - Historical sketch and description of the present situation

Journal of Mathematical Physics

Volumn 41, Issue 6, 2000, Pages 3867-3891

  Kastler, Daniel ^a  

^a CENTRE DE PHYSIQUE THÉORIQUE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034379806     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.533330     Document Type: Article

References (71)

1. (a) A. Connes, Noncommutarive Geometry (Academic, New York, 1994);
2. (b) "C*-algèbres et géométrie différentielle," C. R. Acad. Sci. Paris Ser. A-B 290, A689;A604 (1980).
3. M. F. Atiyah, "Global elliptic operators," Proceedings of the International Conference on Functional Analysis (Tokyo University Press, Tokyo, Japan, 1970).
4. G. Kasparov, "The operator K-functor and extensions of C*-algebras." Izv. Akad. Nauk. SSSR Ser. Math. 44, 571 (1980).
5. A. S. Mishenko and A. T. Fomenko, The index of elliptic operators over C*-algebras, Math. USSR-Izv. 15, 87 (1980).
6. (a) A. Connes, "Non-commutative differential geometry. I. The Chern character in K-homology II. De Rham homology and non-commutative algebra," Publ. Math. IHES 62, 257 (1985);
7. (b) "Spectral sequence and homology of currents for operator algebras," Math. Forschungsinst. Oberwolfach, Tagungsbericht 42181, Funktionalanalyse-C*-Algebren, 27/9-3/10 (1981).
8. A. Connes, "The action functional in non-commutative geometry," Commun. Math. Phys. 117, 673 (1988).
9. A. Connes, "Trace de Dixrnier, modules de Fredholm et géometrie riemannienne," Nucl. Phys. B (Proc. Suppl.) 5B, 65 (1988).
10. (a) A. Connes and J. Lott, "Particle models and non-commutative geometry," Nucl. Phys. B (Proc. Suppl.) 18B (1990), Suppl. 29-47 (1991);
11. (b) "The metric aspect of non-commutative geometry," Proceedings of the July 1991 Cargèse School of Theoretical Physics;
12. (c) D. Kastler, "A detailed account of Alain Connes' version of the standard model in non-commutative differential geometry. I-II," Rev. Math. Phys. 5, 477 (1993); ibid. III, Rev. Math. Phys. 8, 103 (1996).
13. (c) D. Kastler, "A detailed account of Alain Connes' version of the standard model in non-commutative differential geometry. I-II," Rev. Math. Phys. 5, 477 (1993); ibid. III, Rev. Math. Phys. 8, 103 (1996).
14. A. Connes, "Essay on physics and non-commutative geometry," The Interface of Mathematics and Particle Physics (Clarendon, Oxford, 1990).
15. P. De la Harpe and G. Skandalis, "Déterminant associé à une trace sur une algèbre de Banach," Annales Inst. Fourier 34, 241 (1984).
16. R. Asquith, "Noncommutative geometry and the strong force," Phys. Lett. B 366, 220 (1996).
17. B. Iochum and T. Schücker, "Yang-Mills-Higgs versus Connes-Lott," Commun. Math. Phys. 178, 1 (1996).
18. D. Kastler and T. Schücker, "Remarks on Alain Connes' approach to the standard model in non-commutative geometry," Theor. Math. Phys. 92, 523 (1992).
19. D. Kastler, Towards extracting physical predictions from Alain Connes' version of the standard model (the new grand unification?). Istambul workshop Operator algebras, Mathematical physics and Low dimensional topology, A. K. Peters, 1993.
20. A. Connes, "Non-commutative geometry and physics," Proceedings of 1992 Les Houches School of Theoretical Physics (North-Holland, Amsterdam, 1995).
21. (a) D. Kastler and T. Schücker, "A detailed account of Alain Connes' version of the standard model in non-commutative differential geometry. IV," Rev. Math. Phys. 8, 205 (1996);
22. (b) "The standard model a la Connes-Lott," J. Geom. Phys. 17, 1 (1997).
23. (a) I. Iochum, D. Kastler, and T. Schücker, "Fuzzy mass relations for the Higgs," J. Math. Phys. 36, 232 (1995);
24. (b) "Standard model and fuzzy mass relations, geometry and nature," edited by H. Nencka and J. P. Bourguignon, Contemp. Math. 203, 175 (1997).
25. D. Kastler, "Dual pairs of quantum spaces," Lett. Math. Phys. 37, 329 (1996).
26. E. Alvarez, J. Gracia-Bondia, and C. P. Martin, "Anomaly cancellation and the gauge group of the standard model in noncommutative geometry," Phys. Lett. B 364, 33 (1995).
27. A. Connes, "Noncommutative geometry and reality," J. Math. Phys. 36, 6194 (1995).
28. M. Takesaki, Tomita's Theory of Modular Hilbert Algebras and its Applications, Lecture Notes in Math. 128 (Springer-Verlag, Berlin, 1970).
29. M. F. Atiyah, "K-theory and reality," Q. J. Math. 17, 367 (1966).
30. D. Kastler and D. Testard, "Quantum forms of tensor products," Commun. Math. Phys. 155, 135 (1993).
31. L. Carminati, B. Jochum, D. Kastler, and T. Schücker, "The noncommutative constraints on the standard model à la Connes," J. Math. Phys. 38, 1269 (1997).
32. L. Carminati, B. Iochum, D. Kastler, and T. Schücker, "Noncommutative Yang-Mills and noncommutative relativity." A bridge over troubled water; hep-th/9706105.
33. A. Connes, "Brisure de symmétrie spontanée et géométrie du point de vue spectral," Exposé Bourbaki 816, (1995-96).
34. T. Krajewski, J. Geom. Phys. 28, 1 (1997).
35. M. Paschke and A. Sitarz, Discrete spectral triples and their symmetries, q-alg/9612029.
36. D. Kastler, "The Dirac operator and gravitation," Commun. Math. Phys. 166, 633 (1995).
37. W. Kalau and M. Walze, "Gravity, noncommutative geometry and the Wodzicki residue," J. Geom. Phys. 16, 327 (1995).
38. A. Connes, "Gravity coupled with matter and the foundation of noncommutative geometry," Commun. Math. Phys. 117, 673 (1996).
39. A. Chamseddine and A. Connes, "The spectral action principle," Commun. Math. Phys. 186, 731 (1997).
40. B. Iochum, D. Kastler, and T. Schücker, "On the universal Chamseddine-Connes action," J. Math. Phys. 38, 10 (1997).
41. (a) L. Carminati, B. Iochum, D. Kastler, and T. Schücker, in On Conne's new principle of general relativity: can spinors hear the forces of space-time? Operator Algebras and Quantum Field Theory, edited by S. Doplicher, R. Longo, and J. Roberts (International Press, 1997);
42. (b) "Relativity noncommmutative geometry, renormalization and particle physics," Rep. Math. Phys. 43, 53 (1999).
43. P. Gilkey, Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem (CRC. Boca Raton, FL, 1995).
44. exp(2iπ/3)(2)-Induced Frobenius Homomorphisms. Quantum Groups, Noncommutative Geometry and Fundamental Physical Interactions, edited by D. Kastler, M. Rosso, and T. Schücker (Nova Science Publishers, Commack, New York).
45. q(2) (q is a root of unit)," St. Petersburg Math. J. 6, 969 (1994).
46. q(2). Quantum Groups, Noncommutative Geometry and Fundamental Physical Interactions, edited by D. Kastler, M. Rosso, and T. Schücker (Nova Science Publishers, Inc., Commack, New York;
47. 2) at Fourth Root of Unity. Proceedings of the Luminy Conference "Analyse sur les groupes et algèbres de Lie de dimension infinie," Luminy 15-19 Sept 1997, edited by H. Heyer and J. Marion (World Scientific, Singapore, 1997).
48. 1 hep-th/9610114 (1996).
49. 3(3,C)," SISSA 63/97/FM, Int. J. Mod. Phys. (1997).
50. A. H. Chamseddine, J. Fröhlich, and O. Grandjean, "The gravitational sector in the Connes-Lott formulation of the standard model," J. Math. Phys. 36, 6255 (1995).
51. X. Estrada, J. M. Gracia-Bondia, and J. C. Varilly (unpublished).
52. H. Grosse, C. Klimčik, and P. Prešnajder, "Topologically nontrivial field configurations in noncommutative geometry," Commun. Math. Phys. 178, 507 (1996).
53. H. Grosse, C. Klimčik, and P. Prešnajder, On finite 4D quantum field theory in noncommutative geometry. CERN-THH/ 96-51,IHES/P/96/12.
54. S. Doplicher, K. Fredenhagen, and J. Roberts (unpublished).
55. J. Madore, Noncommutative Geometry and its Physical Applications, London Math. Soc. Lecture Notes Series 206 (Cambridge University Press, Cambridge, UK, 1995).
56. D. Kastler, J. Madore, and D. Testard, "Connections of bimodules in noncommutative geometry," Contemp. Math. 203, 159 (1997).
57. M. Dubois-Violette and T. Masson, "On the first-order operators in bimodules," Lett. Math. Phys. 37, 467 (1995).
58. W. Kalau, N. A. Papadopoulos, J. Plass, and J. M. Warzecha, "Differential algebras in noncommutative geometry," J. Geom. Phys. 16, 149 (1995).
59. T. Brzezinski and S. Majid, "Quantum group gauge theory on quantum spaces," Commun. Math. Phys. 157, 591 (1993); 167, 235 (1995).
60. T. Brzezinski and S. Majid, "Quantum group gauge theory on quantum spaces," Commun. Math. Phys. 157, 591 (1993); 167, 235 (1995).
61. M. Durdewic, "Geometry of quantum principal bundles," Commun. Math. Phys. 175, 457 (1995).
62. P. M. Hajac, "Strong connections on quantum principal bundles," Commun. Math. Phys. 182, 579 (1996).
63. L. Dabrowski, P. M. Hajac, G. Landi, and P. Siniscalco, "Metrics and pairs of left and right connections of bimodules," J. Math. Phys. 37, 4635 (1996).
64. S. Majid, Foundations of Quantum Group Theory (Cambridge University Press, Cambridge, UK, 1995).
65. C. Kassel, Quantum Groups, Graduate Texts in Math. 155 (Springer, New York, 1995).
66. G. Landi and C. Rovelli, "General relativity in terms of Dirac eigenvalues," Phys. Rev. Lett. 78, 3051 (1997); Mod. Phys. Lett. A 13, 479 (1998).
67. G. Landi and C. Rovelli, "General relativity in terms of Dirac eigenvalues," Phys. Rev. Lett. 78, 3051 (1997); Mod. Phys. Lett. A 13, 479 (1998).
68. R. Brout, Notes on Conne's construction of the standard model, Bruxelles, hep-th/9706200 1996.
69. q(Sl2 ) for |q|= 1 as the complexification of real Kopf algebra, Nach. Akad. Göttingen, II. Jahrgang 1999, 1. Vandenhoeck & Ruprecht, Göttingen.
70. D. Kastler, "Noncommutative geometry and basic physics," Springer Lecture Notes in Physics 543 (Springer Verlag, Berlin).
71. A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem I: The Hopf algebra structure of graphs and the main theorem, hep-th/9912012. II: the β-function, diffeomorphisms and the renormalization group hep-th/003188.