Noncommutative Yang-Mills and noncommutative relativity: A bridge over troubled water

European Physical Journal C

Volumn 8, Issue 4, 1999, Pages 697-709

  Carminati, L ^a   Iochum, B ^a   Schiucker T ^a  

^a Univ Toulon Var Ctr Phys Theor   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0006417201     PISSN: 14346044     EISSN: None     Source Type: Journal    
DOI: 10.1007/s100529900943     Document Type: Article

References (47)

1. A. Connes, Noncommutative Geometry, Academic Press, 1994
2. A. Connes, J. Lott, The metric aspect of noncommutative geometry, in the proceedings of the 1991 Cargèese Summer Conference, eds.: J. Fröhlich et al., Plenum Press, 1992
3. A. Connes, Noncommutative geometry and reality, J. Math. Phys. 36, 6194 (1995)
4. A. Connes, Gravity coupled with matter and the foundation of noncommutative geometry, hep-th/9603053, Comm. Math. Phys. 155, 109 (1996)
5. A. Chamseddine, A. Connes, The spectral action principle, hep-th/9606001
6. B. Iochum, T. Schücker, Yang-Mills-Higgs versus Connes-Lott, hep-th/9501142, Comm. Math. Phys. 178, 1 (1996)
7. T. Krajewski, Classification of finite spectral triples, hep-th/9701081, J. Geom. Phys., to appear
8. M. Paschke, A. Sitarz, Discrete spectral triples and their symmetries, q-alg/9612029
9. for other approaches see: M. Dubois-Violette, R. Kerner, J. Madore, Gauge bosons in a noncommutative geometry, Phys. Lett. 217B, 485 (1989); J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, Cambridge University Press, 1995; R. Coquereaux, C. Esposito-Farèse, G. Vaillant, Higgs fields as Yang-Mills fields and discrete symmetries, Nucl. Phys. B 353, 689 (1991)
10. for other approaches see: M. Dubois-Violette, R. Kerner, J. Madore, Gauge bosons in a noncommutative geometry, Phys. Lett. 217B, 485 (1989); J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, Cambridge University Press, 1995; R. Coquereaux, C. Esposito-Farèse, G. Vaillant, Higgs fields as Yang-Mills fields and discrete symmetries, Nucl. Phys. B 353, 689 (1991)
11. for other approaches see: M. Dubois-Violette, R. Kerner, J. Madore, Gauge bosons in a noncommutative geometry, Phys. Lett. 217B, 485 (1989); J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, Cambridge University Press, 1995; R. Coquereaux, C. Esposito-Farèse, G. Vaillant, Higgs fields as Yang-Mills fields and discrete symmetries, Nucl. Phys. B 353, 689 (1991)
12. A. Connes, Noncommutative geometry and physics, in the proceedings of the 1992 Les Houches Summer School, eds.: B. Julia, J. Zinn-Justin, North Holland (1995); D. Kastler, The Dirac operator and gravitation, Comm. Math. Phys. 166, 633 (1995); W. Kalau, M. Walze, Gravity, non-commutative Geometry and the Wodzicki residue, gr-qc/9312031, J. Geom. Phys. 16, 327 (1995); for other approaches see: J. Madore, Kaluza-Klein aspects of noncommutative geometry, in the proceedings of the 1988 conference on Differential Geometric Methods in Theoretical Physics, ed.: A. Solomon, World Scientific, 1989; A. Chamseddine, J. Fröhlich, O. Grandjean, The gravitational sector in the Connes-Lott forrnulation of the standard model, hep-th/9503093, J. Math. Phys. 36, 6255 (1995); T. Ackermann, J. Tolksdorf, Unification of gravity and Yang-Mills-Higgs gauge theories, hep-th/9503180; T. Ackermann, Dirac operators and Clifford geometries - new unifying principles in particle physics?, hep-th/9605129; J. Tolksdorf, The Einstein-Hilbert-Yang-Mills-Higgs action and the Dirac-Yukawa operator, hep-th/9612149; H. Figueroa, J.M. Gracia-Bondía, F. Lizzi, J.C. Virilly, A nonperturbative form of the spectral action principle in noncommutative geometry, hep-th/9701179
13. A. Connes, Noncommutative geometry and physics, in the proceedings of the 1992 Les Houches Summer School, eds.: B. Julia, J. Zinn-Justin, North Holland (1995); D. Kastler, The Dirac operator and gravitation, Comm. Math. Phys. 166, 633 (1995); W. Kalau, M. Walze, Gravity, non-commutative Geometry and the Wodzicki residue, gr-qc/9312031, J. Geom. Phys. 16, 327 (1995); for other approaches see: J. Madore, Kaluza-Klein aspects of noncommutative geometry, in the proceedings of the 1988 conference on Differential Geometric Methods in Theoretical Physics, ed.: A. Solomon, World Scientific, 1989; A. Chamseddine, J. Fröhlich, O. Grandjean, The gravitational sector in the Connes-Lott forrnulation of the standard model, hep-th/9503093, J. Math. Phys. 36, 6255 (1995); T. Ackermann, J. Tolksdorf, Unification of gravity and Yang-Mills-Higgs gauge theories, hep-th/9503180; T. Ackermann, Dirac operators and Clifford geometries - new unifying principles in particle physics?, hep-th/9605129; J. Tolksdorf, The Einstein-Hilbert-Yang-Mills-Higgs action and the Dirac-Yukawa operator, hep-th/9612149; H. Figueroa, J.M. Gracia-Bondía, F. Lizzi, J.C. Virilly, A nonperturbative form of the spectral action principle in noncommutative geometry, hep-th/9701179
14. A. Connes, Noncommutative geometry and physics, in the proceedings of the 1992 Les Houches Summer School, eds.: B. Julia, J. Zinn-Justin, North Holland (1995); D. Kastler, The Dirac operator and gravitation, Comm. Math. Phys. 166, 633 (1995); W. Kalau, M. Walze, Gravity, non-commutative Geometry and the Wodzicki residue, gr-qc/9312031, J. Geom. Phys. 16, 327 (1995); for other approaches see: J. Madore, Kaluza-Klein aspects of noncommutative geometry, in the proceedings of the 1988 conference on Differential Geometric Methods in Theoretical Physics, ed.: A. Solomon, World Scientific, 1989; A. Chamseddine, J. Fröhlich, O. Grandjean, The gravitational sector in the Connes-Lott forrnulation of the standard model, hep-th/9503093, J. Math. Phys. 36, 6255 (1995); T. Ackermann, J. Tolksdorf, Unification of gravity and Yang-Mills-Higgs gauge theories, hep-th/9503180; T. Ackermann, Dirac operators and Clifford geometries - new unifying principles in particle physics?, hep-th/9605129; J. Tolksdorf, The Einstein-Hilbert-Yang-Mills-Higgs action and the Dirac-Yukawa operator, hep-th/9612149; H. Figueroa, J.M. Gracia-Bondía, F. Lizzi, J.C. Virilly, A nonperturbative form of the spectral action principle in noncommutative geometry, hep-th/9701179
15. A. Connes, Noncommutative geometry and physics, in the proceedings of the 1992 Les Houches Summer School, eds.: B. Julia, J. Zinn-Justin, North Holland (1995); D. Kastler, The Dirac operator and gravitation, Comm. Math. Phys. 166, 633 (1995); W. Kalau, M. Walze, Gravity, non-commutative Geometry and the Wodzicki residue, gr-qc/9312031, J. Geom. Phys. 16, 327 (1995); for other approaches see: J. Madore, Kaluza-Klein aspects of noncommutative geometry, in the proceedings of the 1988 conference on Differential Geometric Methods in Theoretical Physics, ed.: A. Solomon, World Scientific, 1989; A. Chamseddine, J. Fröhlich, O. Grandjean, The gravitational sector in the Connes-Lott forrnulation of the standard model, hep-th/9503093, J. Math. Phys. 36, 6255 (1995); T. Ackermann, J. Tolksdorf, Unification of gravity and Yang-Mills-Higgs gauge theories, hep-th/9503180; T. Ackermann, Dirac operators and Clifford geometries - new unifying principles in particle physics?, hep-th/9605129; J. Tolksdorf, The Einstein-Hilbert-Yang-Mills-Higgs action and the Dirac-Yukawa operator, hep-th/9612149; H. Figueroa, J.M. Gracia-Bondía, F. Lizzi, J.C. Virilly, A nonperturbative form of the spectral action principle in noncommutative geometry, hep-th/9701179
16. A. Connes, Noncommutative geometry and physics, in the proceedings of the 1992 Les Houches Summer School, eds.: B. Julia, J. Zinn-Justin, North Holland (1995); D. Kastler, The Dirac operator and gravitation, Comm. Math. Phys. 166, 633 (1995); W. Kalau, M. Walze, Gravity, non-commutative Geometry and the Wodzicki residue, gr-qc/9312031, J. Geom. Phys. 16, 327 (1995); for other approaches see: J. Madore, Kaluza-Klein aspects of noncommutative geometry, in the proceedings of the 1988 conference on Differential Geometric Methods in Theoretical Physics, ed.: A. Solomon, World Scientific, 1989; A. Chamseddine, J. Fröhlich, O. Grandjean, The gravitational sector in the Connes-Lott forrnulation of the standard model, hep-th/9503093, J. Math. Phys. 36, 6255 (1995); T. Ackermann, J. Tolksdorf, Unification of gravity and Yang-Mills-Higgs gauge theories, hep-th/9503180; T. Ackermann, Dirac operators and Clifford geometries - new unifying principles in particle physics?, hep-th/9605129; J. Tolksdorf, The Einstein-Hilbert-Yang-Mills-Higgs action and the Dirac-Yukawa operator, hep-th/9612149; H. Figueroa, J.M. Gracia-Bondía, F. Lizzi, J.C. Virilly, A nonperturbative form of the spectral action principle in noncommutative geometry, hep-th/9701179
17. A. Connes, Noncommutative geometry and physics, in the proceedings of the 1992 Les Houches Summer School, eds.: B. Julia, J. Zinn-Justin, North Holland (1995); D. Kastler, The Dirac operator and gravitation, Comm. Math. Phys. 166, 633 (1995); W. Kalau, M. Walze, Gravity, non-commutative Geometry and the Wodzicki residue, gr-qc/9312031, J. Geom. Phys. 16, 327 (1995); for other approaches see: J. Madore, Kaluza-Klein aspects of noncommutative geometry, in the proceedings of the 1988 conference on Differential Geometric Methods in Theoretical Physics, ed.: A. Solomon, World Scientific, 1989; A. Chamseddine, J. Fröhlich, O. Grandjean, The gravitational sector in the Connes-Lott forrnulation of the standard model, hep-th/9503093, J. Math. Phys. 36, 6255 (1995); T. Ackermann, J. Tolksdorf, Unification of gravity and Yang-Mills-Higgs gauge theories, hep-th/9503180; T. Ackermann, Dirac operators and Clifford geometries - new unifying principles in particle physics?, hep-th/9605129; J. Tolksdorf, The Einstein-Hilbert-Yang-Mills-Higgs action and the Dirac-Yukawa operator, hep-th/9612149; H. Figueroa, J.M. Gracia-Bondía, F. Lizzi, J.C. Virilly, A nonperturbative form of the spectral action principle in noncommutative geometry, hep-th/9701179
18. A. Connes, Noncommutative geometry and physics, in the proceedings of the 1992 Les Houches Summer School, eds.: B. Julia, J. Zinn-Justin, North Holland (1995); D. Kastler, The Dirac operator and gravitation, Comm. Math. Phys. 166, 633 (1995); W. Kalau, M. Walze, Gravity, non-commutative Geometry and the Wodzicki residue, gr-qc/9312031, J. Geom. Phys. 16, 327 (1995); for other approaches see: J. Madore, Kaluza-Klein aspects of noncommutative geometry, in the proceedings of the 1988 conference on Differential Geometric Methods in Theoretical Physics, ed.: A. Solomon, World Scientific, 1989; A. Chamseddine, J. Fröhlich, O. Grandjean, The gravitational sector in the Connes-Lott forrnulation of the standard model, hep-th/9503093, J. Math. Phys. 36, 6255 (1995); T. Ackermann, J. Tolksdorf, Unification of gravity and Yang-Mills-Higgs gauge theories, hep-th/9503180; T. Ackermann, Dirac operators and Clifford geometries - new unifying principles in particle physics?, hep-th/9605129; J. Tolksdorf, The Einstein-Hilbert-Yang-Mills-Higgs action and the Dirac-Yukawa operator, hep-th/9612149; H. Figueroa, J.M. Gracia-Bondía, F. Lizzi, J.C. Virilly, A nonperturbative form of the spectral action principle in noncommutative geometry, hep-th/9701179
19. A. Connes, Noncommutative geometry and physics, in the proceedings of the 1992 Les Houches Summer School, eds.: B. Julia, J. Zinn-Justin, North Holland (1995); D. Kastler, The Dirac operator and gravitation, Comm. Math. Phys. 166, 633 (1995); W. Kalau, M. Walze, Gravity, non-commutative Geometry and the Wodzicki residue, gr-qc/9312031, J. Geom. Phys. 16, 327 (1995); for other approaches see: J. Madore, Kaluza-Klein aspects of noncommutative geometry, in the proceedings of the 1988 conference on Differential Geometric Methods in Theoretical Physics, ed.: A. Solomon, World Scientific, 1989; A. Chamseddine, J. Fröhlich, O. Grandjean, The gravitational sector in the Connes-Lott forrnulation of the standard model, hep-th/9503093, J. Math. Phys. 36, 6255 (1995); T. Ackermann, J. Tolksdorf, Unification of gravity and Yang-Mills-Higgs gauge theories, hep-th/9503180; T. Ackermann, Dirac operators and Clifford geometries - new unifying principles in particle physics?, hep-th/9605129; J. Tolksdorf, The Einstein-Hilbert-Yang-Mills-Higgs action and the Dirac-Yukawa operator, hep-th/9612149; H. Figueroa, J.M. Gracia-Bondía, F. Lizzi, J.C. Virilly, A nonperturbative form of the spectral action principle in noncommutative geometry, hep-th/9701179
20. A. Connes, Noncommutative geometry and physics, in the proceedings of the 1992 Les Houches Summer School, eds.: B. Julia, J. Zinn-Justin, North Holland (1995); D. Kastler, The Dirac operator and gravitation, Comm. Math. Phys. 166, 633 (1995); W. Kalau, M. Walze, Gravity, non-commutative Geometry and the Wodzicki residue, gr-qc/9312031, J. Geom. Phys. 16, 327 (1995); for other approaches see: J. Madore, Kaluza-Klein aspects of noncommutative geometry, in the proceedings of the 1988 conference on Differential Geometric Methods in Theoretical Physics, ed.: A. Solomon, World Scientific, 1989; A. Chamseddine, J. Fröhlich, O. Grandjean, The gravitational sector in the Connes-Lott forrnulation of the standard model, hep-th/9503093, J. Math. Phys. 36, 6255 (1995); T. Ackermann, J. Tolksdorf, Unification of gravity and Yang-Mills-Higgs gauge theories, hep-th/9503180; T. Ackermann, Dirac operators and Clifford geometries - new unifying principles in particle physics?, hep-th/9605129; J. Tolksdorf, The Einstein-Hilbert-Yang-Mills-Higgs action and the Dirac-Yukawa operator, hep-th/9612149; H. Figueroa, J.M. Gracia-Bondía, F. Lizzi, J.C. Virilly, A nonperturbative form of the spectral action principle in noncommutative geometry, hep-th/9701179
21. P.B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Publish or Perish (1984); S.A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time, Cambridge University Press, 1989
22. P.B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Publish or Perish (1984); S.A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time, Cambridge University Press, 1989
23. J. Madore, Geometric methods in classical field theory, Phys. Rep. 75, 125 (1981); G. Vilkovisky, The gospel according to DeWitt, in Quantum Theory of Gravity, ed.: S.M. Christensen, Adam Hilger (1984)
24. J. Madore, Geometric methods in classical field theory, Phys. Rep. 75, 125 (1981); G. Vilkovisky, The gospel according to DeWitt, in Quantum Theory of Gravity, ed.: S.M. Christensen, Adam Hilger (1984)
25. A.D. Sakharov, Spectral density of eigenvalues of the wave equation and vacuum polarization, Teor. Mat. Fiz. 23, 178 (1975)
26. A.D. Sakharov, Collected Works, Marcel Dekker, 1982; see however S. Adler, A formula for the induced gravitational constant, Phys. Lett. 95B, 241 (1980)
27. A.D. Sakharov, Collected Works, Marcel Dekker, 1982; see however S. Adler, A formula for the induced gravitational constant, Phys. Lett. 95B, 241 (1980)
28. U. Bonse, T. Wroblewski, Measurement of neutron quantum interference in noninertial frames, Phys. Rev. Lett. 1, 1401 (1983)
29. A. Connes, H. Moscovici, The local index formula in noncommutative geometry, Geometric And Functional Analysis 5, 174 (1995)
30. R. Estrada, J.M. Gracia-Bondía, J.C. Virilly, On summability of distributions and spectral geometry, functan/9702001
31. L. Carminati, B. Iochum, D. Kastler, T. Schücker, On Connes' new principle of general relativity: can spinors hear the forces of space-time?, hep-th/9612228, Operator Algebras and Quantum Field Theory, eds.: S. Doplicher et al., International Press, 1997
32. T. Schücker, Geometries and forces, hep-th/9712095, to appear in the proceedings of the Summer School on Noncommutative Geometry and Applications, Portugal, September 1997, ed.: Paulo Almeida; L. Carminati, B. Iochum, D. Kastler, T. Schücker, Relativity, noncommutative geometry, renormalization and particle physics, Phys. Rep., to appear
33. T. Schücker, Geometries and forces, hep-th/9712095, to appear in the proceedings of the Summer School on Noncommutative Geometry and Applications, Portugal, September 1997, ed.: Paulo Almeida; L. Carminati, B. Iochum, D. Kastler, T. Schücker, Relativity, noncommutative geometry, renormalization and particle physics, Phys. Rep., to appear
34. A. Connes, The action functional in non-commutative geometry, Comm. Math. Phys. 117, 673 (1988)
35. R. Asquith, Non-commutative geometry and the strong force, hep-th/9509163, Phys. Lett. B 366, 220 (1996)
36. D. Kastler, A detailed account of Alain Connes' version of the standard model in non-commutative geometry, I and II, Rev. Math. Phys. 5, 477 (1993); D. Kastler, A detailed account of Alain Connes' version of the standard model in non-commutative geometry, III, Rev. Math. Phys. 8, 103 (1996), D. Kastler, M. Mebkhout, Lectures on Non-Commutative Differential Geometry, World Scientific, to be published; J.C. Várilly, J.M. Gracia-Bondía, Connes' noncommutative differential geometry and the standard model, J. Geom. Phys. 12, 223 (1993) ; C.P. Martin, J.M. Gracia-Bondía, J.C. Várilly, The standard model as a non-commutative geometry: the low mass regime, hep-th/9605001, Phys. Rep., to appear
37. D. Kastler, A detailed account of Alain Connes' version of the standard model in non-commutative geometry, I and II, Rev. Math. Phys. 5, 477 (1993); D. Kastler, A detailed account of Alain Connes' version of the standard model in non-commutative geometry, III, Rev. Math. Phys. 8, 103 (1996), D. Kastler, M. Mebkhout, Lectures on Non-Commutative Differential Geometry, World Scientific, to be published; J.C. Várilly, J.M. Gracia-Bondía, Connes' noncommutative differential geometry and the standard model, J. Geom. Phys. 12, 223 (1993) ; C.P. Martin, J.M. Gracia-Bondía, J.C. Várilly, The standard model as a non-commutative geometry: the low mass regime, hep-th/9605001, Phys. Rep., to appear
38. D. Kastler, A detailed account of Alain Connes' version of the standard model in non-commutative geometry, I and II, Rev. Math. Phys. 5, 477 (1993); D. Kastler, A detailed account of Alain Connes' version of the standard model in non-commutative geometry, III, Rev. Math. Phys. 8, 103 (1996), D. Kastler, M. Mebkhout, Lectures on Non-Commutative Differential Geometry, World Scientific, to be published; J.C. Várilly, J.M. Gracia-Bondía, Connes' noncommutative differential geometry and the standard model, J. Geom. Phys. 12, 223 (1993) ; C.P. Martin, J.M. Gracia-Bondía, J.C. Várilly, The standard model as a non-commutative geometry: the low mass regime, hep-th/9605001, Phys. Rep., to appear
39. D. Kastler, A detailed account of Alain Connes' version of the standard model in non-commutative geometry, I and II, Rev. Math. Phys. 5, 477 (1993); D. Kastler, A detailed account of Alain Connes' version of the standard model in non-commutative geometry, III, Rev. Math. Phys. 8, 103 (1996), D. Kastler, M. Mebkhout, Lectures on Non-Commutative Differential Geometry, World Scientific, to be published; J.C. Várilly, J.M. Gracia-Bondía, Connes' noncommutative differential geometry and the standard model, J. Geom. Phys. 12, 223 (1993) ; C.P. Martin, J.M. Gracia-Bondía, J.C. Várilly, The standard model as a non-commutative geometry: the low mass regime, hep-th/9605001, Phys. Rep., to appear
40. D. Kastler, A detailed account of Alain Connes' version of the standard model in non-commutative geometry, I and II, Rev. Math. Phys. 5, 477 (1993); D. Kastler, A detailed account of Alain Connes' version of the standard model in non-commutative geometry, III, Rev. Math. Phys. 8, 103 (1996), D. Kastler, M. Mebkhout, Lectures on Non-Commutative Differential Geometry, World Scientific, to be published; J.C. Várilly, J.M. Gracia-Bondía, Connes' noncommutative differential geometry and the standard model, J. Geom. Phys. 12, 223 (1993) ; C.P. Martin, J.M. Gracia-Bondía, J.C. Várilly, The standard model as a non-commutative geometry: the low mass regime, hep-th/9605001, Phys. Rep., to appear
41. L. Carminati, B. Iochum, T. Schücker, The noncommutative constraints on the standard model à la Connes, hep-th/9604169, J. Math. Phys. 38, 1269 (1997)
42. D. Kastler, T. Schücker, The standard model a la Connes-Lott, hep-th/9412185, J. Geom. Phys. 388, 1 (1996); D. Kastler, T. Schücker, A detailed account of Alain Connes' version of the standard model in non-commutative geometry, IV, Rev. Math. Phys. 8, 205 (1996)
43. D. Kastler, T. Schücker, The standard model a la Connes-Lott, hep-th/9412185, J. Geom. Phys. 388, 1 (1996); D. Kastler, T. Schücker, A detailed account of Alain Connes' version of the standard model in non-commutative geometry, IV, Rev. Math. Phys. 8, 205 (1996)
44. N. Cabibbo, L. Maiani, G. Parisi, R. Petronzio, Bounds on the fermions and Higgs boson masses in grand unified theories, Nucl. Phys. B 158, 295 (1979)
45. C. Ford, D.R.T. Jones, P.W. Stephenson, M.B. Einhorn, The effective potential and the renormalisation group, Nucl. Phys. B 395, 17 (1993); B. Schrempp, M. Wimmer, Top quark and Higgs Boson Masses: Interplay between Infrared and Ultraviolet Physics, Progress in Particle and Nuclear Physics 37 (1996)
46. C. Ford, D.R.T. Jones, P.W. Stephenson, M.B. Einhorn, The effective potential and the renormalisation group, Nucl. Phys. B 395, 17 (1993); B. Schrempp, M. Wimmer, Top quark and Higgs Boson Masses: Interplay between Infrared and Ultraviolet Physics, Progress in Particle and Nuclear Physics 37 (1996)
47. Particule Data Group, Review of Particles Physics, Phys. Review D 54, 1 (1996)