Abstract
Einstein derived general relativity from Riemannian geometry. Connes extends this derivation to noncommutative geometry and obtains electro–magnetic, weak, and strong forces. These are pseudo forces, that accompany the gravitational force just as in Minkowskian geometry the magnetic force accompanies the electric force. The main physical input of Connes’ derivation is parity violation. His main output is the Higgs boson which breaks the gauge symmetry spontaneously and gives masses to gauge and Higgs bosons.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
1. A. Connes, A. Lichnérowicz and M. P. Schützenberger, Triangle de Pensées, O. Jacob (2000), English version: Triangle of Thoughts, AMS (2001)
2. G. Amelino-Camelia, Are we at the dawn of quantum gravity phenomenology?, Lectures given at 35th Winter School of Theoretical Physics: From Cosmology to Quantum Gravity, Polanica, Poland, 1999, gr-qc/9910089
3. S. Weinberg, Gravitation and Cosmology, Wiley (1972) R. Wald, General Relativity, The University of Chicago Press (1984)
4. J. D. Bjørken and S. D. Drell, Relativistic Quantum Mechanics, McGraw–Hill (1964)
5. L. O’Raifeartaigh, Group Structure of Gauge Theories, Cambridge University Press (1986)
6. M. Göckeler and T. Schücker, Differential Geometry, Gauge Theories, and Gravity, Cambridge University Press (1987)
7. R. Gilmore, Lie Groups, Lie Algebras and some of their Applications, Wiley (1974) H. Bacry, Lectures Notes in Group Theory and Particle Theory, Gordon and Breach (1977)
8. N. Jacobson, Basic Algebra I, II, Freeman (1974,1980)
9. J. Madore, An Introduction to Noncommutative Differential Geometry and its Physical Applications, Cambridge University Press (1995) G. Landi, An Introduction to Noncommutative Spaces and their Geometry, hep-th/9701078, Springer (1997)
10. J. M. Gracia-Bondía, J. C. Várilly and H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser (2000)
11. J. W. van Holten, Aspects of BRST quantization, hep-th/0201124, in this volume
12. J. Zinn-Justin, Chiral anomalies and topology, hep-th/0201220, in this volume
13. The Particle Data Group, Particle Physics Booklet and http://pdg.lbl.gov
14. G. ’t Hooft, Renormalizable Lagrangians for Massive Yang–Mills Fields, Nucl. Phys. B35 (1971) 167 G. ’t Hooft and M. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B44 (1972) 189 G. ’t Hooft and M. Veltman, Combinatorics of Gauge Fields, Nucl. Phys. B50 (1972) 318 B. W. Lee and J. Zinn-Justin, Spontaneously broken gauge symmetries I, II, III and IV, Phys. Rev. D5 (1972) 3121, 3137, 3155; Phys. Rev. D7 (1973) 1049
15. S. Glashow, Partial-symmetries of weak interactions, Nucl. Phys. 22 (1961) 579 A. Salam in Elementary Particle Physics: Relativistic Groups and Analyticity, Nobel Symposium no. 8, page 367, eds.: N. Svartholm, Almqvist and Wiksell, Stockholm 1968 S. Weinberg, A model of leptons, Phys. Rev. Lett. 19 (1967) 1264
16. J. Iliopoulos, An introduction to gauge theories, Yellow Report, CERN (1976)
17. G. Esposito-Farèse, Théorie de Kaluza–Klein et gravitation quantique, Thése de Doctorat, Université d’Aix-Marseille II, 1989
18. A. Connes, Noncommutative Geometry, Academic Press (1994)
19. A. Connes, Noncommutative Geometry and Reality, J. Math. Phys. 36 (1995) 6194
20. A. Connes, Gravity coupled with matter and the foundation of noncommutative geometry, hep-th/9603053, Comm. Math. Phys. 155 (1996) 109
21. H. Rauch, A. Zeilinger, G. Badurek, A. Wilfing, W. Bauspiess and U. Bonse, Verification of coherent spinor rotations of fermions, Phys. Lett. 54A (1975) 425
22. E. Cartan, Leçons sur la théorie des spineurs, Hermann (1938)
23. A. Connes, Brisure de symétrie spontanée et géométrie du point de vue spectral, Séminaire Bourbaki, 48ème année, 816 (1996) 313 A. Connes, Noncommutative differential geometry and the structure of space time, Operator Algebras and Quantum Field Theory, eds.: S. Doplicher et al., International Press, 1997
24. T. Schücker, Spin group and almost commutative geometry, hep-th/0007047
25. J.-P. Bourguignon and P. Gauduchon, Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys. 144 (1992) 581
26. U. Bonse and T. Wroblewski, Measurement of neutron quantum interference in noninertial frames, Phys. Rev. Lett. 1 (1983) 1401
27. R. Colella, A. W. Overhauser and S. A. Warner, Observation of gravitationally induced quantum interference, Phys. Rev. Lett. 34 (1975) 1472
28. A. Chamseddine and A. Connes, The spectral action principle, hep-th/9606001, Comm. Math. Phys.186 (1997) 731
29. G. Landi and C. Rovelli, Gravity from Dirac eigenvalues, gr-qc/9708041, Mod. Phys. Lett. A13 (1998) 479
30. 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)
31. B. Iochum, T. Krajewski and P. Martinetti, Distances in finite spaces from noncommutative geometry, hep-th/9912217, J. Geom. Phys. 37 (2001) 100
32. M. Dubois-Violette, R. Kerner and J. Madore, Gauge bosons in a noncommutative geometry, Phys. Lett. 217B (1989) 485
33. A. Connes, Essay on physics and noncommutative geometry, in The Interface of Mathematics and Particle Physics, eds.: D. G. Quillen et al., Clarendon Press (1990) A. Connes and J. Lott, Particle models and noncommutative geometry, Nucl. Phys. B 18B (1990) 29 A. Connes and J. Lott, The metric aspect of noncommutative geometry, in the proceedings of the 1991 Cargèse Summer Conference, eds.: J. Fröhlich et al., Plenum Press (1992)
34. J. Madore, Modification of Kaluza Klein theory, Phys. Rev. D 41 (1990) 3709
35. P. Martinetti and R. Wulkenhaar, Discrete Kaluza–Klein from Scalar Fluctuations in Noncommutative Geometry, hep-th/0104108, J. Math. Phys. 43 (2002) 182
36. T. Ackermann and J. Tolksdorf, A generalized Lichnerowicz formula, the Wodzicki residue and gravity, hep-th/9503152, J. Geom. Phys. 19 (1996) 143 T. Ackermann and J. Tolksdorf, The generalized Lichnerowicz formula and analysis of Dirac operators, hep-th/9503153, J. reine angew. Math. 471 (1996) 23
37. R. Estrada, J. M. Gracia-Bondía and J. C. Várilly, On summability of distributions and spectral geometry, funct-an/9702001, Comm. Math. Phys. 191 (1998) 219
38. B. Iochum, D. Kastler and T. Schücker, On the universal Chamseddine–Connes action: details of the action computation, hep-th/9607158, J. Math. Phys. 38 (1997) 4929 L. Carminati, B. Iochum, D. Kastler and 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
39. M. Paschke and A. Sitarz, Discrete spectral triples and their symmetries, q-alg/9612029, J. Math. Phys. 39 (1998) 6191 T. Krajewski, Classification of finite spectral triples, hep-th/9701081, J. Geom. Phys. 28 (1998) 1
40. S. Lazzarini and T. Schücker, A farewell to unimodularity, hep-th/0104038, Phys. Lett. B 510 (2001) 277
41. B. Iochum and T. Schücker, A left-right symmetric model à la Connes–Lott, hep-th/9401048, Lett. Math. Phys. 32 (1994) 153 F. Girelli, Left-right symmetric models in noncommutative geometry? hep-th/0011123, Lett. Math. Phys. 57 (2001) 7
42. F. Lizzi, G. Mangano, G. Miele and G. Sparano, Constraints on unified gauge theories from noncommutative geometry, hep-th/9603095, Mod. Phys. Lett. A11 (1996) 2561
43. W. Kalau and M. Walze, Supersymmetry and noncommutative geometry, hep-th/9604146, J. Geom. Phys. 22 (1997) 77
44. D. Kastler, Introduction to noncommutative geometry and Yang–Mills model building, Differential geometric methods in theoretical physics, Rapallo (1990), 25 — , A detailed account of Alain Connes’ version of the standard model in non-commutative geometry, I, II and III, Rev. Math. Phys. 5 (1993) 477, Rev. Math. Phys. 8 (1996) 103 D. Kastler and T. Schücker, Remarks on Alain Connes’ approach to the standard model in non-commutative geometry, Theor. Math. Phys. 92 (1992) 522, English version, 92 (1993) 1075, hep-th/0111234 — , A detailed account of Alain Connes’ version of the standard model in non-commutative geometry, IV, Rev. Math. Phys. 8 (1996) 205 — , The standard model à la Connes–Lott, hep-th/9412185, J. Geom. Phys. 388 (1996) 1 J. C. Várilly and J. M. Gracia-Bondía, Connes’ noncommutative differential geometry and the standard model, J. Geom. Phys. 12 (1993) 223 T. Schücker and J.-M. Zylinski, Connes’ model building kit, hep-th/9312186, J. Geom. Phys. 16 (1994) 1 E. Alvarez, J. M. Gracia-Bondía and C. P. Martín, Anomaly cancellation and the gauge group of the Standard Model in Non-Commutative Geometry, hep-th/9506115, Phys. Lett. B364 (1995) 33 R. Asquith, Non-commutative geometry and the strong force, hep-th/9509163, Phys. Lett. B 366 (1996) 220 C. P. Martín, J. M. Gracia-Bondía and J. C. Várilly, The standard model as a noncommutative geometry: the low mass regime, hep-th/9605001, Phys. Rep. 294 (1998) 363 L. Carminati, B. Iochum and T. Schücker, The noncommutative constraints on the standard model à la Connes, hep-th/9604169, J. Math. Phys. 38 (1997) 1269 R. Brout, Notes on Connes’ construction of the standard model, hep-th/9706200, Nucl. Phys. Proc. Suppl. 65 (1998) 3 J. C. Várilly, Introduction to noncommutative geometry, physics/9709045, EMS Summer School on Noncommutative Geometry and Applications, Portugal, september 1997, ed.: P. Almeida T. Schücker, Geometries and forces, hep-th/9712095, EMS Summer School on Noncommutative Geometry and Applications, Portugal, september 1997, ed.: P. Almeida J. M. Gracia-Bondía, B. Iochum and T. Schücker, The Standard Model in Noncommutative Geometry and Fermion Doubling, hep-th/9709145, Phys. Lett. B 414 (1998) 123 D. Kastler, Noncommutative geometry and basic physics, Lect. Notes Phys. 543 (2000) 131 — , Noncommutative geometry and fundamental physical interactions: the Lagrangian level, J. Math. Phys. 41 (2000) 3867 K. Elsner, Noncommutative geometry: calculation of the standard model Lagrangian, hep-th/0108222, Mod. Phys. Lett. A16 (2001) 241
45. R. Jackiw, Physical instances of noncommuting coordinates, hep-th/0110057
46. N. Cabibbo, L. Maiani, G. Parisi and R. Petronzio, Bounds on the fermions and Higgs boson masses in grand unified theories, Nucl. Phys. B158 (1979) 295
47. L. Carminati, B. Iochum and T. Schücker, Noncommutative Yang–Mills and noncommutative relativity: A bridge over troubled water, hep-th/9706105, Eur. Phys. J. C8 (1999) 697
48. B. Iochum and T. Schücker, Yang–Mills–Higgs versus Connes–Lott, hep-th/9501142, Comm. Math. Phys. 178 (1996) 1 I. Pris and T. Schücker, Non-commutative geometry beyond the standard model, hep-th/9604115, J. Math. Phys. 38 (1997) 2255 I. Pris and T. Krajewski, Towards a Z’ gauge boson in noncommutative geometry, hep-th/9607005, Lett. Math. Phys. 39 (1997) 187 M. Paschke, F. Scheck and A. Sitarz, Can (noncommutative) geometry accommodate leptoquarks? hep-th/9709009, Phys . Rev. D59 (1999) 035003 T. Schücker and S. ZouZou, Spectral action beyond the standard model, hep-th/0109124
49. A. Connes and H. Moscovici, Hopf algebra, cyclic cohomology and the transverse index theorem, Comm. Math. Phys. 198 (1998) 199 D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, q-alg/9707029, Adv. Theor. Math. Phys. 2 (1998) 303 A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. 1. The Hopf algebra structure of graphs and the main theorem, hep-th/9912092, Comm. Math. Phys. 210 (2000) 249 A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann–Hilbert problem. 2. The beta function, diffeomorphisms and the renormalization group, hep-th/0003188, Comm. Math. Phys. 216 (2001) 215 for a recent review, see J. C. Várilly, Hopf algebras in noncommutative geometry, hep-th/010977
50. S. Majid and T. Schücker, Z2 x Z2 Lattice as Connes–Lott–quantum group model, hep-th/0101217, J. Geom. Phys. 43 (2002) 1
51. J. C. Várilly and J. M. Gracia-Bondía, On the ultraviolet behaviour of quantum fields over noncommutative manifolds, hep-th/9804001, Int. J. Mod. Phys. A14 (1999) 1305 T. Krajewski, Géométrie non commutative et interactions fondamentales, Thése de Doctorat, Université de Provence, 1998, math-ph/9903047 C. P. Martín and D. Sanchez-Ruiz, The one-loop UV divergent structure of U(1) Yang–Mills theory on noncommutative R4, hep-th/9903077, Phys. Rev. Lett. 83 (1999) 476 M. M. Sheikh-Jabbari, Renormalizability of the supersymmetric Yang–Mills theories on the noncommutative torus, hep-th/9903107, JHEP 9906 (1999) 15 T. Krajewski and R. Wulkenhaar, Perturbative quantum gauge fields on the noncommutative torus, hep-th/9903187, Int. J. Mod. Phys. A15 (2000) 1011 S. Cho, R. Hinterding, J. Madore and H. Steinacker, Finite field theory on noncommutative geometries, hep-th/9903239, Int. J. Mod. Phys. D9 (2000) 161
52. M. Rieffel, Irrational Rotation C*-Algebras, Short Comm. I.C.M. 1978 A. Connes, C* algèbres et géométrie différentielle, C.R. Acad. Sci. Paris, Ser. A-B (1980) 290, English version hep-th/0101093 A. Connes and M. Rieffel, Yang–Mills for non-commutative two-tori, Contemp. Math. 105 (1987) 191
53. J. Bellissard, K-theory of C*-algebras in solid state physics, in: Statistical Mechanics and Field Theory: Mathematical Aspects, eds.: T. C. Dorlas et al., Springer (1986) J. Bellissard, A. van Elst and H. Schulz-Baldes, The noncommutative geometry of the quantum Hall effect, J. Math. Phys. 35 (1994) 5373
54. A. Connes and G. Landi, Noncommutative manifolds, the instanton algebra and isospectral deformations, math.QA/0011194, Comm. Math. Phys. 216 (2001) 215 A. Connes and M. Dubois-Violette, Noncommutative finite-dimensional manifolds I. Spherical manifolds and related examples, math.QA/0107070
55. M. Chaichian, P. Prešnajder, M. M. Sheikh-Jabbari and A. Tureanu, Noncommutative standard model: Model building, hep-th/0107055 X. Calmet, B. Jurčo, P. Schupp, J. Wess and M. Wohlgenannt, The standard model on non-commutative space-time, hep-ph/0111115
56. A. Connes and C. Rovelli, Von Neumann algebra Automorphisms and time-thermodynamics relation in general covariant quantum theories, gr-qc/9406019, Class. Quant. Grav. 11 (1994) 1899 C. Rovelli, Spectral noncommutative geometry and quantization: a simple example, gr-qc/9904029, Phys. Rev. Lett. 83 (1999) 1079 M. Reisenberger and C. Rovelli, Spacetime states and covariant quantum theory, gr-qc/0111016
57. W. Kalau, Hamiltonian formalism in non-commutative geometry, hep-th/9409193, J. Geom. Phys. 18 (1996) 349 E. Hawkins, Hamiltonian gravity and noncommutative geometry, gr-qc/9605068, Comm. Math. Phys. 187 (1997) 471 T. Kopf and M. Paschke, A spectral quadruple for the De Sitter space, math-ph/0012012 A. Strohmaier, On noncommutative and semi-Riemannian geometry, math-ph/0110001
Author information
Authors and Affiliations
Editor information
Rights and permissions
About this chapter
Cite this chapter
Schücker, T. Forces from Connes’ Geometry. In: Bick, E., Steffen, F.D. (eds) Topology and Geometry in Physics. Lecture Notes in Physics, vol 659. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31532-2_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-31532-2_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23125-7
Online ISBN: 978-3-540-31532-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)