Skip to main content
SpringerLink
Log in

The Interface of Noncommutative Geometry and Physics

  • Chapter
Book cover Clifford Algebras

Part of the book series: Progress in Mathematical Physics ((PMP,volume 34))

Abstract

As a mathematical theory, noncommutative geometry (NCG) is by now well established. From the beginning, its progress has been crucially influenced by quantum physics: we briefly review this development in recent years.

The standard model of fundamental interactions, with its central role for the Dirac operator, led to several formulations culminating in the concept of a real spectral triple. String theory then came into contact with NCG, leading to an emphasis on Moyal-like algebras and formulations of quantum field theory on noncommutative spaces. Hopf algebras have yielded an unexpected link between the noncommutative geometry of foliations and perturbative quantum field theory.

The quest for a suitable foundation of quantum gravity continues to promote fruitful ideas, among them the spectral action principle and the search for a better understanding of “noncommutative spaces”.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Connes, C*-algebres et geometrie differentielle, C. R. Acad. Sci. Paris 290 (1980), 599–604.

    MathSciNet  MATH  Google Scholar 

  2. M.A. Rieffel, C*-algebras associated with irrational rotations, Pac. J. Math. 93 (1981), 415–429.

    MathSciNet  MATH  Google Scholar 

  3. A. Connes, Spectral sequence and homology of currents for operator algebras, Tagungsbericht 42/81, Mathematisches Forschungszentrum Oberwolfach, 1981.

    Google Scholar 

  4. A. Connes, Cohomologie cyclique et foncteurs Extn, C. R. Acad. Sci. Paris 296 (1983),953–958.

    MathSciNet  MATH  Google Scholar 

  5. A. Connes, Noncommutative differential geometry, Publ. Math. IHES 39 (1985), 257–360.

    MathSciNet  Google Scholar 

  6. A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, in Geometric Methods in Operator Algebras Eds. H. Araki and E. G. Effros; Pitman Research Notes in Math. 123 (1986), pp. 52–144.

    Google Scholar 

  7. A. Connes and M. Karoubi, Caracterè multiplicatif d’un module de Fredholm, K-Theory 2 (1988), 431–463.

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Araki, Schwinger terms and cyclic cohomology, in Quantum Theories and Geometry Eds. M. Cahen and M. Flato, Kluwer, Dordrecht, 1988; pp. 1–22.

    Google Scholar 

  9. P. Baum and R. G. Douglas, Index theory, bordism and K-holnology, in Operator Algebras and K-Theory Eds. R. G. Douglas and C. Schochet; Contemp. Math. 10 (1982), pp. 1–31.

    Google Scholar 

  10. N. Higson and J. Roe, Analytic K-Homology Oxford University Press, Oxford, 2000.

    MATH  Google Scholar 

  11. M. F. Atiyah, R. Bott and A. Shapiro, Clifford modules, Topology 3 (1964), 3–38.

    Article  MathSciNet  Google Scholar 

  12. A. Connes and J. Lott, Particle models and noncommutative geometry, Nucl. Phys. B (Proc. Suppl.) 18 (1990), 29–47.

    Article  MathSciNet  Google Scholar 

  13. J. C. Várilly and J. M. Gracia-Bondía, Connes’ noncommutative differential geometry and the standard model, J. Geom. Phys. 12 (1993), 223–301.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Kastler and T. Schücker, A detailed account of Alain Connes’ version of the standard model in noncommutative differential geometry. IV, Rev. Math. Phys. 8 (1996), 205–228.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Connes, Noncommutative geometry and reality, J. Math. Phys. 36 (1995), 6194–6231.

    Article  MathSciNet  MATH  Google Scholar 

  16. C. P. Martín, J. M. Gracia-Bondía and J. C. Várilly, The standard model as a noncommutative geometry: the low energy regime, Phys. Reports 294 (1998), 363–406.

    Article  Google Scholar 

  17. E. Álvarez, J. M. Gracia-Bondía and C. P. Martín, Parameter constraints in a noncommutative geometry model do not survive standard quantum corrections, Phys. Lett. B306 (1993), 55–58.

    Google Scholar 

  18. F. Scheck, The standard model within noncommutative geometry: A comparison of models, Talk at the Ninth Max Born Symposium, Karpacz, Poland, September 1996; hep-thl9701073, Mainz, 1997.

    Google Scholar 

  19. M. Takesaki, Tomita’s Theory of Modular Hilbert Algebras Springer, Berlin, 1970.

    MATH  Google Scholar 

  20. M. F. Atiyah, K-theory and reality, Quart. J. Math. 17 (1966), 367–386.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Connes, La notion de variété et les axiomes de la géométrie, Cours au Collège de France, Paris, January - March 1996.

    Google Scholar 

  22. J. M. Gracia-Bondía, J. C. Várilly and H. Figueroa, Elements of Noncommutative Geometry Birkhauser, Boston, 2001.

    MATH  Google Scholar 

  23. A. Connes, Gravity coupled with matter and foundation of noncommutative geometry, Commun. Math. Phys. 182 (1996), 155–176.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. H. Chamseddine and A. Connes, The spectral action principle, Commun. Math. Phys. 186 (1997), 731–750.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Estrada, J. M. Gracia-Bondía and J. C. Várilly, On summability of distributions and spectral geometry, Commun. Math. Phys. 191 (1998), 219–248.

    Article  MATH  Google Scholar 

  26. R. Wulkenhaar, Nonrenormalizability of θ-expanded noncommutative QED, J. High Energy Phys. 0203 (2002), 024.

    Article  MathSciNet  Google Scholar 

  27. E. Langmann, Generalized Yang-Mills actions from Dirac operator determinants, J. Math. Phys. 42 (2001), 5238–5256.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Rennie, Poincare duality and spine structures for complete noncommutative manifolds, math-ph10107013, Adelaide, 2001.

    Google Scholar 

  29. A. Rennie, Smoothness and locality for nonunital spectral triples, K-Theory 28 (2003), 127–165.

    Article  MathSciNet  MATH  Google Scholar 

  30. H. S. Snyder, Quantized space-time, Phys. Rev. 71 (1947), 38–41.

    Article  MATH  Google Scholar 

  31. S. Doplicher, K. Fredenhagen and J. E. Roberts, The quantum structure of spacetime at the Planck scale and quantum fields, Commun. Math. Phys. 172 (1995), 187–220.

    Article  MathSciNet  MATH  Google Scholar 

  32. M. M. Sheikh-Jabbari, Open strings in a B-field background as electric dipoles, Phys. Lett. B455 (1999), 129–134.

    MathSciNet  Google Scholar 

  33. N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High Energy Phys. 9 (1999), 032.

    Article  MathSciNet  Google Scholar 

  34. V. Schomerus, D-branes and deformation quantization, J. High Energy Phys. 9906 (1999), 030.

    Article  MathSciNet  Google Scholar 

  35. A. Connes, M. R. Douglas and A. Schwartz, Noncommutative geometry and Matrix theory: compactification on tori, J. High Energy Phys. 9802 (1998), 003.

    Article  Google Scholar 

  36. M. R. Douglas and C. M. Hull, D-branes and the noncommutative torus, J. High Energy Phys. 9802 (1998), 008.

    Article  MathSciNet  Google Scholar 

  37. G. Landi, F. Lizzi and R. J. Szabo, String geometry and the noncommutative torus, Commun. Math. Phys. 206 (1999), 603–637.

    Article  MathSciNet  MATH  Google Scholar 

  38. R. Jackiw and S.-Y. Pi, Noncommutative l-cocycle in the Seiberg-Witten map, Phys. Lett. B 534 (2002), 181–184.

    Article  MathSciNet  MATH  Google Scholar 

  39. B. Jureo, P. Schupp and J. Wess, Noncommutative line bundle and Morita equivalence, Lett. Math. Phys. 61 (2002), 171–186.

    Article  MathSciNet  Google Scholar 

  40. R. Estrada, J. M. Gracia-Bondía and J. C. Várilly, On asymptotic expansions of twisted products, J. Math. Phys. 30 (1989), 2789–2796.

    Article  MathSciNet  MATH  Google Scholar 

  41. J. E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45 (1949), 99–124.

    Article  MathSciNet  MATH  Google Scholar 

  42. J. M. Gracia-Bondía and J. C. Várilly, Algebras of distributions suitable for phasespace quantum mechanics. I, J. Math. Phys. 29 (1988), 869–879.

    Article  MathSciNet  MATH  Google Scholar 

  43. J. C. Várilly and J. M. Gracia-Bondía, Algebras of distributions suitable for phasespace quantum mechanics. II. Topologies on the Moyal algebra, J. Math. Phys. 29 (1988), 880–887.

    Article  MathSciNet  MATH  Google Scholar 

  44. J. M. Gracia-Bondía, F. Lizzi, G. Marmo and P. Vitale, Infinitely many star-products to play with, J. High Energy Phys. 0204 (2002), 026.

    Article  Google Scholar 

  45. J. C. Várilly and J. M. Gracia-Bondía, On the ultraviolet behaviour of quantum fields over noncommutative manifolds, Int. J. Mod. Phys. A14 (1999), 1305–1323.

    Google Scholar 

  46. T. Filk, Divergences in a field theory on quantum space, Phys. Lett. B376 (1996), 53–58.

    MathSciNet  Google Scholar 

  47. A. González-Arroyo and C. P. Korthals-Altes, Reduced model for large N continuum field theories, Phys. Lett. B131 (1983), 396–398.

    Google Scholar 

  48. S. Minwalla, M. V. Raamsdonk and N. Seiberg, Noncommutative perturbative dynamics, J. High Energy Phys. 0002 (2000), 020.

    Article  Google Scholar 

  49. J. Gomis and T. Mehen, Space-time noncommutative field theories and unitarity, Nucl. Phys. B591 (2000), 265–270.

    Article  MathSciNet  Google Scholar 

  50. D. Bahns, S. Doplicher, K. Fredenhagen and G. Piacitelli, On the unitarity problem in space/time noncommutative theories, Phys. Lett. B533 (2002), 178–181.

    MathSciNet  Google Scholar 

  51. H. Cheng, How to quantize Yang-Mills theory, in Chen Ning Yang: A Great Physicist of the Twentieth Century Eds. C. S. Liu and S.-T. Yau, International Press, Cambridge, MA, 1995; pp. 49–57.

    Google Scholar 

  52. C. Rim and J. H. Yee, Unitarity in space-time noncommutative field theories, hepth/0205193, Chonbuk, Korea, 2002.

    Google Scholar 

  53. Y. Liao and K. Sibold, Time-ordered perturbation theory on noncommutative spacetime I: basic rules, Eur. Phys. J. C25 (2002), 469–477; II: unitarity, Eur. Phys. J. C25 (2002), 479–486.

    MathSciNet  Google Scholar 

  54. M. R. Douglas and N. A. Nekrasov, Noncommutative field theory, Rev. Mod. Phys. 73 (2002), 977–1029.

    Article  MathSciNet  Google Scholar 

  55. R. J. Szabo, Quantum field theory on noncommutative spaces, Physics Reports 378 (2003), 207–299.

    Article  MathSciNet  MATH  Google Scholar 

  56. K. Morita, Connes’ gauge theory on noncommutative spacetimes, hep-th/0011080, Nagoya, 2000.

    Google Scholar 

  57. M. Chaichian, P. Prešnajder, M. M. Sheikh-Jabbari and A. Tureanu, Noncommutative Standard Model: model building, Eur. Phys. J. C29 (2003), 413–432.

    Google Scholar 

  58. X. Calmet, B. Jurčo, P. Schupp, J. Wess and M. Wohlgenannt, The standard model on noncommutative spacetime, Eur. Phys. J. C23 (2002), 363–376.

    Google Scholar 

  59. J. M. Gracia-Bondía and C. P. Martfn, Chiral gauge anomalies on noncommutative R4, Phys. Lett. B479 (2000), 321–328.

    Google Scholar 

  60. J. M. Gracia-Bondía, Noncommutative geometry and fundamental interactions: the first ten years, Ann. Phys. (Leipzig) 11 (2002), 479–495.

    Article  MATH  Google Scholar 

  61. A. Connes, A short survey of noncommutative geometry, J. Math. Phys. 41 (2000), 3832–3866.

    Article  MathSciNet  MATH  Google Scholar 

  62. A. Connes and G. Landi, Noncommutative manifolds, the instanton algebra and isospectral deformations, Commun. Math. Phys. 221 (2001), 141–159.

    Article  MathSciNet  MATH  Google Scholar 

  63. A. Connes and M. Dubois-Violette, Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples, Commun. Math. Phys. 230 (2002), 539–579.

    Article  MathSciNet  MATH  Google Scholar 

  64. J. C. Várilly, Quantum symmetry groups of noncommutative spheres, Commun. Math. Phys. 221 (2001), 511–523.

    Article  MATH  Google Scholar 

  65. M. A. Rieffel, Deformation Quantization for Actions of Rd Memoirs of the AMS 506, Providence, RI, 1993.

    Google Scholar 

  66. P. S. Chakraborty and A. Pal, Equivariant spectral triples on the quantum SU(2) group, K-Theory 28 (2003), 107–126.

    Article  MathSciNet  MATH  Google Scholar 

  67. A. Connes, Cyclic cohomology, quantum group symmetries and the local index formula for SU q (2), math.QA/0209142, IHES, 2002.

    Google Scholar 

  68. A. Connes, Talk at the Third Meeting on Nichtkommutative Geometrie, Oberwolfach, March 2002.

    Google Scholar 

  69. 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, Commun. Math. Phys. 210 (2000), 249–273.

    Article  MathSciNet  MATH  Google Scholar 

  70. A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem II: the β-function, diffeomorphisms and the renormalization group, Commun. Math. Phys. 216 (2001), 215–241.

    Article  MathSciNet  MATH  Google Scholar 

  71. J. M. Gracia-Bondía and S. Lazzarini, Connes-Kreimer-Epstein-Glaser renormalization, hep-th/0006106, Marseille and Mainz, 2000.

    Google Scholar 

  72. W. Zimmermann, Remark on equivalent formulations for Bogoliubov’s method of renormalization, in Renormalization Theory G. Velo and A. S. Wightman, eds., NATO ASI Series C 23 (D. Reidel, Dordrecht, 1976).

    Google Scholar 

  73. J. C. Várilly, Hopf algebras in noncommutative geometry, in Geometrical and Topological Methods in Quantum Field Theory Eds. A. Cardona, H. Ocampo and S. Paycha, World Scientific, Singapore, 2003; hep-th/0109077.

    Google Scholar 

  74. F. Girelli, P. Martinetti and T. Krajewski, The Hopf algebra of Connes and Kreimer and wave function renormalization, Mod. Phys. Lett. A16 (2001), 299–303.

    MathSciNet  Google Scholar 

  75. A. Connes and R. Moscovici, Ropf algebras, cyclic cohomology and the transverse index theorem, Commun. Math. Phys. 198 (1998), 198–246.

    Article  MathSciNet  Google Scholar 

  76. A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys. 199 (1998), 203–242.

    Article  MathSciNet  MATH  Google Scholar 

  77. A. Connes and D. Kreimer, Insertion and elimination: the doubly infinite Lie algebra of Feynman graphs, Ann. Henri Poincaré 3 (2002), 411–433.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidad de Costa Rica, 2060 San José, Costa Rica

    Joseph C. Várilly

Authors
  1. Joseph C. Várilly
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Tennessee Technological University, 5054, TN 38505, Cookeville, USA

    Rafał Abłamowicz

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Birkhäuser Boston

About this chapter

Cite this chapter

Várilly, J.C. (2004). The Interface of Noncommutative Geometry and Physics. In: Abłamowicz, R. (eds) Clifford Algebras. Progress in Mathematical Physics, vol 34. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2044-2_15

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2044-2_15

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-3525-1

  • Online ISBN: 978-1-4612-2044-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation