Skip to main content
SpringerLink
Log in

Statistics of Fields, the Yang-Baxter Equation, and the Theory of Knots and Links

  • Chapter
  • First Online:
Book cover Nonperturbative Quantum Field Theory

Part of the book series: Nato Science Series B: ((NSSB,volume 185))

Abstract

In these notes we describe an analysis of the statistics problem in quantum field theory. It has been known for some time that in two space-time dimensions there is more to the problem of statistics of local fields than Bose- or Fermi statistics [1], In three-dimensional space-time, local quantum fields obey Bose- or Fermi statistics, but “extended particles”, coupled to the vacuum by fields localized on cones, may exhibit intermediate statistics, so called O-statistics [2].

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. R.F. Streater and A.S. Wightman,’TCT, Spin and Statistics and All That 9 Reading, Mass: Benjamin 1918. (This book contains many references to the early literature on solitons and statistics in two dimensions.) R.F. Streater and LP. Wilde, Nuclear Physics B 24, 561 (1970).

    Google Scholar 

  2. Y.S. Wu, Phys. Rev. Lett. 52, 2103 (1984); 53, HI (1984).

    Google Scholar 

  3. J. Frohlich, Commun. Math. Phys. 47, 269 (1976).

    Article  ADS  Google Scholar 

  4. Acta Phys. Austriaca, Suppl. XV, 133 (1976).

    Google Scholar 

  5. J. Bellissard, J. Frohlich and B. Gidas, Commun. Math. Phys. 60, 37 (1978).

    Article  ADS  Google Scholar 

  6. J. Frohlich, in “Invariant Wave Equations”, Berlin-Heidelberg-New York: Springer-Verlag, Lecture Notes in Physics 73, 1978.

    Google Scholar 

  7. J. Frohlich, in : “Recent Developments in Gauge Theories” (Cargese 1979). G.’t Hooft et al. (eds.), New York: Plenum Press, 1980.

    Google Scholar 

  8. E.C. Marino and J.A. Swieca, Nucl. Phys. B170 [FS1],175 (1980).

    Article  ADS  Google Scholar 

  9. E.C. Marino, B. Schroer and J.A. Swieca, Nucl. Phys. B200 [FS4],473 (1982).

    Article  ADS  Google Scholar 

  10. R. Koberle and E. C. Marino, Phys. Lett. 126B, 475 (1983).

    Article  ADS  Google Scholar 

  11. J. Frohlich and P.-A. Marchetti, “Bosonization, Topological Solitons and Fractional Charges in Two-Dimensional Quantum Field Theory” submitted to Commun. Math. Phys.. and Commun. Math. Phys. 112, 343-383 (1987)

    Google Scholar 

  12. A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B241, 333 (1984).

    Article  ADS  Google Scholar 

  13. V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B247, 83 (1984).

    Article  ADS  Google Scholar 

  14. D. Friedan, E. Martinec and S. Shenker, Phys. Lett. B160, 55 (1985).

    Article  ADS  Google Scholar 

  15. D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B271, 93 (1986).

    Article  ADS  Google Scholar 

  16. V.G. Knizhnik, Phys. Lett. B160, 403 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  17. V.G. Knizhnik, Phys. Lett. B178, 21 (1986).

    Article  ADS  MathSciNet  Google Scholar 

  18. G. Segal, “The Definition of Pertubative Conformal Field Theory”, notes, 1987. J. Frohlich, unpublished notes, Bures-sur-Yvette, 1987.

    Google Scholar 

  19. V.F.R. Jones, Bull. Amer. Math. Soc. 12, 103 (1985); Notes on a Talk in Atiyah’s Seminar (1986); paper in preparation.

    Google Scholar 

  20. L. Kadanoffand H. Ceva, Phys. Rev. B 11, 3918 (1971).

    Article  ADS  Google Scholar 

  21. K. Osterwalder and R. Schrader, Commun. Math. Phys. 31, 83 (1973).

    Article  ADS  Google Scholar 

  22. K. Osterwalder and R. Schrader, Commun. Math. Phys. 42, 281 (1975).

    Article  ADS  Google Scholar 

  23. V. Glaser, Commun. Math. Phys. 37, 257 (1974).

    Article  ADS  Google Scholar 

  24. J. Frohlich, K. Osterwalder and E. Seiler, Ann. Math. 118, 461 (1983).

    Article  MathSciNet  Google Scholar 

  25. M. Sato, T. Miwa and M. Jimbo, Publ. RIMS, Kyoto University, 15. 871 (1979).

    Google Scholar 

  26. F. Wegner, J. Math. Phys. 12, 2259 (1971).

    Article  ADS  Google Scholar 

  27. K.G. Wilson, Phys. Rev. D 10, 2445 (1974).

    ADS  Google Scholar 

  28. G.’t Hooft, Nucl. Phys. B 138, 1 (1978).

    Article  ADS  Google Scholar 

  29. F. Wilczek and A. Zee, Phys. Rev. Lett. 51, 2250 (1983).

    Article  ADS  MathSciNet  Google Scholar 

  30. J. Frohlich and P.-A. Marchetti, “Field Theory of Anyons and 0-Statistics”, to appear.

    Google Scholar 

  31. S. Doplicher, R. Haag and J. Roberts, Commun. Math. Phys. 23, 199 (1971); 35, 49 (1974). (See also ref. 39.)

    Google Scholar 

  32. J. Glimm and A. Jaffe, “Quantum Physics”(2nd edition), Berlin-Heidelberg-New York: Springer-Verlag, 1987.

    Book  Google Scholar 

  33. J. Imbrie, Commun. Math. Phys. 82, 261 (1981), 82, 305 (1981).

    Article  ADS  Google Scholar 

  34. R.F. Streater and A.S. Wightman, see ref. 1. R. “Jost,The General Theory of Quantized Fields”, Providence, R.I.: Publ. Amer. Math. Soc., 1965.

    Google Scholar 

  35. T. Kohno, Nagoya Math. J. 92, 21 (1983); Invent. Math. 82, 57 (1985).

    Google Scholar 

  36. T. Kohno, “Linear Representations of Braid Groups and Classical Yang-Baxter Equations”, to appear in “Contemporary Mathematics: Artin’s Braid Group”, Santa Cruz, 1987.

    MATH  Google Scholar 

  37. A.A. Belavin and V.G. Drinfel’d, Funct.Anal.Appl. 16, 1 (1982).

    Article  Google Scholar 

  38. A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241, 333 (1984).

    Article  ADS  Google Scholar 

  39. G. Segal, ref. 10.

    Google Scholar 

  40. V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B 247, 83 (1984).

    Article  ADS  Google Scholar 

  41. B. Schroer, Lecture Notes, Cargese 1987, to appear in the proceedings; and refs. given there.

    Google Scholar 

  42. E. Witten, Commun. Math. Phys. 92, 455 (1984).

    Article  ADS  Google Scholar 

  43. A.M. Polyakov and P.B. Wiegmann, Phys. Lett. B 131, 121 (1983).

    Article  ADS  Google Scholar 

  44. A. Tsuchiya and Y. Kanie, Lett. Math. Phys. 13, 303 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  45. V.F.R. Jones, Invent. Math. 72, 1983.

    Google Scholar 

  46. V.F.R. Jones, unpubl. seminar notes, and paper in preparation.

    Google Scholar 

  47. V.G. Turaev, “The Yang-Baxter Equation and Invariants of Links ”,LOMI Preprint E-3-1987.

    Google Scholar 

  48. L.H. Kauffman, “Statistical Mechanics and the Jones Polynomial” Preprint, University of Illinois, 1987.

    Google Scholar 

  49. V. Pasquier, Preprints, Saclay SPh T/87-014, SPh T/ 87-62.

    Google Scholar 

  50. D. Buchholz and K. Fredenhagen, Commun. Math. Phys. 84, 1 (1982).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretical Physics, ETH-Hönggerberg, CH-8093, Zürich, Germany

    Jürg Fröhlich

Authors
  1. Jürg Fröhlich
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Utrecht, Utrecht, The Netherlands

    G. ’t Hooft

  2. Harvard University, Cambridge, Massachusetts, USA

    A. Jaffe

  3. University of Hamburg, Hamburg, Federal Republic of Germany

    G. Mack

  4. University of Paris VI, Paris, France

    P. K. Mitter

  5. LAPP, Annecy-le-Vieux, France

    R. Stora

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Plenum Press, New York

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Fröhlich, J. (1988). Statistics of Fields, the Yang-Baxter Equation, and the Theory of Knots and Links. In: ’t Hooft, G., Jaffe, A., Mack, G., Mitter, P.K., Stora, R. (eds) Nonperturbative Quantum Field Theory. Nato Science Series B:, vol 185. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0729-7_4

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-0729-7_4

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-8053-8

  • Online ISBN: 978-1-4613-0729-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation