Communications in Mathematical Physics

, Volume 125, Issue 2, pp 201–226 | Cite as

Superselection sectors with braid group statistics and exchange algebras

I. General theory
  • K. Fredenhagen
  • K. H. Rehren
  • B. Schroer


The theory of superselection sectors is generalized to situations in which normal statistics has to be replaced by braid group statistics. The essential role of the positive Markov trace of algebraic quantum field theory for this analysis is explained, and the relation to exchange algebras is established.


Neural Network Statistical Physic Field Theory Normal Statistic Complex System 
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  1. 1.
    Doplicher, S., Roberts, J.E.: Proceedings of the International Conference on Mathematical Physics, Marseille, July 16–23 (1986)Google Scholar
  2. 1a.
    Doplicher, S., Roberts, J.E.: MonoidalC*-categories and a new duality theory for compact groups URSL-DM/NS-8. University of Rome preprint and references thereinGoogle Scholar
  3. 2.
    Borchers, H.J.: Commun. Math. Phys.1, 281 (1965)Google Scholar
  4. 2a.
    Doplicher, S., Haag, R., Roberts, J.E.: Commun. Math. Phys.13, 1 (1965);15, 173 (1969)Google Scholar
  5. 2b.
    Doplicher, S., Haag, R., Roberts, J.E.: Commun. Math. Phys.23, 199 (1971);35, 49 (1974), also referred to as DHRI, IIGoogle Scholar
  6. 3.
    Haag, R., Kastler, D.: J. Math. Phys.5, 848 (1964)Google Scholar
  7. 4.
    Jones, V.: Bull. Am. Math. Soc.12, 103 (1985)Google Scholar
  8. 5.
    Rehren, K.H., Schroer, B.: Einstein causality and Artin braids. FU Berlin preprint, to be published in Nucl. Phys. B. The concept of exchange algebras was first introduced in Rehren, K.H., Schroer, B.: Phys. Lett. B198, 480 (1987)Google Scholar
  9. 6.
    Buchholz, D., Mack, G., Todorov, I.: The current algebra on the circle as a germ of local field theories. University of Hamburg preprint 1988Google Scholar
  10. 7.
    Fröhlich, J.: Statistics of fields, the Yang Baxter equation and the theory of knots and links. Proceedings of the 1987 Carèse SchoolGoogle Scholar
  11. 8.
    Bisognano, J.J., Wichmann, E.H.: J. Math. Phys.17, 303 (1976)Google Scholar
  12. 9.
    Roberts, J.E.: Spontaneously broken gauge symmetries and super-selection rules. Proceedings of the International School of Mathematical Physics, University of Camerino, 1974, Gallavotti, G. (ed.) 1976Google Scholar
  13. 10.
    Borchers, H.J.: Commun. Math. Phys.4, 315 (1967)Google Scholar
  14. 11.
    Artin, E.: Collected papers, pp. 416–498, Lang, S., Tate, J.E. (eds.). Reading, MA: Addison-Wesley 196xGoogle Scholar
  15. 12.
    Ocneanu, A.: Unpublished, see Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc.12, 103, (1988);Google Scholar
  16. 12a.
    Wenzl, H.: Invent. Math.92, 349 (1988)Google Scholar
  17. 13.
    Birman, J.S., Wenzl, H.: Braids, links polynomials and a new algebra. New York: Columbia University Press 1987Google Scholar
  18. 14.
    Murakami, J.: The representation of theq-analogue of Brauer's centralizer algebra and the Kaufman polynomial of links, Department of Mathematics, Osaka University preprint 1987Google Scholar
  19. 15.
    Doplicher, S., Roberts, J.E.: Commun. Math. Phys.15, 173 (1972)Google Scholar
  20. 16.
    Ocneanu, A.: Path algebras, A.M.S. meeting in Santa Cruz, 1986Google Scholar
  21. 17.
    Witten, E.,: Nucl. Phys. B268, 353 (1986).Google Scholar
  22. 17a.
    Gross, D.J., Jevicki, A.: Nucl. Phys. B283, 1 (1987)Google Scholar
  23. 18.
    Schroer, B., Swieca, J.A.: Phys. Rev. D10, 480 (1974).Google Scholar
  24. 18a.
    Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Nucl. Phys. B241, 333 (1984)Google Scholar
  25. 19.
    Lieb, E.H., Wu, F.Y.: In: Phase transitions and critical phenomena. Domb, C., Green, M.S. (eds.) London: Academic Press 1972, Vol. IIGoogle Scholar
  26. 20.
    Fredenhagen, K.: Commun. Math. Phys.79, 141 (1981)Google Scholar
  27. 21.
    Buchholz, D., Fredenhagen, K.: Commun. Math. Phys.84, 1 (1982)Google Scholar
  28. 22.
    Streater, R.F., Wightman, A.S.: PCT, Spin & Statistics, and all that. New York: Benjamin 1964Google Scholar
  29. 23.
    Friedan, D., Qiu, Z., Shenker, S.: Conformal invariance, unitarity, and two-dimensional critical exponents. In: Vertex operators in mathematics and physics, p. 419. Lepowsky, J., Mandelstam, S., Singer, I.M. (eds.). Berlin, Heidelberg, New York: Springer 1984Google Scholar
  30. 23a.
    For a derivation using QFT-techniques: Rehren, K.H., Schroer, B.: Nucl. Phys. B295 [FS21], 229 (1988)Google Scholar
  31. 24.
    Moore, G., Seiberg, N.: Polynomial equation for rational conformal field theories. Institute for Advanced Study, preprint 1988Google Scholar
  32. 24a.
    Brustein, R., Ne'eman, Y., Sternberg, S.: Duality, Crossing and Mac Lane's Coherence, TAUP-1669-88Google Scholar
  33. 25.
    Fröhlich, J., Marchetti, P.: Quantum Field Theory of Anyons, ETH Zürich preprint 1988Google Scholar
  34. 25a.
    Marino, E.C.: Quantum theory of nonlocal vortex fields, PUC Rio de Janeiro preprint 1987Google Scholar
  35. 26.
    Schroer, B.: Nucl. Phys. B295 [FS21], 586 (1988)Google Scholar
  36. 27.
    Yang, C.N.: Phys. Rev. Lett.19, 1312 (1967)Google Scholar
  37. 27a.
    Baxter, R.J.: Exactly solved models in statistical mechanics. London: Academic Press 1982;Google Scholar
  38. 27b.
    Faddeev, L.: Sov. Scient. Rev. C1, 107 (1980)Google Scholar
  39. 28.
    Ya Reshetikhin, N.: Quantized universal enveloping algebra. The Yang-Baxter equations and invariants of links I, II, LOMI preprints 1988Google Scholar
  40. 29.
    Longo, R.: Index of subfactors and statistics of quantum fields. PreprintGoogle Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • K. Fredenhagen
    • 1
  • K. H. Rehren
    • 1
    • 2
  • B. Schroer
    • 1
  1. 1.Institut für Theorie der ElementarteilchenFreie Universität BerlinBerlin 33
  2. 2.Inst. for Theor. Physics, RijksuniversiteitUtrechtThe Netherlands

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