Communications in Mathematical Physics

, Volume 145, Issue 1, pp 123–148 | Cite as

Field operators for anyons and plektons

  • Karl-Henning Rehren


Given its superselection sectors with non-abelian braid group statistics, we extend the algebraA of local observables into an algebra ℱ containing localized intertwiner fields which carry the superselection charges. The construction of the inner degrees of freedom, as well as the study of their transformation properties (quantum symmetry), are entirely in terms of the superselection structure of the observables. As a novel and characteristic feature for braid group statistics, Clebsch-Gordan and commutation “coefficients” generically take values in the algebra ℳ of symmetry operators, much as it is the case with quasi-Hopf symmetry.A, ℱ, and ℳ are allC* algebras, i.e. represented by bounded operators on a Hilbert space with positive metric.


Neural Network Statistical Physic Hilbert Space Complex System Characteristic Feature 
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  1. 1.
    Doplicher, S., Roberts, J. E.: Compact Lie groups associated with endomorphisms ofC *-algebras. Bull. Am. Math. Soc. (New Series)11, 333 (1984); Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys.131, 51 (1990)Google Scholar
  2. 2.
    Doplicher, S., Haag, R., Roberts, J. E.: Local observables and particle statistics. I+II. Commun. Math. Phys.23, 199 (1971) and35, 49 (1974)Google Scholar
  3. 3.
    Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras I: General theory. Commun. Math. Phys.125, 201 (1989)Google Scholar
  4. 4.
    Drinfel'd, V. G.: Quantum groups. In: Proceedings of the International Congress of Mathematicians (Berkeley 1986) Gleason, A. G. (ed.) New York: Academic Press 1987, p. 799; Quasi Hopf algebras and Knizhnik-Zamolodchikov equations. In: Problems of Modern Quantum Field Theory, Proceedings Alushta (USSR) 1989. Belavin, A. A. et al. (eds.) Berlin, Heidelberg, New York: Springer 1989Google Scholar
  5. 5.
    Mack, G., Schomerus, V.: Quasi Hopf quantum symmetry in quantum theory. Preprint DESY 91-037 (Univ. Hamburg 1991), to appear in Nucl. Phys. BGoogle Scholar
  6. 6.
    Fredenhagen, K.: Sum rules for spin in (2+1)-dimensional quantum field theory. In: Quantum Groups. Doebner, H. et al. (eds.) Lecture Notes in Physics vol.370. Berlin, Heidelberg, New York: Springer 1990, p. 340Google Scholar
  7. 7.
    Tsuchiya, A., Kanie, Y.: Vertex operators in conformal field theory on ℙ1 and monodromy representations of braid group. In: Conformal field theory and solvable lattice models. Adv. Stud. Pure Math.16, 297 (1988); Rehren, K.-H., Schroer, B.: Einstein causality and Artin braids. Nucl. Phys.B312, 715 (1989)Google Scholar
  8. 8.
    Fröhlich, J.: Statistics of fields, the Yang-Baxter equation, and the theory of knots and links. In: Nonperturbative quantum field theory, Proceedings Cargèse 1987. G. 't Hooft et al. (eds.) London: Plenum Press 1988, p. 71Google Scholar
  9. 9.
    Roberts, J. E.: Lectures on algebraic quantum field theory. In: Algebraic theory of superselection sectors. Kastler, D. (ed.) Singapore: World Scientific 1990, pp. 1–112Google Scholar
  10. 10.
    Cuntz, J.: SimpleC * algebras generated by isometries. Commun. Math. Phys.57, 173 (1977)Google Scholar
  11. 11.
    Goodman, F. M., delaHarpe, P., Jones, V.: Coxeter Graphs and Towers of Algebras. MSRI Publ. (eds.)14. Berlin, Heidelberg, New York: Springer 1989Google Scholar
  12. 12.
    Pimsner, M. V., Popa, S.: Entropy and index for subfactors. Ann. Sci. Éc. Norm. Sup. 4e sér.19, 57 (1986)Google Scholar
  13. 13.
    Ocneanu, A.: Quantized groups, string algebras, and Galois theory for algebras. In: Operator Algebras and Applications, Vol. 2, Evans, D. E. et al. (eds.) London Math. Soc. Lecture Notes Series vol.135. Cambridge: Cambridge University Press 1988, p. 119Google Scholar
  14. 14.
    Jones, V. F. R.: Index for subfactors. Invent. Math.72, 1 (1983)Google Scholar
  15. 15.
    Longo, R.: Index of subfactors and statistics of quantum fields. I+II. Commun. Math. Phys.126, 217 (1989) and130, 285 (1990)Google Scholar
  16. 16.
    Doplicher, S.: Local observables and the structure of quantum field theory. In: Algebraic Theory of Superselection Sectors. Kastler, D. (ed.) Singapore: World Scientific 1990, p. 230; Doplicher, S., Roberts, J. E.: A new duality theory for compact groups. Invent. Math.98, 157 (1989)Google Scholar
  17. 17.
    Reshetikhin, N. Yu., Turaev, V. G.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys.127, 1 (1990)Google Scholar
  18. 18.
    Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys.123, 177 (1989)Google Scholar
  19. 19.
    Rehren, K.-H.: Space-time fields and exchange fields. Commun. Math. Phys.132, 461 (1990)Google Scholar
  20. 20.
    Buchholz, D., Mack, G., Todorov, I.: The current algebra on the circle as a germ for local field theories. Proceedings Annecy 1988. Nucl. Phys. B (Proc. Suppl.)5B, 20 (1988)Google Scholar
  21. 21.
    Rehren, K.-H.: Quantum symmetry associated with braid group statistics. In: Quantum Groups, Proceedings Clausthal-Zellerfeld 1989. Doebner, H. et al. (eds.) Lecture Notes in Physics vol.370. Berlin, Heidelberg, New York: Springer 1990, p. 318Google Scholar
  22. 22.
    Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351 (1989); Segal, G.: Two-dimensional conformal field theories and modular functors. In: Proceedings of the IXth Intern. Congr. on Mathematical Physics (Swansea 1988), eds. B. Simon et al. (eds.) London: Adam Hilger 1989, p. 22Google Scholar
  23. 23.
    Rehren, K.-H.: Quantum symmetry associated with braid group statistics. II. To appear in: Quantum Groups II, proceedings of the II. Wigner Symposium, Goslar 1991, Doebner, H. et al. (eds.)Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Karl-Henning Rehren
    • 1
  1. 1.II. Institut für Theoretische PhysikUniversität HamburgHamburg 50Federal Republic of Germany

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