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Communications in Mathematical Physics

, Volume 176, Issue 3, pp 541–554 | Cite as

Conformal Haag-Kastler nets, pointlike localized fields and the existence of operator product expansions

  • Klaus Fredenhagen
  • Martin Jörß
Article

Abstract

Starting from a chiral conformal Haag-Kastler net on 2 dimensional Minkowski space we construct associated pointlike localized fields. This amounts to a proof of the existence of operator product expansions.

We derive the result in two ways. One is based on the geometrical identification of the modular structure, the other depends on a “conformal cluster theorem” of the conformal two-point-functions in algebraic quantum field theory.

The existence of the fields then implies important structural properties of the theory, as PCT-invariance, the Bisognano-Wichmann identification of modular operators, Haag duality and additivity.

Keywords

Neural Network Statistical Physic Field Theory Complex System Quantum Field Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [BaW] Baumgaertel, H., Wollenberg, M.: Causal nets of operator algebras. Akademie-Verlag 1992Google Scholar
  2. [BGL] Brunetti, R., Guido, D., Longo, R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys.156, 201 (1993)Google Scholar
  3. [BiW] Bisognano, J.J., Wichmann, E.H.: On the duality condition for a hermitian scalar field. J. Math. Phys.16, 985–1007 (1975); On the duality condition for quantum fields. J. Math. Phys.17, 303–321 (1976)CrossRefGoogle Scholar
  4. [Bor1] Borchers, H.J.: The CPT-Theorem in Two-dimensional Theories of Local Observables. Commun. Math. Phys.134, 315 (1992)CrossRefGoogle Scholar
  5. [Bor2] Borchers, H.J.: On the converse of the Reeh-Schlieder theorem. Commun. Math. Phys.10, 269 (1968)CrossRefGoogle Scholar
  6. [BoY] Borchers, H.J., Yngvason, J.: From quantum fields to local von Neumann algebras. Rev. Math. Phys., Special Issue (1992) pp. 15–47Google Scholar
  7. [BrR] Bratelli, O., Robinson D.W.: Operator algebras and quantum statistical mechanics I. Berlin, Heidelberg, New York: Springer, 1979Google Scholar
  8. [BS-M] Buchholz, D., Schulz-Mirbach, H.: Haag duality in conformal quantum field theory. Rev. Math. Phys.2, 105 (1990)CrossRefGoogle Scholar
  9. [Buc] Buchholz, D.: On the Manifestation of Particles. DESY-preprint 93-155 (1993), and report in the proceedings of the Beer Sheva Conference, 1993Google Scholar
  10. [BuF] Buchholz, D., Fredenhagen, K.: Dilations and interactions. J. Math. Phys.18, 1107–1111 (1977)CrossRefGoogle Scholar
  11. [DSW] Driessler, W., Summers, S.J., Wichmann, E.H.: On the connection between quantum fields and von Neumann algebras of local operators. Commun. Math. Phys.105, 49 (1986)CrossRefGoogle Scholar
  12. [Fre1] Fredenhagen, K.: On the General Theory of Quantized Fields. (1991) Report in the proceedings of the Leipzig conference, 1991Google Scholar
  13. [Fre2] Fredenhagen, K.: A Remark on the Cluster Theorem. Commun. Math. Phys.97, 461 (1985)CrossRefGoogle Scholar
  14. [FröG] Fröhlich, J., Gabbiani, F.: Operator algebras and conformal field theory. Commun. Math. Phys.155, 569 (1993)Google Scholar
  15. [Haag] Haag, R.: Local quantum physics. Berlin, Heidelberg, New York: Springer, 1992Google Scholar
  16. [Jör1] Jörß, M.: Lokale Netze auf dem eindimensionalen Lichtkegel. (1991) diploma thesis, FU BerlinGoogle Scholar
  17. [Jör2] Jörß, M.: On the Existence of Pointlike Localized Fields in Conformally Invariant Quantum Physics DESY-preprint 92-156 (1992) and report in the proceedings of the Cambridge Conference 1992Google Scholar
  18. [Lang] Lang, S.: SL2 (R). Berlin, Heidelberg, New York: Springer, 1975Google Scholar
  19. [Lüs] Lüscher, M.: Operator product expansions on the vacuum in conformal quantum field theory in two space-time dimensions. Commun. Math. Phys.50,23–52 (1976)CrossRefGoogle Scholar
  20. [Mac] Mack, G.: Convergence of Operator Product Expansions on the vacuum in Conformally Invariant Quantum Field Theory. Commun. Math. Phys.53, 155 (1976)CrossRefGoogle Scholar
  21. [ReS] Reeh, H., Schlieder, S.: Nuovo Cimento22, 1051 (1961) Bemerkungen zur Unitäräquivalenz von Lorentzinvarianten FeldernGoogle Scholar
  22. [Rig] Rigotti, C.: On the essential duality condition for hermitian scalar fields. Preprint at the University of Marseille, 1977Google Scholar
  23. [SSV] Schroer, B., Swieca, J.A., Völkel, A.H.: Global operator expansions in conformally invariant relativistic quantum field theory. Phys. Rev. D11, 6, 1509 (1974)CrossRefGoogle Scholar
  24. [StW] Streater, R., Wightman, A.S.: PCT, Spin and Statistics, and All That. New York: Benjamin, 1964Google Scholar
  25. [Tak] Takesaki, M.: Tomita's theory of modular Hilbert algebras and its applications. Berlin, Heidelberg, New York: Springer, 1970Google Scholar
  26. [Tre] Treves, F.: Topological Vector Spaces, Distributions and Kernels. New York, London: Academic Press, 1967Google Scholar
  27. [Wil] Wilson, R.: Non-Lagrangian Models of Current Algebras. Phys. Rev.179, 5, 1499 (1969)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Klaus Fredenhagen
    • 1
  • Martin Jörß
    • 1
  1. 1.II. Institut für Theoretische Physik der Universität HamburgHamburgGermany

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