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Modular structure and duality in conformal quantum field theory

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Making use of a recent result of Borchers, an algebraic version of the Bisognano-Wichmann theorem is given for conformal quantum field theories, i.e. the Tomita-Takesaki modular group associated with the von Neumann algebra of a wedge region and the vacuum vector coincides with the evolution given by the rescaled pure Lorentz transformations preserving the wedge. A similar geometric description is valid for the algebras associated with double cones. Moreover essential duality holds on the Minkowski spaceM, and Haag duality for double cones holds provided the net of local algebras is extended to a pre-cosheaf on the superworld≈M, i.e. the universal covering of the Dirac-Weyl compactification ofM. As a consequence a PCT symmetry exists for any algebraic conformal field theory in even spacetime dimension. Analogous results hold for a Poincaré covariant theory provided the modular groups corresponding to wedge algebras have the expected geometrical meaning and the split property is satisfied. In particular the Poincaré representation is unique in this case.

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  1. Araki, H.: A lattice of von Neumann algebras associated with the quantum field theory of a free Bose field. J. Math. Phys.4, 1343–1362 (1963)

    Article  Google Scholar 

  2. Bisognano, J., Wichmann, E.: On the duality condition for a Hermitian scalar field. J. Math. Phys.16, 985–1007 (1975)

    Article  Google Scholar 

  3. Borchers, H.J.: The CPT theorem in two-dimensional theories of local observables. Commun. Math. Phys.143, 315–332 (1992)

    Article  Google Scholar 

  4. Buchholz, D.: On the structure of local quantum fields with non-trivial interaction. In: Proc. of the Int. Conf. on Operator Algebras, Ideals and their Applications in Theoretical Physics, Baumgärtel, Lassner, Pietsch, Uhlmann, (eds), Leipzig: Teubner Verlagsgesellschaft 1978, pp. 146–153

    Google Scholar 

  5. Buchholz, D., D'Antoni, C.: Private communication

  6. Buchholz, D., Schulz-Mirbach, H.: Haag duality in conformal quantum field theory. Rev. Math. Phys.2, 105 (1990)

    Article  Google Scholar 

  7. Dirac, P.A.M.: Wave equations in conformal space. Ann. Math.37, 429 (1936)

    MathSciNet  Google Scholar 

  8. Doplicher, S., Longo, R.: Standard and split inclusions of von Neumann algebras. Invent. Math.73, 493 (1984)

    Article  Google Scholar 

  9. Haag, R.: Local Quantum Physics. Berlin, Heidelberg, New York: Springer 1992

    Google Scholar 

  10. Fredenhagen, K.: Generalization of the theory of superselection sectors. In: The Algebraic Theory of Superselection Sectors. Kastler, D. (ed.) Singapore: World Scientific 1990

    Google Scholar 

  11. Fredenhagen, K., Jörss, M.: In preparation, see: Jörss, M.: On the existence of point-like localized fields in conformally invariant quantum physics. Desy Preprint

  12. Gabbiani, F., Fröhlich, J.: Operator algebras and Conformal Field Theory. Preprint

  13. Gilman, L., Jerison, M.: Rings of continuous functions. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

  14. Guido, D., Longo, R.: Relativistic invariance and charge conjugation in quantum field theory. Commun. Math. Phys.148, 521–551 (1992)

    Article  Google Scholar 

  15. Hislop, P., Longo, R.: Modular structure of the local observables associated with the free massless scalar field theory. Commun. Math. Phys.84, 84 (1982)

    Article  Google Scholar 

  16. Longo, R.: Algebraic and modular structure of von Neumann algebra of physics. Proc. of Symposia in Pure Mathematics38, 551–566 (1982)

    Google Scholar 

  17. Mack, G., Lüscher, M.: Global conformal invariance in quantum field theory. Commun. Math. Phys.41, 203 (1975)

    Article  Google Scholar 

  18. Postnikov, M.: Leçons de geometrie. Groupes et algebres de Lie. Moscou: Éditios Mir 1985

    Google Scholar 

  19. Schroer, B.: Recent developments of algebraic methods in quantum field theory. Int. J. Modern Phys. B6, 2041–2059 (1992)

    Article  Google Scholar 

  20. Schroer, B., Swieca, J.A.: Conformal transformations for quantized fields. Phys. Rev. D10, 480 (1974)

    Article  Google Scholar 

  21. Segal, I.E.: Causally oriented manifolds and groups. Bull. Am. Math. Soc.77, 958 (1971)

    Google Scholar 

  22. Strâtilâ, S., Zsido, L.: Lectures on von Neumann algebras. England: Abacus Press 1979

    Google Scholar 

  23. Streater, R.F., Wightman, A.S.: PCT, spin and statistics, and all that. Reading, MA: Addison Wesley 1989

    Google Scholar 

  24. Todorov, I.T., Mintchev, M.C., Petkova, V.B.: Conformal invariance in quantum field theory. Publ. Scuola Normale Superiore, Pisa 1978

    Google Scholar 

  25. Weyl, H.: Space-Time-Matter. Dover Publications 1950

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Communicated by H. Araki

Dedicated to Eyvind H. Wichmann on the occasion of his 65th birthday

Supported in part by Ministero della Ricerca Scientifica and CNR-GNAFA

Supported in part by INFN, sez. Napoli

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Brunetti, R., Guido, D. & Longo, R. Modular structure and duality in conformal quantum field theory. Commun.Math. Phys. 156, 201–219 (1993).

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