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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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Authors and Affiliations

  1. Dipartimento di Fisica, Università di Napoli, “Federico II”, Mostra d'Oltremare, Pad. 19, I-40185, Napoli, Italy

    R. Brunetti

  2. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133, Roma, Italy

    D. Guido & R. Longo

Authors
  1. R. Brunetti
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  2. D. Guido
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  3. R. Longo
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Additional information

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). https://doi.org/10.1007/BF02096738

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