Modular structure and duality in conformal quantum field theory
Abstract
Making use of a recent result of Borchers, an algebraic version of the BisognanoWichmann theorem is given for conformal quantum field theories, i.e. the TomitaTakesaki 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 space M, and Haag duality for double cones holds provided the net of local algebras is extended to a precosheaf on the superworld ≈M, i.e. the universal covering of the DiracWeyl compactification of M. 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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 September 1993
 DOI:
 10.1007/BF02096738
 arXiv:
 arXiv:functan/9302008
 Bibcode:
 1993CMaPh.156..201B
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Operator Algebras
 EPrint:
 23 pages, plain TeX, TVM26121992