Conformal covariance, modular structure, and duality for local algebras in free massless quantum field theories. (English) Zbl 0644.46057
Summary: The Tomita modular operators and the duality property for the local von Neumann algebras in quantum field models describing free massless particles with arbitrary helicity are studied. It is proved that the representation of the Poincaré group in each model extends to a unitary representation of SU(2,2), a covering group of the conformal group. An irreducible set of “standard” linear fields is shown to be covariant with respect to this representation. The von Neumann algebras associated with wedge, double-cone, and lightcone regions generated by these fields are proved to be unitarily equivalent. The modular operators for these algebras are obtained in explicit form using the conformal covariance and the results of Bisognano and Wichmann on the modular structure of the wedge algebras. The modular automorphism groups are implemented by one- parameter groups of conformal transformations. The modular conjugation operators are used to prove the duality property for the double-cone algebras and the timelike duality property for the lightcone algebras.
MSC:
| 46N99 | Miscellaneous applications of functional analysis |
| 46L60 | Applications of selfadjoint operator algebras to physics |
| 81T05 | Axiomatic quantum field theory; operator algebras |
Keywords:
Tomita modular operators; duality property for the local von Neumann algebras in quantum field models describing free massless particles with arbitrary helicity; representation of the Poincaré group; covering group of the conformal group; modular automorphism groups; one-parameter groups of conformal transformations; modular conjugation operators; double-cone algebras; timelike duality propertyReferences:
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