Conformal covariance, modular structure, and duality for local algebras in free massless quantum field theories

Abstract

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 studies. 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.