On the modular structure of local algebras of observables (Q1068334)

According to the Reeh-Schlieder-theorem the vacuum vector \(\Omega\) is cyclic and separating for each local observable algebra \({\mathfrak A}({\mathcal O})\). It is an interesting problem to determine the modular operators \(\Delta_{{\mathcal O},\Omega}\) which are associated to the pairs (\({\mathfrak A}({\mathcal O}),\Omega)\) according to the Tomita-Takesaki-theory. This has been solved only for special cases, the most important one being the case of the wedge \({\mathcal O}=W=\{x\in {\mathbb{R}}^ 4|\) \(| x^ 0| <x^ 1\}\). Here \textit{J. J. Bisognano} and \textit{E. H. Wichmann} [J. Math. Phys. 16, 985-1007 (1975; Zbl 0316.46002)] were able to show that ln \(\Delta\) \({}_{W,\Omega}\) is the generator of Lorentz boosts in \(x^ 1\)-direction. This result is directly related via the KMS-condition to the Hawking-Unruh-effect of thermal radiation during accelerated motion. In the present paper it is shown that the modular operators \(\Delta_{{\mathcal O},\Omega}\) for double cones \({\mathcal O}\) such that \({\mathcal O}\) touches the boundary of W at the origin are approximately equal to \(\Delta_{W,\Omega}\) if applied to \({\mathfrak A}(\lambda {\mathcal O})\Omega\), \(\lambda >0\) sufficiently small. This can be used to calculate the spectrum of \(\Delta_{{\mathcal O},\Omega}\) in asymptotically dilation invariant theories, i.e. theories where the Wightman functions have a scaling limit. It is found that the spectrum is always equal to \({\mathbb{R}}_+\), thus all algebras \({\mathfrak A}({\mathcal O})\) in such a theory are of type \(III_ 1\) in the classification of \textit{A. Connes} [Ann. Sci. Ec. Norm. Sup., IV. Ser. 6, 133-252 (1973; Zbl 0274.46050)]. More recently it has been shown [\textit{D. Buchholz}, \textit{K. Fredenhagen} and \textit{C. D'Antoni}, to be published] that under a condition on the density of states proposed by \textit{D. Buchholz} and \textit{E. H. Wichmann} [DESY 86-011, to be published] the algebras are also hyperfinite, thus according to the recent uniqueness result of \textit{U. Haagerup} [Odense Preprints 1984, No.10] on hyperfinite type \(III_ 1\) factors the local algebras of observables are essentially unique (up to a possible nontrivial center).