Polarization-free generators, i.e. ``interacting'' Heisenberg operators which are localized in wedge-shaped regions of Minkowski space and generate single particle states from the vacuum, are a novel tool in the analysis and synthesis of two-dimensional integrable quantum field theories. In the present article, the status of these generators is analyzed in a general setting. It is shown that such operators exist in any theory and in any number of spacetime dimensions. But in more than two dimensions they have rather delicate domain properties in the presence of interaction. If, for example, they are defined and temperate on a translation-invariant, dense domain, then the underlying theory yields only trivial scattering. In two-dimensional theories, these domain properties are consistent with non-trivial interaction, but they exclude particle production. Thus the range of applications of polarization-free generators seems to be limited to the realm of two-dimensional theories.Comment: Dedicated to the memory of Harry Lehmann, 19 pages; revised version (proof of Lemma 3.4 corrected

In the book of Haag ͓Local Quantum Physics ͑Springer Verlag, Berlin, 1992͔͒ about local quantum field theory the main results are obtained by the older methods of C*-and W*-algebra theory. A great advance, especially in the theory of W*-algebras, is due to Tomita's discovery of the theory of modular Hilbert algebras ͓Quasi-standard von Neumann algebras, Preprint ͑1967͔͒. Because of the abstract nature of the underlying concepts, this theory became ͑except for some sporadic results͒ a technique for quantum field theory only in the beginning of the nineties. In this review the results obtained up to this point will be collected and some problems for the future will be discussed at the end. In the first section the technical tools will be presented. Then in the second section two concepts, the half-sided translations and the half-sided modular inclusions, will be explained. These concepts have revolutionized the handling of quantum field theory. Examples for which the modular groups are explicitly known are presented in the third section. One of the important results of the new theory is the proof of the PCT theorem in the theory of local observables. Questions connected with the proof are discussed in Sec. IV. Section V deals with the structure of local algebras and with questions connected with symmetry groups. In Sec. VI a theory of tensor product decompositions will be presented. In the last section problems that are closely connected with the modular theory and that should be treated in the future will be discussed.

Let Jί be a von Neumann algebra with cyclic and separating vector Ω, and let U(a) be a continuous unitary representation of R with positive generator and Ω as fixed point. If these unitaries induce for positive arguments endomorphisms of Jί then the modular group act as dilatations on the group of unitaries. Using this it will be shown that every theory of local observables in two dimensions, which is covariant under translations only, can be imbedded into a theory of local observables covariant under the whole Poincare group. This theory is also covariant under the CPT-transformation.

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