ABSTRACT. Making use of a recent result of Borchers, an algebraic version of the Bisognano-Wichmann theorem is given for conformal quantum field theories, i.e. the Tomita-Takesaki modular group associated with the von Neumann algebra of a wedge region and the vacuum vector concides 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 pre-cosheaf on the superworld ˜ M, i.e. the universal covering of the Dirac-Weyl 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.

We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = F-Vect, where F is a field. An object X ∈ A with two-sided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗-categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3-manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1

Dedicated to Hans Borchers on the occasion of his seventieth birthday

Abstract. The Bisognano-Wichmann property on the geometric behavior of the modular group of the von Neumann algebras of local observables associated to wedge regions in Quantum Field Theory is shown to provide an intrinsic sufficient criterion for the existence of a covariant action of the (universal covering of) the Poincaré group. In particular this gives, together with our previous results, an intrinsic characterization of positive-energy conformal pre-cosheaves of von Neumann algebras. To this end we adapt to our use Moore theory of central extensions of locally compact groups by polish groups, selecting and making an analysis of a wider class of extensions with natural measurable properties and showing henceforth that the universal covering of the Poincaré group has only trivial central extensions (vanishing of the first and second order cohomology) within our class.

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In this paper we present a generalization of the Radon-Nikodym theorem proved by Pedersen and Takesaki in [5]. Given a normal, semifinite and faithful (n.s.f.) weight ϕ on a von Neumann algebra M and a strictly positive operator δ, affiliated with M and satisfying a certain relative invariance property with respect to the modular automorphism group σ ϕ of ϕ, with a strictly positive operator as the invariance factor, we construct the n.s.f. weight ϕ(δ 1 2 · δ 1 2). All the n.s.f. weights on M whose modular automorphisms commute with σ ϕ are of this form, the invariance factor being affiliated with the centre of M. All the n.s.f. weights which are relatively invariant under σ ϕ are of this form, the invariance factor being a scalar.

Starting with a conformal Quantum Field Theory on the real line, we show that the Haag dual net is still conformal with respect to a new representation of the Möbius group. We infer from this that every conformal net is normal and conormal, namely the local von Neumann algebra associated with an interval coincides with its double relative commutant inside the local von Neumann algebra associated with any larger interval. The net and the dual net give together rise to an infinite dimensional symmetry group, of which we study a class of positive energy irreducible representations. We mention how superselsection sectors extend to the dual net and we illustrate by examples how, in general, this process generates solitonic sectors. We describe the free theories associated with the lowest weight n representations of PSL(2, R), showing that they violate 3-regularity for n> 2. When n ≥ 2, we obtain examples of non Möbius-covariant sectors of a 3-regular (non 4-regular) net.

In this note we compare two applications of the Fermi Golden Rule to the study of a small quantum system $ interacting with a reservoir. The first application is the Van Hove (weak coupling) limit for dynamics reduced to the subsystem $. This application yields a dynamical semigroup generated by the Davies generator. The second application is computation of the Level Shift Operator for the Liouvillean, which is used to predict location of resonances and eigenvalues of the Liouvillean to the 2rid order of perturbation theory. In the case of thermal equilibrium we show that the Davies generator is conjugate to the Level Shift Operator for the standard Liouvillean. In general, we show that the Davies generator coincides with the Level Shift Operator for the C-Liouvillean.

We define concepts of amenability and co-amenability for algebraic quantum groups in the sense of A. Van Daele [23]. We show that coamenability of an algebraic quantum group always implies amenability of its dual. Various necessary and/or sufficient conditions for amenability or co-amenability are obtained. Co-amenability is shown to have interesting consequences for the modular theory in the case that the algebraic quantum group is of compact type.