Comment on: Modular Theory and Geometry

In this note we comment on part of a recent article by B. Schroer and H.-W. Wiesbrock. Therein they calculate some new modular structure for the U(1)-current-algebra (Weyl-algebra). We point out that their findings are true in a more general setting. 1 e-mail: [email protected] I We would like to add a point to a recent inspiring work of B. Schroer and H.-W. Wiesbrock [1]. For the U(1)-current-algebra in two-dimensional spacetime they construct states invariant under higher representations of the Möbius-group, generated by the modes L−n,0,n. These new states fulfill the KMS-property with respect to modified dilations. Taking von Neumann algebras with disjoint localization regions these modified dilations are identified with the modular group associated to these algebras and states. The localization properties of the new states are different from the vacuumsetting. Regions, originally causally disjoint, become causally interdependent with respect to these states. In section 2 we sketch briefly the ansatz and result concerning the modular structure in the case of the U(1)-current-algebra. Section 3 contains our point to add, providing a more general point of view of the aforementioned results of Schroer and Wiesbrock. We try in section 4 to clarify the localization properties of the new theories. Some additional remarks and a short summary are given in Section 5. II Conformal field theory in two dimensions (CFT2) [12] provides a well suited realm for algebraic quantum field theory [2] , especially for problems concerning the geometric identification of the modular structure [3, 4]. Minkowskian CFT2 may be represented on the product of two circles, S 1 × Sspacetime (the ”compactpicture”). The global symmetry group of the CFT2 is the Möbius-group PSU(1, 1)× PSU(1, 1). We will concentrate on one of the groups, being realized on one of the circles: PSU(1, 1) := SU(1, 1)/{±1}, SU(1, 1) := {(