On the net of von Neumann algebras associated with a wedge and wedge-causal manifolds

Hans-J├╝rgen Borchers
December 12, 2009
A wedge in a flat or curved ordered space can be defined with help of two light-rays passing through a point and the double-cones spanned between these light-rays. Only special manifolds have the property that the space-like complement of a wedge is again a wedge as in the flat situation. Such manifolds will be called wedge-causal. Starting from a wedge and its associated von Neumann algebra then its properties will be investigated in the flat and the wedge-causal situation. It will be shown, that in the flat situation, all local algebras are of von Neumann type III, and that they are all of the same Connes-von Neumann-type III$_1$ . Here the types can be determined, because the modular group of the wedge-algebra acts local. For the situation of the Minkowski space we will show how to construct from the wedge-algebra the algebra of the double cones. In addition we will show how to construct from a double-cone algebra the algebra of larger double cones and of the wedge. For this we will use either the translations or the modular group of the wedge-algebra and the double cone theorem. All these investigations are dimension independent. Moreover, we will develop new methods determining the von Neumann and the Connes types for the wedge- and double-cone algebras.
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modular theory, QFT on curved spacetimes, wedges, geometric modular action