Nets of Subfactors

A subtheory of a quantum field theory specifies von~Neumann subalgebras $\aa(\oo)$ (the `observables' in the space-time region $\oo$) of the von~Neumann algebras $\bb(\oo)$ (the `fields' localized in $\oo$). Every local algebra being a (type $\III_1$) factor, the inclusion $\aa(\oo) \subset \bb(\oo)$ is a subfactor. The assignment of these local subfactors to the space-time regions is called a `net of subfactors'. The theory of subfactors is applied to such nets. In order to characterize the `relative position' of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows to characterize, and reconstruct, local extensions $\bb$ of a given theory $\aa$ in terms of the observables. Various non-trivial examples are given.