Nets of Subfactors

Abstract

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.Comment: Plain TeX, 32 pages. Several unnecessarily restrictive assumptions have been relaxed. Proposition 4.10. has been reformulated in a more natural way. Sect. 3 has been rearranged and a too general statement has been adjusted. Some further minor change