Nets of Subfactors

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A subtheory of a quantum field theory specifies von Neumann subalgebras å() (the `observables' in the space-time region) of the von Neumann algebras () (the `fields' localized in). Every local algebra being a (type ₁) factor, the inclusion å() ⊂ () 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 of a given theory å in terms of the observables. Various non-trivial examples are given.