NETS OF SUBFACTORS

A subtheory of a quantum field theory specifies von Neumann subalgebras A(phi) (the 'observables' in the space-time region phi) of the von Neumann algebras B(phi) (the 'fields' localized in phi). Every local algebra being a (type III1) factor, the inclusion A(phi) subset of B(phi) 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 single 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 one to characterize, and reconstruct, local extensions B of a given theory A in terms of the observables. Various non-trivial examples are given. Several results go beyond the quantum field theoretical application.