Nets of Subfactors

Roberto Longo, Karl-Henning Rehren
November 10, 1994
A subtheory of a quantum field theory specifies von Neumann subalgebras ${\cal A(O)}$ (the `observables' in the space-time region ${\cal O}$) of the von Neumann algebras ${\cal B(O)}$ (the `fields' localized in ${\cal O}$). Every local algebra being a (type III$_1$) factor, the inclusion ${\cal A(O)} \subset {\cal B(O)}$ 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 ${\cal B}$ of a given theory ${\cal A}$ in terms of the observables. Various non-trivial examples are given.