Local nets and self-adjoint extensions of quantum field operators (Q1174254)

This is an addendum to authors' paper in Math. Nachr. 145, 111-117 (1990). It is remarked that two closed Hermitian operators which commute strongly can be extended to strongly commuting selfadjoint operators in a larger Hilbert space. This is used to show that, if a Wightman field \(\Phi\) is associated with a net \(\{{\mathcal A}(R): R \text{ domain in }R^ d\}\) of von Neumann algebras (i.e. \(\Phi(f)\), \(supp f\subset R\), have closed extensions affiliated with \({\mathcal A}(R)\)) satisfying local commutativity then the Wigthman functional fulfills the positivity condition in the paper quoted above and, therefore, by the argument there, the field operators have selfadjoint extensions in a larger Hilbert space, such that local commutativity and cyclicity are preserved.