On the structure of the von Neumann algebras generated by local functions of the free bose field

Abstract

It is shown that the von Neumann algebra$R_\mathfrak{B}$(B) generated by any scalar local functionB(x) of the free fieldA ₀(x) is equal either to$R_\mathfrak{B}$(A ₀) or to$R_\mathfrak{B}$(:A ²₀ :). The latter statement holds if the state space space$\mathfrak{H}_B$ obtained from the vacuum state by repeated application ofB(x) is orthogonal to the one particle subspace. In the proof of these statements, space-time limiting techniques are used.