On the Locality Ideal In the Algebra of Test Functions for Quantum Fields

Abstract

Some basic properties of the locality ideal in Borchers's tensor algebra are established. It is shown that the ideal is a prime ideal and that the corresponding quotient algebra has a faithful Hilbert space representation. A topology is determined for which the positive cone in the quotient algebra is normal, and it is shown that every n-point distribution satisfying the locality condition is a linear combination of positive functional which also satisfy that condition.