Localization Regions of Local Observables

Exploiting the properties of the Jost-Lehmann-Dyson representation, it is shown that in 1+2 or more spacetime dimensions, a nonempty smallest localization region can be associated with each local observable (except for the c-numbers) in a theory of local observables in the sense of Araki, Haag, and Kastler. Necessary and sufficient conditions are given that observables with spacelike separated localization regions commute (locality of the net alone does not yet imply this).