Two-point function and generalized free fields

Abstract

Several theorems are proven which relate to the possibility of constructing a noninteracting field with an arbitrary two-point Wightman function. They are: (a) if φ(x) is a complete local field, and [φ(x), φ(y)]= D(x-y), whereD is an arbitrary operator depending onx andy only through their difference, thenD is a c-number function; (b) such fields are generalized free fields, as defined by Greenberg; (c) any generalized free field is unitarily equivalent to a superposition of Klein Gordon fields, and moreover the asymptotic condition and unitarity restrict this to a superposition of ordinary fields with different discrete masses.