All massless, scalar fields with trivialS-matrix are wick-polynomials

Abstract

We extend a result about non-interacting fields given by Buchholz and Fredenhagen. Consider a massless, scalar field o in 3 + 1 dimensional space-time which does not interact. The corresponding Hilbert space is assumed to be the FockspaceH of the free massless fieldA. This implies — as we show in the first part — that alln-point-functions are rational functions of their arguments. In the second part we use this fact to construct a symmetric, traceless tensorfield φμ1...μn, relatively local to the original field o, and connecting the vacuum with the one particle states. In the last part we prove φμ1...μn to be relatively local to the free fieldA.