Construction of Relativistic Quantum Fields in the Framework of White Noise Analysis

We construct a class of Euclidean invariant distributions H indexed by a function H holomorphic at zero. These generalized functions can be considered as generalized densities w.r.t. the white noise measure and their moments fullll all Osterwalder-Schrader axioms except for reeection positivity. The case where F (s) = ?(H(is)+ 1 2 s 2); s 2 R, is a L evy characteristic is considered in AGW96]. Under this assumption the moments of the Euclidean invariant distributions H can be represented as moments of a generalized white noise measure P H. Here we enlarge this class by convolution with kernels G coming from Euclidean invariant operators G. The moments of the resulting Euclidean invariant distributions G H also fullll all Osterwalder-Schrader axioms except for reeection positivity. For no nontrivial case we succeeded in proving reeection positiv-ity. Nevertheless, an analytic extension to Wightman functions can be performed. These functions fullll all Wightman axioms except for the positivity condition. Moreover, we can show that they fullll the Hilbert space structure condition and therefore the modiied Wight-man axioms of indeenite metric quantum eld theory, see Str93].