Construction of relativistic quantum fields in the framework of white noise analysis

We construct a class of Euclidean invariant distributions Phi(H) indexed by a function H holomorphic at zero. These generalized functions can be consi dered as generalized densities w.r.t. the white noise measure, and their mo ments fulfill all Osterwalder-Schrader axioms, except for reflection positi vity. The case where F(s)=-(H(is)+ 1/2 s(2)), s is an element of R, is a Le vy characteristic is considered in Rev. Math. Phys. 8, 763 (1996). Under th is assumption the moments of the Euclidean invariant distributions Phi(H) c an 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 dis tributions Phi(H)(G) also fulfill all Osterwalder-Schrader axioms except fo r reflection positivity. For no nontrivial case we succeeded in proving ref lection positivity. Nevertheless, an analytic extension to Wightman functio ns can be performed. These functions fulfill all Wightman axioms except for the positivity condition. Moreover, we can show that they fulfill the Hilb ert space structure condition and therefore the modified Wightman axioms of indefinite metric quantum field theory [Dynamics of Complex and Irregular Systems (World Scientific, Singapore, 1993)]. (C) 1999 American Institute o f Physics. [S0022-2488(99)02608-0].