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

Abstract

We construct a class of Euclidean invariant distributions ΦH 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 fulfill all Osterwalder–Schrader axioms, except for reflection positivity. The case where F(s)=−(H(is)+12s2), s∈R, is a Lévy characteristic is considered in Rev. Math. Phys. 8, 763 (1996). Under this assumption the moments of the Euclidean invariant distributions ΦH can be represented as moments of a generalized white noise measure PH. Here we enlarge this class by convolution with kernels G coming from Euclidean invariant operators 𝒢. The moments of the resulting Euclidean invariant distributions ΦHG also fulfill all Osterwalder–Schrader axioms except for reflection positivity. For no nontrivial case we succeeded in proving reflection positivity. Nevertheless, an analytic extension to Wightman functions can be performed. These functions fulfill all Wightman axioms except for the positivity condition. Moreover, we can show that they fulfill the Hilbert 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)].