Measures on Hilbert-Schmidt operators and algebraic quantum field theory

We present a general construction of non-Gaussian probability measures on the space of distributional kernels obeying a natural extension of the Osterwalder-Schrader axioms of Euclidean quantum field theory in arbitrary space-time dimension $d$. These measures may be interpreted as corresponding to scalar massive quantum fields with polynomial self-interaction. As a consequence, we obtain examples of non-free models satisfying the Haag-Kastler axioms of algebraic quantum field theory for arbitrary $d$. When $d<4$ we are able to transfer the measures to the space of distributions and verify the standard Osterwalder-Schrader axioms, hence, by a well-known reconstruction theorem, we also obtain quantum field theory models satisfying the axioms of Wightman.