Measures on Hilbert-Schmidt operators and algebraic quantum field theory

TL;DR

This paper constructs non-Gaussian probability measures on distributional kernels that extend Euclidean quantum field theory axioms, leading to new scalar quantum field models satisfying algebraic and Wightman axioms across various dimensions.

Contribution

It introduces a general method to build non-Gaussian measures fulfilling Euclidean axioms, resulting in novel quantum field models in arbitrary space-time dimensions.

Findings

Constructed non-Gaussian measures satisfying Osterwalder-Schrader axioms.

Produced scalar quantum field models obeying Haag-Kastler axioms.

Verified models satisfy Wightman axioms for dimensions less than 4.

Abstract

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.