A Second-Quantized Kolmogorov-Chentsov Theorem via the Operator Product Expansion

TL;DR

This paper connects probability theory and quantum field theory by establishing a theorem for constructing products of random distributions using operator product expansion, accommodating complex features like logarithmic corrections and anomalous dimensions.

Contribution

It introduces a general theorem linking the operator product expansion to the pointwise multiplication of random distributions, extending the scope of regularity structures and paracontrolled distributions.

Findings

Proves a theorem for almost sure construction of products of random distributions.

Accommodates logarithmic corrections and anomalous scaling.

Applies to fractional Gaussian fields in conformal field theory.

Abstract

We establish a direct connection between two fundamental topics: one in probability theory and one in quantum field theory. The first topic is the problem of pointwise multiplication of random Schwartz distributions which has been the object of recent progress thanks to Hairer's theory of regularity structures and the theory of paracontrolled distributions introduced by Gubinelli, Imkeller and Perkowski. The second topic is Wilson's operator product expansion which is a general property of models of quantum field theory and a cornerstone of the bootstrap approach to conformal field theory. Our main result is a general theorem for the almost sure construction of products of random distributions by mollification and suitable additive as well as multiplicative renormalizations. The hypothesis for this theorem is the operator product expansion with precise bounds for pointwise correlations. We conjecture these bounds to be universal features of quantum field theories with gapped dimension spectrum. Our theorem can accommodate logarithmic corrections, anomalous scaling dimensions and even lack of translation invariance. However, it only applies to fields with short distance singularities that are milder than white noise. As an application, we provide a detailed treatment of a scalar conformal field theory of mean field type, i.e., the fractional massless free field also known as the fractional Gaussian field.