Exact renormalization groups and transportation of measures

TL;DR

This paper presents a novel perspective on Polchinski's exact renormalization group, showing how it can generate Lipschitz transport maps between Gaussian free fields and interacting field theories, enabling the verification of various functional inequalities.

Contribution

It introduces a new approach linking renormalization groups with transportation methods to establish functional inequalities for quantum and statistical field theories.

Findings

Lipschitz transport maps can be constructed between Gaussian and interacting fields.

Functional inequalities are verified for complex field theories using this approach.

The method extends current results beyond traditional techniques.

Abstract

This note provides a new perspective on Polchinski's exact renormalization group, by explaining how it gives rise, via the multiscale Bakry-\'Emery criterion, to Lipschitz transport maps between Gaussian free fields and interacting quantum and statistical field theories. Consequently, many functional inequalities can be verified for the latter field theories, going beyond the current known results.