Renormalization Group Flow as Optimal Transport

TL;DR

This paper reveals that the exact renormalization group flow can be understood as an optimal transport problem, linking information theory and RG, and introduces new numerical methods and neural network connections.

Contribution

It establishes the equivalence between Polchinski's RG equation and optimal transport gradient flow, providing a novel information-theoretic perspective on RG.

Findings

RG flow is equivalent to optimal transport gradient flow.

A regularized relative entropy acts as an RG monotone.

Reformulation of RG as a variational problem enables new numerical techniques.

Abstract

We establish that Polchinski's equation for exact renormalization group flow is equivalent to the optimal transport gradient flow of a field-theoretic relative entropy. This provides a compelling information-theoretic formulation of the exact renormalization group, expressed in the language of optimal transport. A striking consequence is that a regularization of the relative entropy is in fact an RG monotone. We compute this monotone in several examples. Our results apply more broadly to other exact renormalization group flow equations, including widely used specializations of Wegner-Morris flow. Moreover, our optimal transport framework for RG allows us to reformulate RG flow as a variational problem. This enables new numerical techniques and establishes a systematic connection between neural network methods and RG flows of conventional field theories.