Renormalization group flow as optimal transport

Jordan Cotler, Semon Rezchikov

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


We establish that Polchinski's equation for exact renormalization group (RG) 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.

Original languageEnglish (US)
Article number025003
JournalPhysical Review D
Issue number2
StatePublished - Jul 15 2023
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics


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