Renormalization group flow as optimal transport
Jordan Cotler (Harvard Society of Fellows, Cambridge, Massachusetts 02138 USA, Black Hole Initiative, Harvard University, Cambridge, Massachusetts 02138 USA, Department of Physics, Harvard University, Cambridge, Massachusetts 02138 USA); Semon Rezchikov (Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138 USA)
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.
