Renormalization group flow as optimal transport

Abstract

This highly original paper relates the $e\phantom{\rule{0}{0ex}}x\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}c\phantom{\rule{0}{0ex}}t$ $r\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}z\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n$ $g\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}p$ (ERG) $f\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}w$ to $o\phantom{\rule{0}{0ex}}p\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l$ $g\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}d\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}t$ $f\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}w$ which naturally leads to the (previously known) interpretation of the ERG as a flow that minimizes the relative entropy of a probability distribution. This provides an elegant explanation of otherwise opaque features of ERG schemes and establishes a clear and intriguing link to information theory. The intuitive picture that emerges is that the coarse-graining of the RG flow produces entropy and this entropy production determines the flow itself. All these basic relations are established nonperturbatively.

Topics & Concepts