Mutual information as a measure of renormalizability

TL;DR

This paper introduces a mutual information-based measure to assess the renormalizability of quantum field theories, applicable both in and out of equilibrium, by analyzing correlations in momentum space.

Contribution

It proposes a novel mutual information criterion for renormalizability that works for out-of-equilibrium quantum field theories in various spacetimes.

Findings

Logarithmic derivative of mutual information indicates renormalizability class

Mutual information relaxes to equilibrium values after quenches

Behavior consistent across Minkowski and de Sitter spacetimes

Abstract

Renormalization is an essential technique in field-theoretic descriptions of natural phenomena, where the absence of a UV-complete description yields an abundance of divergent quantities. While the renormalization prescription has been thoroughly refined for equilibrium systems, consistently extending it to out-of-equilibrium systems is an active area of research. In this paper, we identify a mutual information-based measure of renormalizability that applies to quantum field theories both in and out of equilibrium. Specifically, we use mutual information to characterize correlations between infinitesimal shells in momentum space and show that the logarithmic derivative of mutual information with mode separation, at large mode separation, is a measure of renormalizability. We first consider Minkowski spacetime, where we introduce dynamics by performing an interaction quench, initializing the field in the free theory vacuum and then turning on the interaction. We show that the late-time mutual information relaxes to that for the interacting vacuum and the logarithmic derivative at large mode separation is negative for super-renormalizable theories, zero for renormalizable (marginal) theories, and positive for non-renormalizable theories. We then consider a conformally-coupled scalar field on the Poincar\'{e} patch of de Sitter spacetime, initializing the field in the Bunch-Davies vacuum in the asymptotic past. For different self-interactions and at any finite time, we find that the resulting mutual information has the same qualitative behavior as a function of mode separation, demonstrating that it can be used as a reliable indicator of renormalizability.