Mutual information as a measure of renormalizability

TL;DR

The paper develops a universal, out-of-equilibrium diagnostic for renormalizability in quantum field theories by measuring the mutual information between infinitesimal momentum shells. The approach analyzes the large-separation scaling of the MI density, $\lim_{r\to\infty}\frac{d}{d\ln r}\ln \mathcal{I}_r(\infty)$, which assumes negative, zero, or positive values for super-renormalizable, renormalizable, and non-renormalizable theories, respectively, with the mass-dimension relation $[\lambda]=2-\frac{1}{2}(d-1)(n-2)$ governing the scaling. The method is demonstrated in Minkowski spacetime after an interaction quench for $\lambda\phi^3$, $\lambda\phi^4$, and $\lambda\phi^6$, and extended to a conformally coupled scalar in de Sitter spacetime with BD initial conditions, where the same qualitative large-$r$ behavior persists. The results provide a direct information-theoretic link to UV-IR correlations and RG structure, suggesting MI as a diagnostic for renormalizability and potentially for beta functions and fixed points in QFTs, even away from equilibrium.

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 Poincaré 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.

Figures (5)

• Figure 1: (Left) Bipartite partition of the Hilbert space, used to calculate the entanglement entropy of region $A$. (Right) Tripartite partition of the Hilbert space, used to calculate the mutual information between $A$ and $B$.
• Figure 2: (Left) Time-dependent mutual information density as a function of time for a massless $\lambda \phi^3$ theory in $d = 3$, $5$, and $7$ spatial dimensions for the $r = 1$ case, normalized by the time-independent result. We find the same qualitative behavior for any $r$. (Right) Time-independent mutual information density as a function of mode separation for the same theory, normalized to unity at $r = 1$.
• Figure 3: (Left) Time-dependent mutual information density as a function of time for a massless $\lambda \phi^4$ theory in $d = 2$ and $3$ spatial dimensions for the $r = 1$ case, normalized by the time-independent result. Solid and dashed lines indicate analytic and numerical results, respectively. We find the same qualitative behavior for any $r$. (Right) Time-independent mutual information density as a function of mode separation for the same theory, normalized to unity at $r = 1$. For the additional $\lambda \phi^4$ in $5+1$D case, we regularized the integrals using dimensional regularization with a regularization parameter $\xi = 10^{-3}$.
• Figure 4: Time-independent mutual information density as a function of mode separation for a massless $\lambda \phi^6$ theory in $2+1$D, normalized to unity at $r = 1$. All integrals were computed numerically for different choices of UV cutoffs $\Lambda$ as shown.
• Figure 5: Time-dependent mutual information density as a function of mode separation in de Sitter spacetime for a (left) massless $\lambda \phi^3$ theory for $k_{A} = H$, normalized by the $r = 1$ result at $H \eta = -1$ and (right) massless $\lambda \phi^4$ theory for arbitrary $k_{A}$, normalized by the $r = 1$ result at $\eta \to 0^-$.