Relative Entropy and Mutual Information in Gaussian Statistical Field Theory

TL;DR

This paper investigates the properties of relative entropy and mutual information in Gaussian scalar field theories, revealing their dependence on spatial dimension, boundary conditions, and region separation, with implications for understanding correlations in field theories.

Contribution

It provides a detailed analysis of how relative entropy and mutual information behave in Gaussian scalar fields, including conditions for finiteness and divergence, and connects these properties to the Markov property.

Findings

Mutual information is finite between separated regions and follows an area law.

Relative entropy depends critically on the spatial dimension and boundary conditions.

Mutual information can become infinite when regions are touching.

Abstract

Relative entropy is a powerful measure of the dissimilarity between two statistical field theories in the continuum. In this work, we study the relative entropy between Gaussian scalar field theories in a finite volume with different masses and boundary conditions. We show that the relative entropy depends crucially on $d$, the dimension of Euclidean space. Furthermore, we demonstrate that the mutual information between two disjoint regions in $\mathbb{R}^d$ is finite if the two regions are separated by a finite distance and satisfies an area law. We then construct an example of "touching" regions between which the mutual information is infinite. We argue that the properties of mutual information in scalar field theories can be explained by the Markov property of these theories.