  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 and only if the two regions are separated by a
finite distance. We argue that the properties of mutual information in scalar
field theories can be explained by the Markov property of these theories.
