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Abstract

This thesis is concerned with the study of classical statistical and relativistic quantum field theories on spatial subregions. Restricting a state of a field theory to a spatial subregion immediately yields two questions which we address in this work: the information theory of such reduced states and their time evolution.

With regard to the information theory of reduced states, we consider states of Gaussian statistical field theories. Such theories occur in the description of classical thermodynamic systems close to a second-order phase transition or as the high-temperature limit of quantum field theories. Due to the infinite number of degrees of freedom in such theories, we employ the concept of relative entropies and demonstrate that they are well-defined for systems with a continuum of degrees of freedom. In particular, we investigate the relative entropy between field theories in a finite volume with different masses and boundary conditions. We demonstrate how the relative entropy depends crucially on the dimension of Euclidean space. Furthermore, we show 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 this result is due to the Markov property of such theories.

Regarding the time evolution of states restricted to spatial subregions, we turn our attention to states of relativistic quantum field theories. Starting from a lattice regularized scalar field theory, we demonstrate how the derivative term in the action linearly couples the interior and the exterior degrees of freedom, leading to non-unitary open system dynamics for the reduced theory. Furthermore, we incorporate initial state correlations by considering local excitations of global thermal states. We show that, due to the local structure of the theory, the non-unitary contributions to the time evolution of the reduced state are entirely contained in an effective boundary action.