Local temperatures and local terms in modular Hamiltonians

Abstract

We show there are analogs to the Unruh temperature that can be defined for any quantum field theory and region of the space. These local temperatures are defined using relative entropy with localized excitations. We show that important restrictions arise from relative entropy inequalities and causal propagation between Cauchy surfaces. These suggest a large amount of universality for local temperatures, especially the ones affecting null directions. For regions with any number of intervals in two spacetime dimensions, the local temperatures might arise from a term in the modular Hamiltonian proportional to the stress tensor. We argue this term might be universal, with a coefficient that is the same for any theory, and check analytically and numerically that this is the case for free massive scalar and Dirac fields. In dimensions d≥3, the local terms in the modular Hamiltonian producing these local temperatures cannot be formed exclusively from the stress tensor. For a free scalar field, we classify the structure of the local terms.