Modular Hamiltonians and relative entropy

In a very general setting, entropy quantifies the amount of information about a system that an observer has access to. However, in contrast to quantum mechanics, in quantum field theory naive measures of entropy are divergent.

To obtain finite results, one needs to consider measures such as relative entropy, which can be computed from the modular Hamiltonian using Tomita--Takesaki theory. In this talk, I will explain the connection between the quantum-mechanical expressions for (relative) entropy, the modular Hamiltonian and Tomita--Takesaki theory. I then present two examples 
of modular Hamiltonians that were recently derived: one for conformal fields in diamond regions of conformally flat spacetimes (including de Sitter), and one for fermions of small mass in diamond regions of two-dimensional Minkowski spacetime. I will give results for the relative entropy between the de Sitter vacuum state and a coherent excitation thereof in diamonds and wedges and show explicitly that the result satisfies the expected properties for a relative entropy.

Based on arXiv:2308.14797, 2310.12185, 2311.13990 and 2312.04629.