Mass, Entropy and Modular Hamiltonian

Abstract

The Tomita-Takesaki theory of modular automorphisms has found many uses in quantum field theory, in particular in computing relative entropies. However, the modular Hamiltonian is only known in some cases.
I present results for the full modular data (flow, Hamiltonian and involution) for conformal scalar fields in the de Sitter (Bunch-Davies) vacuum state, restricted to diamonds. I further show that in the limit where the diamonds become large and fill the full static patch, the modular Hamiltonian generates time translations in the static patch, recovering the results of Sewell and Kay for this case.
Using these results, I show that the relative entropy for fields inside de Sitter diamonds and wedges is equal to the Noether charge corresponding to the de Sitter symmetries to which the modular flow corresponds, and that it satisfies the expected properties of positivity and convexity. Lastly, I present results on the modular Hamiltonian for massive fermions in diamonds in Minkowski space.
Based on arXiv:2308.14797, arXiv:2310.12185, arXiv:2311.13990 and arXiv:2312.04629.