Modular Hamiltonian for de Sitter diamonds

TL;DR

This paper computes the modular Hamiltonian for conformal field theories in de Sitter spacetime regions, linking it to time translations and enabling entanglement entropy calculations.

Contribution

It provides the first explicit determination of the Tomita-Takesaki modular data for CFTs in de Sitter diamonds, extending known Minkowski results.

Findings

Modular automorphisms become time translations in the static patch.

Enables computation of relative entanglement entropies for CFTs in de Sitter regions.

Provides pedagogical derivation of Minkowski modular Hamiltonians.

Abstract

We determine the Tomita-Takesaki modular data for CFTs in double cone and light cone regions in conformally flat spacetimes. This includes in particular the modular Hamiltonian for diamonds in the de Sitter spacetime. In the limit where the diamonds become large, we show that the modular automorphisms become time translations in the static patch. As preparation, we also provide a pedagogical rederivation of the known results for Minkowski spacetime. With our results and using the Araki formula, it becomes possible to compute relative entanglement entropies for CFTs in these regions.