Relating the modular Hamiltonian to two-point functions

TL;DR

The paper establishes a direct link between the modular Hamiltonian $\ln\Delta$ and the two-point function for a free scalar field in Gaussian states by working within the standard subspace formalism and analyzing the one-particle modular data. It provides a compact arcoth-based expression for the restricted modular operator, an explicit resolvent-integral representation, and a concrete two-point-function decomposition that yields the blocks $M$ and $N$ in terms of $X$, $\Pi$, and $B$, reproducing Casini–Huerta’s results in the discretized limit and generalizing to general CCR algebras. It also extends the framework to general CCR algebras via $E$ and $V$, giving a modular Hamiltonian $K$ in terms of $V$, and derives a KMS-based route that yields $K = -\ln\big(\mathbb{1}-G^{-1} i\epsilon\big)$, aligning with the resolvent approach. Together, these results provide practical, implementable tools for computing modular Hamiltonians from two-point data in curved spacetime QFT and clarify the connection to established results in the literature.

Abstract

We consider the modular Hamiltonian associated to standard subspaces for a free scalar field in a globally hyperbolic spacetime in an arbitrary Gaussian state. We show how the modular Hamiltonian is related to the two-point function of the theory. For the restriction of the modular Hamiltonian to the subspace, we recover formulas that were obtained previously by Peschel, Casini and Huerta. We also show how the same results can be obtained more directly from the KMS condition, and generalize our results to general CCR algebras.