Modular Hamiltonians for Euclidean Path Integral States

TL;DR

This paper derives a Lorentzian formula for modular Hamiltonians in excited states of quantum field theories, providing all-order perturbative results for shape deformations and exploring universal properties of modular flow in conformal field theories.

Contribution

It introduces a simple, Lorentzian perturbative formula for modular Hamiltonians for excited states created by local sources, applicable to shape deformations and null cuts.

Findings

Derived all-order perturbative modular Hamiltonian formula for shape deformations.

Resummed the series for null shape deformations, matching known results.

Analyzed universal properties of modular flow in conformal field theories.

Abstract

We study half-space/Rindler modular Hamiltonians for excited states created by turning on sources for local operators in the Euclidean path integral in relativistic quantum field theories. We derive a simple, manifestly Lorentzian formula for the modular Hamiltonian to all orders in perturbation theory in the sources. We apply this formula to the case of shape-deformed half spaces in the vacuum state, and obtain the corresponding modular Hamiltonian to all orders in the shape deformation in terms of products of half-sided null energy operators, i.e., stress tensor components integrated along the future and past Rindler horizons. In the special case where the shape deformation is purely null, our perturbation series can be resummed, and agrees precisely with the known formula for vacuum modular Hamiltonians for null cuts of the Rindler horizon. Finally, we study some universal properties of modular flow (corresponding to Euclidean path integral states) of local operators inside correlation functions in conformal field theories. In particular, we show how the flow becomes the local boost in the limit where the operator being flowed approaches the entanglement cut.