Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition

TL;DR

This paper derives the modular Hamiltonian for deformed half-spaces in quantum field theories, uses it to prove the averaged null energy condition, and connects these results to conformal field theory bounds and holography.

Contribution

It introduces a first-order deformation of the modular Hamiltonian involving stress tensor components and applies this to prove energy conditions and bounds in CFTs.

Findings

Proved the averaged null energy condition in Minkowski space.

Generalized perturbative entanglement entropy methods to excited states.

Connected modular Hamiltonians to holographic duals in AdS/CFT.

Abstract

We study modular Hamiltonians corresponding to the vacuum state for deformed half-spaces in relativistic quantum field theories on $\mathbb{R}^{1,d-1}$. We show that in addition to the usual boost generator, there is a contribution to the modular Hamiltonian at first order in the shape deformation, proportional to the integral of the null components of the stress tensor along the Rindler horizon. We use this fact along with monotonicity of relative entropy to prove the averaged null energy condition in Minkowski space-time. This subsequently gives a new proof of the Hofman-Maldacena bounds on the parameters appearing in CFT three-point functions. Our main technical advance involves adapting newly developed perturbative methods for calculating entanglement entropy to the problem at hand. These methods were recently used to prove certain results on the shape dependence of entanglement in CFTs and here we generalize these results to excited states and real time dynamics. We also discuss the AdS/CFT counterpart of this result, making connection with the recently proposed gravitational dual for modular Hamiltonians in holographic theories.