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

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.

40 pages, 5 figures

References in corpus (12)

• Towards a derivation of holographic entanglement entropy
• Conformal collider physics: Energy and charge correlations
• Relative entropy and the Bekenstein bound
• Higher Derivative Gravity, Causality and Positivity of Energy in a UV complete QFT
• Relative Entropies in Conformal Field Theory
• Holographic entanglement entropy and gravitational anomalies
• Bulk Emergence and the RG Flow of Entanglement Entropy
• Entanglement Entropy: A Perturbative Calculation
• Relative entropy of excited states in two dimensional conformal field theories
• Entanglement Entropy for Relevant and Geometric Perturbations
• Energy Flux Positivity and Unitarity in CFTs
• Relative Entropy, Mixed Gauge-Gravitational Anomaly and Causality

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