Local Modular Hamiltonians from the Quantum Null Energy Condition

Abstract

The vacuum modular Hamiltonian $K$ of the Rindler wedge in any relativistic quantum field theory is given by the boost generator. Here we investigate the modular Hamiltoninan for more general half-spaces which are bounded by an arbitrary smooth cut of a null plane. We derive a formula for the second derivative of the modular Hamiltonian with respect to the coordinates of the cut which schematically reads $K"=T_{vv}$. This formula can be integrated twice to obtain a simple expression for the modular Hamiltonian. The result naturally generalizes the standard expression for the Rindler modular Hamiltonian to this larger class of regions. Our primary assumptions are the quantum null energy condition --- an inequality between the second derivative of the von Neumann entropy of a region and the stress tensor --- and its saturation in the vacuum for these regions. We discuss the validity of these assumptions in free theories and holographic theories to all orders in $1/N$.

arXiv comment6 pages, 1 figure

References in corpus (22)

• Holographic Derivation of Entanglement Entropy from AdS/CFT
• Aspects of Holographic Entanglement Entropy
• A Covariant Holographic Entanglement Entropy Proposal
• Towards a derivation of holographic entanglement entropy
• Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime
• Quantum corrections to holographic entanglement entropy
• Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality
• Relative entropy equals bulk relative entropy
• Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy
• The Gravity Dual of a Density Matrix
• Causality & holographic entanglement entropy
• The Ryu-Takayanagi Formula from Quantum Error Correction
• A Quantum Focussing Conjecture
• Entanglement hamiltonians in two-dimensional conformal field theory
• A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices
• Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition
• Proof of the Quantum Null Energy Condition
• Causal Holographic Information
• Light Front Quantization
• Proof of a Quantum Bousso Bound
• Entropy on a null surface for interacting quantum field theories and the Bousso bound
• Holographic Proof of the Quantum Null Energy Condition

Cited by in corpus (24)

• Modular Hamiltonians on the null plane and the Markov property of the vacuum state
• A General Proof of the Quantum Null Energy Condition
• The entanglement spectrum of chiral fermions on the torus
• Energy is Entanglement
• Ignorance is Cheap: From Black Hole Entropy To Energy-Minimizing States In QFT
• Anisotropic Unruh temperatures
• Modular Hamiltonians in flat holography and (W)AdS/WCFT
• Comments on black hole interiors and modular inclusions
• Surface Theory: the Classical, the Quantum, and the Holographic
• Entanglement Dynamics in 2D CFT with Boundary: Entropic origin of JT gravity and Schwarzian QM
• Proof of the quantum null energy condition for free fermionic field theories
• Modular operator for null plane algebras in free fields
• Resolving modular flow: a toolkit for free fermions
• Saturation of the Quantum Null Energy Condition in Far-From-Equilibrium Systems
• Global aspects of conformal symmetry and the ANEC in dS and AdS
• Positive gravitational subsystem energies from CFT cone relative entropies
• The holographic dual of Rényi relative entropy
• Properties of Modular Hamiltonians on Entanglement Plateaux
• Explicit reconstruction of the entanglement wedge via the Petz map
• Entanglement Hamiltonian of Interacting Systems: Local Temperature Approximation and Beyond
• How Much NEC Breaking Can the Universe Endure?
• Modular Hamiltonians for Euclidean Path Integral States
• Virasoro algebras, kinematic space and the spectrum of modular Hamiltonians in CFT${}_2$
• Local Geometric Bounds on Generalized Entropy Evolution along Null Horizons