Modular Hamiltonians for deformed half-spaces and the averaged null energy condition

By T. Faulkner, R. Leigh, Onkar Parrikar, Huajia Wang. 2016

287 citations recorded by the source corpus.

A bstractWe study modular Hamiltonians corresponding to the vacuum state for deformed half-spaces in relativistic quantum field theories on ℝ1,d−1$$ {\mathrm{\mathbb{R}}}^{1,d-1} $$. 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.

Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8317f274dd3dd86955e89a62b1ebc8a4d07362c2

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Citations in the corpus

Entropy, extremality, euclidean variations, and the equations of motion (2017, background): “The variation can be carried out [44]: 15”
Nonlinear gravity from entanglement in conformal field theories (2017, methodology): “Our calculations make use of techniques developed in [13] (see also [14, 15]) for perturbative CFT calculations of entanglement entropy and methods developed in [16, 17] for the relevant gravity calculations.”
A general proof of the quantum null energy condition (2017, methodology): “ning the important connection between gravity and quantum information. Recently two dierent proofs of the Averaged Null Energy Condition (ANEC) in any UV complete QFT have appeared in the literature [18,19]. In Minkowski space this is the { 1 { positivity constraint: Z 1 1 dx T (x ;x+ = 0;y) 0 (1.1)”
The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole (2019, unknown): “Moreover, in JT gravity traversable wormholes would require violations of the ANEC [37, 38] which, in our context, are forbidden by extending the results of [39] to Killing horizons in curved space as described in that reference.”
Averaged null energy condition from causality (2016, background): “Therefore these constraints follow from monotonicity of relative entropy [9].”
Bulk locality from modular flow (2017, background): “Even if the position of the surface changes by a small amount, the modular hamiltonian will change everywhere [45], but the previous arguments seems to suggest that one can still work in perturbation theory.”
The conformal bootstrap: Theory, numerical techniques, and applications (2018, methodology): “Two general proofs of the ANEC were given recently, one via quantum information (Faulkner et al., 2016), and one by causality (Hartman et al., 2017).44 Specializing to CFTs, the causality argument makes it clear that the ANEC is not an extra assumption but follows from other CFT axioms such as…”
Light-ray operators in conformal field theory (2018, methodology): “The ANEC was proven in [45] using information theory and in [46] using causality.”
Modular Hamiltonians on the null plane and the Markov property of the vacuum state (2017, background): “From the ANEC we get the positivity of all operators Pf = ∫ dd−2x⊥ f(x⊥) ∫ dλTλλ(λ, x ⊥) , (5.18) where f(x⊥) > 0.”
Entanglement entropy: Holography and renormalization group (2018, methodology): “Moreover, the monotonic properties of entanglement measures have been successfully applied to derive non-trivial constraints on energy and entropy in recent studies (Balakrishnan et al., 2017; Bousso et al., 2014, 2015, 2016; Faulkner et al., 2016).”
Energy conditions in general relativity and quantum field theory (2020, methodology): “Using somewhat similar arguments, Faulkner et al have argued that the ANEC can be derived directly using modular Hamiltonians [140].”
Traversable wormholes via a double trace deformation (2016, result): “(There is also evidence that these results extend to more general field theories [18, 25, 24, 30, 13]).”