Proof of the Quantum Null Energy Condition

By R. Bousso, Zachary Kenneth Fisher, Jason Koeller, Stefan Leichenauer, Aron C. Wall. 2015

206 citations recorded by the source corpus.

We prove the quantum null energy condition (QNEC), a lower bound on the stress tensor in terms of the second variation in a null direction of the entropy of a region. The QNEC arose previously as a consequence of the quantum focusing conjecture, a proposal about quantum gravity. The QNEC itself does not involve gravity, so a proof within quantum field theory is possible. Our proof is somewhat nontrivial, suggesting that there may be alternative formulations of quantum field theory that make the QNEC more manifest. Our proof applies to free and super-renormalizable bosonic field theories, and to any points that lie on stationary null surfaces. An example is Minkowski space, where any point p and null vector ka define a null plane N (a Rindler horizon). Given any codimension-2 surface Σ that contains p and lies on N, one can consider the von Neumann entropy Sout of the quantum state restricted to one side of Σ. A second variation Sout′′ can be defined by deforming Σ along N, in a small neighborhood of p with area A. The QNEC states that ⟨Tkk(p)⟩≥ℏ2πlimA→0Sout′′/A.

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

View Paper: https://www.semanticscholar.org/paper/7488cd94f66ba83831c0b4d5ba5ecaf4d5d4c163

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

Modular Hamiltonians for deformed half-spaces and the averaged null energy condition (2016, unknown): “See for example [27–29].”
A general proof of the quantum null energy condition (2017, background): “It would also be interesting to study this in free theories extending the proof of the QNEC for free QFTs [24].”
Modular Hamiltonians on the null plane and the Markov property of the vacuum state (2017, methodology): “From this point of view, our results imply that the QNEC is saturated for a small deformation of the vacuum state in QFT.”
Recovering the QNEC from the ANEC (2018, methodology): “Previous proofs [1, 7] used ideas that are hard to make mathematically rigorous in general, such as path integrals and the replica trick.”
Light-ray operators in conformal field theory (2018, background): “How do they behave under modular flow? Can they appear in OPEs of entangling twist defects? The ANEC can be improved to the quantum null energy condition (QNEC) [104, 105], which was recently proven in [106] together with a higher integer spin generalization.”
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).”
Averaged null energy condition from causality (2016, background): “Recent work on the quantum null energy condition [14, 27, 28] is one example, and they are also linked by c-theorems for renormalization group flows in various dimensions.”
Energy conditions in general relativity and quantum field theory (2020, background): “Although the naive formulation of this condition suffers from the fact that the von Neumann entropy is ill-defined, it has nevertheless been argued that under suitable circumstances the QNEC implies the ANEC [139] and, conversely, that the ANEC implies this QNEC [23].”