A general proof of the quantum null energy condition

Srivatsan Balakrishnan,Huajia Wang,Thomas Faulkner,Zuhair U Khandker

doi:10.1007/jhep09(2019)020

Abstract

We prove a conjectured lower bound on 〈T__(x)〉ψ in any state ψ of a CFT on Minkowski space, dubbed the Quantum Null Energy Condition (QNEC). The bound is given by the second order shape deformation, in the null direction, of the geometric entanglement entropy of an entangling cut passing through x. Our proof involves a combination of the two independent methods that were used recently to prove the weaker Averaged Null Energy Condition (ANEC). In particular the properties of modular Hamiltonians under shape deformations for the state ψ play an important role, as do causality considerations. We study the two point function of a “probe” operator mathcal{O} in the state ψ and use a lightcone limit to evaluate this correlator. Instead of causality in time we consider causality on modular time for the modular evolved probe operators, which we constrain using Tomita-Takesaki theory as well as certain generalizations pertaining to the theory of modular inclusions. The QNEC follows from very similar considerations to the derivation of the chaos bound and the causality sum rule. We use a kind of defect Operator Product Expansion to apply the replica trick to these modular flow computations, and the displacement operator plays an important role. We argue that the proof extends to more general relativistic QFT with an interacting UV fixed point and also prove a higher spin version of the QNEC. Our approach was inspired by the AdS/CFT proof of the QNEC which follows from properties of the Ryu-Takayanagi (RT) surface near the boundary of AdS, combined with the requirement of entanglement wedge nesting. Our methods were, as such, designed as a precise probe of the RT surface close to the boundary of a putative gravitational/stringy dual of any QFT with an interacting UV fixed point.

Highlights

Introduction and summaryBounds on the stress tensor T−− of a QFT have important consequences for the semiclassical limit of gravity — using these bounds we can rule out pathological spacetimes that might arise when coupling gravity to matter in the form of this QFT
We prove a conjectured lower bound on T−−(x) ψ in any state ψ of a CFT on Minkowski space, dubbed the Quantum Null Energy Condition (QNEC)
Our approach was inspired by the AdS/CFT proof of the QNEC which follows from properties of the Ryu-Takayanagi (RT) surface near the boundary of AdS, combined with the requirement of entanglement wedge nesting

Summary

Introduction

In order to generalize these proofs to the quantum regime several new energy conditions on T−− have been conjectured with various degrees of non-locality [5,6,7,8,9,10,11,12,13] Despite their origin in gravitational physics these generalizations often have a limit which applies directly to the QFT in curved space, and it is interesting to study their validity and consequences even in flat Minkowski space [14, 15]. Where we use u = x− = t − x and v = x+ = t + x for null coordinates adapted to an entangling surface ∂A which passes through the point u = v = yi = 0 (we have set y = 0 relative to the introduction!).

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Conclusion