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Title: A general proof of the quantum null energy condition

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.

Authors:
 [1];  [1];  [1];  [2]
  1. Univ. of Illinois, Urbana, IL (United States)
  2. Univ. of Illinois, Urbana, IL (United States); Univ. of California, Santa Barbara, CA (United States)
Publication Date:
Research Org.:
Univ. of Illinois at Urbana-Champaign, IL (United States)
Sponsoring Org.:
USDOE Office of Science (SC); National Science Foundation (NSF); Defense Advanced Research Projects Agency (DARPA)
OSTI Identifier:
1612303
Grant/Contract Number:  
SC0015655; PHY-1607611; D15AP00108
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Volume: 2019; Journal Issue: 9; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Physics; AdS-CFT Correspondence; Conformal Field Theory

Citation Formats

Balakrishnan, Srivatsan, Faulkner, Thomas, Khandker, Zuhair U., and Wang, Huajia. A general proof of the quantum null energy condition. United States: N. p., 2019. Web. doi:10.1007/jhep09(2019)020.
Balakrishnan, Srivatsan, Faulkner, Thomas, Khandker, Zuhair U., & Wang, Huajia. A general proof of the quantum null energy condition. United States. doi:10.1007/jhep09(2019)020.
Balakrishnan, Srivatsan, Faulkner, Thomas, Khandker, Zuhair U., and Wang, Huajia. Tue . "A general proof of the quantum null energy condition". United States. doi:10.1007/jhep09(2019)020. https://www.osti.gov/servlets/purl/1612303.
@article{osti_1612303,
title = {A general proof of the quantum null energy condition},
author = {Balakrishnan, Srivatsan and Faulkner, Thomas and Khandker, Zuhair U. and Wang, Huajia},
abstractNote = {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.},
doi = {10.1007/jhep09(2019)020},
journal = {Journal of High Energy Physics (Online)},
issn = {1029-8479},
number = 9,
volume = 2019,
place = {United States},
year = {2019},
month = {9}
}

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