Boundary as Unified Stage: From Time Translation Operator, Null--Modular Double Cover to GHY Boundary Term

Haobo Ma, Wenlin Zhang

Published Nov 23, 2025• 0 citations

Abstract

Construct unified framework with boundary as sole fundamental stage, gluing three mature but usually separate structures as three projections of same object: (i) Based on Birman--Krein--Friedel--Wigner--Smith framework, ``time translation operator'' characterized by scattering phase, spectral shift function, Wigner--Smith time delay matrix; (ii) Based on Tomita--Takesaki modular theory and recent results on causal diamond/null surface modular Hamiltonians, Markov property, QNEC, Null--Modular do

DOI: 10.5281/zenodo.17692295

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Publisher

Abstract

DOI: 10.5281/zenodo.17692295

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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.