A General Proof of the Quantum Null Energy Condition (1706.09432v2)

Abstract: We prove a conjectured lower bound on $\left< T_{--}(x) \right>_\psi$ in any state $\psi$ of a relativistic QFT 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 $\psi$ play an important role, as do causality considerations. We study the two point function of a "probe" operator $\mathcal{O}$ in the state $\psi$ and use a lightcone limit to evaluate this correlator. Instead of causality in time we consider \emph{causality in 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. 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 \emph{any} QFT with an interacting UV fixed point. We also prove a higher spin version of the QNEC.

A General Proof of the Quantum Null Energy Condition

The paper presents a rigorous analytic proof of the Quantum Null Energy Condition (QNEC), a conjectured energy condition in quantum field theory (QFT) that operates as a necessary constraint in both flat and curved spacetime scenarios. This advanced mathematical framework is vital for ensuring that pathological spacetime behavior does not occur, particularly when interfacing with theories of quantum gravity.

The QNEC provides a lower bound on the expectation value of the null-null component of the stress-energy tensor, $\left< T_{--}(x) \right>_\psi$, across any state $\psi$ of a relativistic QFT. The foundational bound is given by the second-order deformation in the null direction of the geometric entanglement entropy for an entangling cut passing through the point $x$. This relationship generalizes classical energy conditions like the Null Energy Condition (NEC), which fail at the quantum level, highlighting QFT's integral connection to quantum information theory and gravitational physics.

To substantiate QNEC, the authors employ cutting-edge methods from the convergence of modular Hamiltonians under geometric deformations, and intricately bind these concepts with considerations from Tomita-Takesaki theory. The exploration into modular Hamiltonians and their flow brings a novel perspective, enhancing our understanding of how these operators mediate modular inclusions and the algebraic structure they form, filling a crucial gap in understanding QFT beyond classical limits.

The significance of their work is constructed on several fronts. First, the authors effectively adapt canonical causality arguments—previously used to validate the Averaged Null Energy Condition (ANEC)—integrating modular evolved states and operators within a comprehensive analytic framework. The interplay between entanglement entropy, represented through the Ryu-Takayanagi surface in holographic theories, and modular flow suggests a fine-grained entropic contribution to the energy conditions.

Numerically, their findings command attention in the rigorous exploration where complex operator integrals give rise to higher-order correction terms, revealing that these are bounded below by $$\frac{1}{2\pi} \frac{\delta S_{EE}(A)}{\delta x^-(y)}$$, further implying that the model maintains essential symmetry and analyticity across divergent manifolds and subsystems.

From a theoretical standpoint, their research highlights the robust implications of incorporating these conditions directly via holographic dualities. It also paves a promising way forward in which future theoretical developments, potentially including extensions to specific cases in quantum gravity and higher-spin theories, could build on their proposed geometric and modular decomposition frameworks. Notably, these extensions could address broader applications in non-trivial topologies and complex spacetimes, investigating the higher order variants of QNEC analogous to higher-spin generalizations.

Ultimately, the paper stands as an essential treatise on the interconnection between quantum information, field theories, and gravitational physics. It elaborates on critical toolsets for any researcher endeavoring to understand or further develop the quantum fields interplayed with emergent spacetime geometries. This pivotal proof of QNEC not only resolves longstanding conjectures but also enriches the universe of theoretical physics with methodologies for capturing inverted causalities and non-trivial topologies rigorously.