Holographic ANEC, QNEC, and Focusing Arguments

ANEC, QNEC, and focusing statements constrain different objects under different hypotheses. Holography can translate them into causal or extremal-surface constraints, but the translation must preserve dimension, smearing, state domain, gravitational order, and renormalization. The quantum focusing conjecture is not interchangeable with the proved domains of QNEC.

Required background. QNEC on Curved Backgrounds supplies the curved-spacetime theorem domains. Boundary Relative Entropy and Bulk Modular Data supplies the holographic information relation.

Helpful background. The Quantum Focusing Conjecture: Dimension, EFT, Smearing, and Status keeps conjecture and theorem distinct. Quantum Null Energy Condition, Quantum Energy Inequalities, and Averaged Null Energy Condition supply the QFT bounds. The Generalized Second Law: Hypotheses and Proven Scope fixes the horizon comparison.

First application. Use QNEC or a focusing inequality to constrain an extremal-surface displacement or boundary energy profile in a controlled AdS state.

Three different statements

For a complete achronal null generator with affine parameter $\lambda$, ANEC has the schematic form

$$\int_{-\infty}^{\infty}d\lambda\, \langle T_{kk}\rangle\ge0.$$

Its validity depends on the QFT, state, spacetime, smearing, and completeness assumptions. It is an averaged stress-tensor statement and contains no entropy derivative.

At a point on a suitably deformed entangling surface, QNEC relates local null energy to a diagonal second shape variation,

$$2\pi\langle T_{kk}\rangle \ge \frac{1}{\sqrt h}\,S''_{kk},$$

with the precise subtraction, normalization, and curvature terms determined by dimension and setting. It is a QFT inequality. The leading holographic proof for Einstein-gravity duals follows from extremal-surface behavior and near-boundary expansion Koeller and Leichenauer 2016, while later QFT proofs establish broader domains with their own assumptions Balakrishnan et al. 2017.

Classical null focusing instead follows from Raychaudhuri’s equation,

$$\frac{d\theta}{d\lambda} =-\frac{\theta^2}{d-1}-\sigma_{ab}\sigma^{ab} -R_{ab}k^ak^b,$$

for a hypersurface-orthogonal congruence in $(d+1)$ dimensions. Einstein’s equation plus a null-energy condition gives a sign for the last term. Quantum fields violate pointwise NEC, so replacing area expansion by generalized-entropy expansion motivates the quantum focusing conjecture. That conjecture requires its own definition and regulator; it is not proved merely because QNEC holds in one limit.

Extremal-surface displacement

Consider a boundary entangling surface deformed along a null direction. The associated bulk extremal surface obeys a Jacobi equation. Near the boundary, its displacement coefficients depend on both the boundary shape and $\langle T_{kk}\rangle$. Entanglement-wedge nesting constrains the relative displacement of nested surfaces. At the relevant order, the resulting inequality is QNEC.

The reasoning needs:

• the correct Fefferman–Graham coefficient and stress-tensor normalization;
• a surface smooth enough for the local variation;
• a fixed extremal-surface branch;
• control of logarithmic and state-independent counterterms;
• the specified large-N and higher-derivative orders.

In higher-curvature gravity, both the entropy functional and bulk equations change. A proof using only minimal area cannot be imported unchanged.

Causal consequences

ANEC can enforce boundary causality and constrain time advances in a holographic effective theory. Relative-entropy monotonicity and wedge nesting can also yield averaged energy constraints. The implication direction matters: a boundary theorem can rule out a bulk model whose dictionary violates it, but one consistent holographic model does not prove the theorem for every QFT.

Similarly, a focusing inequality can constrain where an extremal surface moves under a state perturbation. It does not by itself determine the full bulk stress tensor; different stress profiles can share the tested average or surface response.

Adversarial controls

Remove smearing. A finite smeared inequality need not admit a pointwise limit in an interacting QFT on a curved background.

Change dimension. Counterterms that are harmless in one dimension can contribute to the local entropy variation in another. The same printed formula can cease to be scheme independent.

Replace QFC by QNEC. QNEC is a boundary limit or consequence in important settings; it does not establish the full bulk generalized-entropy focusing statement.

Cross a QES transition. The entropy can be nonsmooth when the dominant surface changes. A single-branch second derivative is then not the derivative of the minimized generalized entropy.

Evidence ceiling

Within their proved domains, ANEC and QNEC are genuine QFT inequalities that holographic models must respect. In controlled Einstein-dual settings, extremal-surface nesting reproduces QNEC and constrains bulk causality. These results do not prove the quantum focusing conjecture in general, fix arbitrary higher-derivative corrections, or reconstruct a unique bulk stress tensor.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Balakrishnan, Souvik, Thomas Faulkner, Zuhair U. Khandker, and Huajia Wang. “A General Proof of the Quantum Null Energy Condition.” arXiv preprint (2017). arXiv:1706.09432.
• Bousso, Raphael, Zachary Fisher, Jason Koeller, Stefan Leichenauer, and Aron C. Wall. “Proof of the Quantum Null Energy Condition.” Physical Review D 93, 024017 (2016). DOI; arXiv:1509.02542.
• Koeller, Jason, and Stefan Leichenauer. “Holographic Proof of the Quantum Null Energy Condition.” Physical Review D 94, 024026 (2016). DOI; arXiv:1512.06109.