Energy Conditions, Inequalities, and Focusing

Energy conditions and quantum inequalities constrain different objects on different domains. A pointwise classical condition, a timelike worldline QEI, ANEC on a complete achronal generator, QNEC at an entangling cut, and a focusing statement for generalized entropy cannot be substituted for one another. This chapter keeps the field and state class, curve or surface, smearing, affine normalization, dimension, curvature, renormalization prescription, and theorem status attached to every conclusion.

Helpful background. Quantum energy inequalities supplies the abstract sampling framework; ANEC and QNEC supply their information-theoretic forms; and null-smeared stress observables fixes the curved-spacetime observable.

From local stress to a qualified consequence

The site uses signature $(+---)$ and

$$R^\rho{}_{\sigma\mu\nu}V^\sigma =[\nabla_\mu,\nabla_\nu]V^\rho.$$

For an affinely parametrized, hypersurface-orthogonal null congruence with tangent $k^\mu=dx^\mu/d\lambda$ in $d$ dimensions,

$$\frac{d\theta}{d\lambda} =-\frac{\theta^2}{d-2} -\sigma_{\mu\nu}\sigma^{\mu\nu} -R_{\mu\nu}k^\mu k^\nu.$$

A nonzero twist adds $+\omega_{\mu\nu}\omega^{\mu\nu}$; most focusing applications here use null hypersurfaces, for which the twist vanishes. With the Einstein equation convention of this volume,

$$R_{\mu\nu}k^\mu k^\nu =8\pi G\,T_{\mu\nu}k^\mu k^\nu$$

because $k^2=0$. The normalization remains data: under $\lambda\mapsto a\lambda+b$, $k^\mu\mapsto k^\mu/a$ and both $T_{kk}$ and the integration measure rescale. A sign statement is invariant only after all factors and endpoints are transformed consistently.

Quantum field theory permits negative renormalized energy density and null stress. It replaces no single classical condition with one universal quantum condition. Timelike QEIs bound smooth averages of specified observables in specified state classes. In four dimensions there is no general state-independent QEI obtained merely by integrating $T_{kk}$ along a null worldline; Fewster and Roman give explicit counterexamples in Minkowski space (Fewster and Roman 2003, §§ II–III). ANEC, QNEC, and generalized-entropy statements recover control only under their own, different hypotheses.

Read the structure map as a sequence of typed choices. The observable and geometric support are chosen first; state, smearing, affine, dimensional, and scheme data then determine which theorem—if any—can be applied.

Structure of an energy or focusing argument. The diagram is schematic and not to scale; arrows are licensed only after the observable, support, state class, normalization, dimension, anomaly, counterterms, and theorem status have been fixed.

The failure map is the complementary stopping rule. Missing hypotheses do not make a theorem approximately true; they change the strongest available statement, often to a model calculation or an open question.

Failure conditions for quantum energy and focusing claims. The diagram is schematic and not to scale; a counterexample to the original pointwise QFC does not by itself refute a restricted or sufficiently smeared proposal, and evidence for one variant does not prove another.

Route by observable and question

Order	Page and task
1	Classical energy conditions and quantum violations: define NEC, WEC, DEC, and SEC and test a renormalized Casimir stress.
2	Raychaudhuri evolution and null focusing: carry signs, shear, initial expansion, and affine normalization through the focusing equation.
3	Quantum energy inequalities in curved spacetime: state a timelike QEI with its field, state, sampler, and curvature domain.
4	QEI sampling, scheme, and state domains: compare samplers and worldlines without changing the width convention or renormalization problem.
5	Averaged null energy conditions: distinguish complete achronal ANEC from finite, chronal, or boundary-reflected null averages.
6	Negative energy and quantum interest: derive compensation bounds for smooth timelike pulses.
7	QNEC on curved backgrounds: compare renormalized $T_{kk}$ with an entropy shape variation in one scheme.
8	The QFC and its status: separate original, stronger, restricted, smeared, and model-specific statements.
9	Quantum expansion and covariant entropy bounds: integrate generalized-entropy change only before the first caustic and within the stated QFC variant.
10	Quantum trapped surfaces and singularity theorems: expose every local-focusing and global-causality hypothesis in an incompleteness argument.
11	Wormhole, chronology, and superluminal constraints: test exotic geometries against only the QEIs and ANEC results whose domains they satisfy.
12	Causal, thermodynamic, and stability constraints: keep passivity, retarded response, ANEC, the GSL, and pole stability as separate tests.
13	Energy and entropy bounds: hypotheses and handoffs: classify bounds without merging incompatible entropy, region, state, or gravity assumptions.

Domain and failure conditions

This is the chapter’s canonical comparison table. Every leaf links back here and supplies the narrower conditions for its application.

Statement	Averaged or varied object	Field and state domain	Geometry, dimension, and normalization	Renormalization data	Status and decisive failure
Pointwise NEC/WEC/DEC/SEC	$T_{kk}$ or observer projections at one point	Classical stress models; not all quantum states	Local tangent vectors in a declared signature	Classical tensor, or a fixed quantum stress prescription	Classical assumption. Renormalized quantum counterexamples prevent a universal pointwise quantum bound.
Timelike worldline QEI	$\int d\tau\,g(\tau)^2\langle T_{uu}\rangle$	The theorem’s field, Hadamard state class, and smooth sampler	Timelike curve; acceleration and curvature enter; dimension fixed	Absolute or reference-state bound and finite stress terms declared	Theorem only on its stated domain. A nonsmooth sampler, non-Hadamard state, or uncontrolled sampling scale invalidates it.
Null-worldline QEI	$\int d\lambda\,f(\lambda)\langle T_{kk}\rangle$	Special low-dimensional or otherwise restricted theories	Null curve and affine normalization	Observable and subtraction fixed	No general state-independent four-dimensional theorem of the timelike-QEI form. Do not infer one by taking a null limit.
ANEC	$\int_{-\infty}^{\infty}d\lambda\,\langle T_{kk}\rangle$	The cited QFT/state class and convergence assumptions	Usually complete achronal generator; affine scale and boundaries explicit	Stress prescription and possible geometric terms fixed	Theorem in selected domains. Incompleteness, chronality, boundary reflections, or divergent tails block transfer.
Quantum interest	Timelike negative pulse plus later positive compensation	The QEI’s field and state class; smooth pulses	One worldline; separation below geometric control scales	Same energy observable and sampler family	Derived/model-dependent compensation bound. Delta pulses or out-of-domain separations are not licensed.
QNEC	$2\pi\langle T_{kk}\rangle$ versus entropy second variation per transverse area	QFT and state covered by the proof	Null deformation of a cut; local stationarity/geometric hypotheses depend on dimension	Stress and entropy counterterms varied together	Theorem in established flat-space and restricted curved settings. Scheme dependence or nonstationary geometry can obstruct the local curved form.
Original pointwise QFC	Functional derivative of quantum expansion along generators	Semiclassical matter plus a generalized-entropy definition	Codimension-two cut; dimension and gravitational EFT fixed	Area/Wald-like terms and $S_{\mathrm{out}}$ renormalized together	Conjecture, with known $d\ge5$ higher-curvature counterexamples and controlled $d=2$ counterexamples.
Restricted QFC	Nonincrease tested where initial quantum expansion vanishes	Model-dependent semiclassical classes	Same null hypersurface and cut data; restrictions explicit	Same generalized entropy throughout	Conjecture generally; proved in specified brane and JT-plus-QFT model classes, not universally.
Smeared focusing proposal	Transverse or affine average of quantum expansion variation	EFT and state class stated	Smearing scale above the higher-derivative cutoff	Higher-curvature entropy terms retained	Controlled proposal/model evidence where demonstrated; not a pointwise theorem.
Quantum covariant entropy bound	Difference of generalized entropy between two cuts	Same state and entropy algebra along the lightsheet	Nonpositive relevant quantum expansion; no intervening caustic	One renormalized $S_{\mathrm{gen}}$ at both cuts	Conditional consequence of the applicable focusing statement. Changing entropy or crossing a caustic breaks the derivation.
Quantum singularity theorem	Quantum-trapped cut plus averaged focusing input	The theorem’s state and semiclassical regime	Global hyperbolicity or stated causal substitute, noncompactness/completeness hypotheses	Generalized entropy and stress fixed	Conditional incompleteness theorem. Removing a global or generalized-entropy sign hypothesis breaks the chain.

What a complete statement reports

Report the exact inequality, field and interactions, state class, observable, curve or surface, smearing function and width convention, affine normalization, dimension, boundary conditions, curvature and acceleration scales, anomaly and finite counterterms, entropy prescription, theorem or conjecture variant, evidence cutoff, and the first failed hypothesis. Separate a proved theorem, controlled model, numerical test, conjecture, counterexample, and open problem.

As of August 2026, the original QFC is not a universal theorem. The $d\ge5$ Gauss–Bonnet counterexample is explicit (Fu, Koeller, and Marolf 2017, §§ 2–3); restricted focusing has been proved in particular $d=2$ JT-plus-QFT models that also violate stronger variants (Franken et al. 2026, abstract and §§ 3–4). Those results must remain attached to their dimensions, actions, hierarchy, smearing, and cut assumptions.

References

• Fewster, C. J., and T. A. Roman. “Null Energy Conditions in Quantum Field Theory.” Physical Review D 67 (2003): 044003; erratum 80 (2009): 069903. DOI.
• Franken, V., S. Kaya, F. Rondeau, A. Shahbazi-Moghaddam, and P. Tran. “Tests of Restricted Quantum Focusing and a New CFT Bound.” Journal of High Energy Physics 2026 (2026): 111. DOI.
• Fu, Z., J. Koeller, and D. Marolf. “Violating the Quantum Focusing Conjecture and Quantum Covariant Entropy Bound in d ≥ 5 Dimensions.” Classical and Quantum Gravity 34 (2017): 175006. DOI.