Energy Conditions, Inequalities, and Focusing

Quantum fields can violate classical pointwise energy conditions while obeying more restricted averaged or entropic bounds. The resulting limits on focusing and negative energy depend on the observable, sampling, geometry, and theorem hypotheses.

Enter this chapter

Begin with Classical Energy Conditions and Quantum Violations to identify which classical energy condition a quantum expectation value violates.

Helpful background. Quantum Energy Inequalities and Averaged Null Energy Condition develop the definitions used by the chapter’s opening pages.

Topics

1. Classical Energy Conditions and Quantum Violations — What do the classical null, weak, dominant, and strong energy conditions constrain, and why do renormalized quantum stress tensors violate their pointwise forms?
2. Raychaudhuri Evolution, Null Focusing, and Renormalized Stress — How does the Raychaudhuri equation translate a renormalized null-stress statement into focusing information, and where do shear, affine choice, and semiclassical corrections enter?
3. Quantum Energy Inequalities in Curved Spacetime — For which field, Hadamard-state domain, timelike curve, sampling function, dimension, curvature scale, and stress prescription does a curved-spacetime QEI hold?
4. QEI Sampling, State Domains, and Scheme Dependence — How do sampling shape, support, observer acceleration, state domain, reference subtraction, and renormalization scheme alter the numerical content of a QEI?
5. Averaged Null Energy in Curved Spacetime — Under which field, state, complete or achronal null geodesic, affine normalization, boundary, smearing, and curvature hypotheses is a curved-spacetime ANEC statement known?
6. Negative Energy and Quantum Interest — When must a negative-energy pulse be overcompensated by positive energy, and how do separation, duration, curvature, state, and sampler determine quantum-interest restrictions?
7. QNEC on Curved Backgrounds — Under which dimensional, geometric, stationarity, field-theoretic, and renormalization hypotheses is a curved-background QNEC statement finite and scheme independent?
8. The Quantum Focusing Conjecture: Dimension, EFT, Smearing, and Status — What distinguishes the original QFC, restricted or smeared variants, model-specific tests, and current conjectural status once higher-curvature counterexamples are included?
9. Quantum Expansion and Covariant Entropy Bounds — How is quantum expansion defined from generalized-entropy shape variation, and under which lightsheet and focusing assumptions can it motivate a covariant entropy bound?
10. Quantum Trapped Surfaces and Semiclassical Singularity Theorems — Which semiclassical singularity results use quantum trapped surfaces or generalized-entropy conditions, and what causality, completeness, regularity, and focusing hypotheses do they require?
11. Wormholes, Chronology, and Superluminal-Travel Constraints — What do QEIs, ANEC results, causality, and semiclassical validity actually constrain about traversable wormholes, chronology violation, and effective superluminal travel?
12. Causal, Thermodynamic, and Stability Constraints — How can causal, thermodynamic, energy-information, and linear-stability conditions be combined without treating logically independent necessary conditions as one sufficient theorem?
13. Energy and Entropy Bounds: Hypotheses and Applications — Which energy–entropy bounds are theorems, conditional semiclassical statements, conjectures, counterexample-sensitive claims, or handoffs to holography and mathematical QFT?

How the ideas fit together

A timelike quantum energy inequality bounds a sampled energy density, whereas an averaged null condition integrates a different contraction along a null curve. Changing the curve or smearing can remove the hypotheses behind an inequality; timelike results cannot simply be converted into null ones Fewster and Roman 2003, §§ II–III.

Focusing adds a geometric evolution equation, and entropy-based conditions add a specified entangling surface and renormalization prescription. A bound at one level therefore does not automatically establish a wormhole obstruction, singularity theorem, or quantum-focusing statement at another.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

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.

Where to go next

Continue to Gravity as Effective Field Theory to identify the gravitational degrees of freedom and expansion parameters.

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.