Quantum Energy Inequalities and Semiclassical Focusing

Quantum fields invalidate pointwise classical energy conditions, but they do not make negative energy arbitrary at every scale. This chapter develops a hierarchy of replacements: timelike quantum energy inequalities constrain sampled duration and magnitude; ANEC and AANEC constrain complete null averages under geometric hypotheses; QNEC relates local null stress to an entropy shape derivative; and quantum focusing proposes a finite-gravity monotonicity of generalized entropy.

Helpful background. Quantum Fields in Curved Spacetime supplies Hadamard states and renormalized stress tensors, Einstein Equations supplies backreaction, Black-Hole Geometry and Mechanics supplies null horizons, and Congruences and Optical Scalars supplies measured energy density and focusing.

Enter the hierarchy of quantum bounds

For an observer with unit tangent $u^a$, the local density is $\rho=T_{ab}u^au^b$. A vacuum-subtracted local observable need not be positive because pair-creation terms interfere between the vacuum and two-particle sectors. The total Hamiltonian remains positive: integration over a complete spatial slice removes those local interference terms and leaves the nonnegative number-operator spectrum.

A worldline QEI replaces pointwise positivity with a sampled lower bound. For a free minimally coupled scalar in four-dimensional Minkowski space,

$$\int dt\,g(t)^2 \langle{:}T_{00}{:}\rangle \ge -\frac{1}{16\pi^2}\int dt\,\lvert g''(t)\rvert^2.$$

If $g_\tau(t)=\tau^{-1/2}g(t/\tau)$, the right side scales as $-\tau^{-4}$. The result allows arbitrarily negative pointwise values while forbidding a fixed large negative density from persisting for an arbitrarily long sampled time. Fewster states the theorem, its hypotheses, and its positive-type proof in Fewster 2012, §§ 1.3 and 2.4–2.5, PDF.

Null averages require a different theorem. A finite segment of a null worldline in four dimensions has no analogous state-independent lower bound. On a complete achronal generator, however, ANEC-type results hold for important theory and geometry classes. Completeness, achronality, convergence prescription, and self-consistency are not technical decorations; dropping them produces known counterexamples.

Entropy enters the local null bound

Let a codimension-two cut be deformed along an affinely parametrized null generator. QNEC states, in the proved regimes,

$$\langle T_{kk}\rangle_{\mathrm{ren}} \ge \frac{1}{2\pi\sqrt h} \frac{\delta^2S_{\mathrm{out}}}{\delta\lambda^2}.$$

The right side is the diagonal coefficient of a localized second shape derivative. It may be negative, so QNEC accommodates local NEC violation. Original null-quantization proofs cover free and suitable superrenormalizable bosonic fields on stationary null surfaces; later modular-flow methods prove broad interacting flat-space cases. Curved-space scheme independence has additional dimension- and congruence-dependent hypotheses. Bousso and collaborators give the original statement and proof architecture in Bousso et al. 2016, §§ 2–5, PDF.

At finite gravitational coupling, form

$$S_{\mathrm{gen}} =S_{\mathrm{grav}}+S_{\mathrm{out}}, \qquad \Theta =\frac{4G}{\sqrt h} \frac{\delta S_{\mathrm{gen}}}{\delta V}.$$

The full QFC proposes that every further forward null deformation makes $\Theta$ nonincreasing. Its off-diagonal component follows from strong subadditivity; its locally stationary weak-gravity diagonal limit is QNEC. The finite-gravity diagonal statement is not a universal theorem. Pointwise higher-curvature counterexamples in $d\ge5$ require an EFT-smearing qualification, while two-dimensional JT examples violate the unrestricted version and satisfy restricted focusing at $\Theta=0$. The quantum-focusing page compares these different regimes and their primary evidence.

Compare the statements before applying them

Bound	Averaging or deformation	Main hypotheses to inspect	Typical output
Worldline QEI	Smooth timelike sampling over duration $\tau$	Field, coupling, dimension, Hadamard state, curve, reference or absolute renormalization	Negative lower bound with a sampling-scale law
ANEC/AANEC	Complete affine null generator; AANEC adds achronality	Geometry, completeness, convergence prescription, QFT class, fixed versus self-consistent background	Nonnegative full null average
QNEC	Localized second null deformation of an entangling cut	Null normalization, expansion/shear restrictions, regulator, entropy subtraction, flat versus curved theorem	Local stress bounded by entropy curvature
QFC/rQFC	First and second variations of renormalized generalized entropy	Semiclassical expansion, gravitational entropy terms, state, smooth cuts, caustics, unrestricted versus $\Theta=0$	Conjectured finite-gravity sign monotonicity

These statements are not ordered simply from weak to strong. A QEI applies on finite timelike intervals but gives a negative lower bound. AANEC gives zero but only on a special complete null curve. QNEC is local in the stress tensor and nonlocal in entropy. QFC includes gravity but has a less settled evidential status.

Kontou and Sanders compare the averaged conditions, QEI derivations, counterexamples, and geometric applications in Kontou and Sanders 2020, §§ 4.1–4.3, PDF.

Route through the chapter

Page	Reader’s question	Result
Negative Energy in Quantum Field Theory	How can local energy be negative while the Hamiltonian is bounded below?	An explicit vacuum–two-particle interference calculation, plus Casimir and squeezed-state interpretations.
Worldline Quantum Energy Inequalities	How does shortening a sampling time change the lower bound?	A theorem with exact hypotheses, positive-type proof, $\tau^{-4}$ scaling, and counterexamples when regularity or timelikeness is dropped.
Null-Averaged Energy Conditions	When is a complete null average nonnegative?	A comparison of ANEC, AANEC, theorem classes, self-consistent variants, and chronal counterexamples.
The Quantum Null Energy Condition	Which entropy deformation bounds local null stress?	The diagonal entropy derivative, stationary proof, general flat proof, and sharply separate curved-space restrictions.
Quantum Focusing and Generalized Expansions	How does generalized entropy replace area in focusing?	Full and restricted QFC, established limits, dated evidence, and controlled toy-model violations.
Constraints on Wormholes, Warp Drives, and Time Machines	What duration, magnitude, and transverse-size tradeoffs constrain exotic support?	A comparison separating rigorous matter bounds from geometry-specific no-go estimates and controlled evasions.

Read in order. The first page removes the false premise of pointwise positivity. The next two establish what timelike and null averaging can actually recover. QNEC introduces entropy only after those distinctions are secure; QFC then adds generalized entropy and gravity. The final page applies the full hierarchy without turning conditional estimates into universal exclusions.

From a bound to a geometric conclusion

First identify the required contraction of the stress tensor from Einstein’s equation and the observer or null generator that probes it. Choose the theorem whose curve, field, state, and dimension match. Declare the sampling function, affine normalization, entropy regulator, or transverse profile before calculating.

Next verify the approximation window. A flat QEI used in curved spacetime needs a sampling scale short compared with every local curvature radius and boundary distance, yet long compared with the field or material cutoff. An AANEC theorem needs a complete achronal generator. A QNEC calculation needs a finite, scheme-independent entropy–stress combination. A QFC calculation needs all gravitational entropy counterterms at the same order as the field equation.

Finally separate three claims: the proposed metric requires a stress tensor; a QEI constrains stress tensors realizable by a stated QFT; and a self-consistent solution must realize both while remaining stable with controlled fluctuations. Passing only one step is not an existence proof.

Review the chapter

A model of radius $R$ requires $\rho=-\alpha/(GR^2)$ during a proper-time crossing $L$. A free-field QEI gives $\rho\ge-C/L^4$. The same model has a complete null curve, but two points on that curve are timelike related. Derive the duration hierarchy and decide whether the worldline QEI, ANEC, or AANEC can be invoked.

Solution

Combining the required density with the QEI gives

$$\frac{\alpha}{GR^2}\le\frac{C}{L^4}, \qquad \frac{L}{R} \le\left(\frac{C}{\alpha}\right)^{1/4} \sqrt{\frac{\ell_{\mathrm P}}{R}}.$$

Thus a macroscopic radius forces a parametrically shorter sampled duration, subject to the QEI’s field, state, curve, and local-flatness hypotheses. The worldline QEI applies if the crossing observer and sampling profile satisfy those hypotheses.

Ordinary ANEC might be considered only if a theorem covers the field, geometry, state, completeness, and convergence prescription. AANEC cannot be invoked for the stated curve because it is chronal: two of its points are timelike related. Completeness does not repair that missing hypothesis. Nor does the timelike QEI by itself supply a transverse-area bound or a self-consistent quantum state for the geometry.

References

• Bousso, R., Fisher, Z., Koeller, J., Leichenauer, S., and Wall, A. C. 2016. “Proof of the Quantum Null Energy Condition.” Physical Review D 93, 024017. DOI. Open PDF.
• Fewster, C. J. 2012. “Lectures on Quantum Energy Inequalities.” arXiv:1208.5399. Open PDF.
• Kontou, E.-A., and Sanders, K. 2020. “Energy Conditions in General Relativity and Quantum Field Theory.” Classical and Quantum Gravity 37, 193001. DOI. Open PDF.