Averaged Null Energy Condition

The averaged null energy condition (ANEC) is an integral inequality along a complete null generator. It is not the pointwise null energy condition and not a timelike QEI with a narrow sampler. Its validity depends on the QFT, state class, renormalized stress tensor, null geometry, and convergence of the complete integral.

Required background. The CFT stress tensor supplies the null component, Hilbert-space positivity supplies the positivity input, and shape-deformation perturbation supplies the modular route to averaged null energy.

Helpful background. Quantum energy inequalities provide a contrasting timelike sampled bound.

The complete null integral

Let $x^\mu(\lambda)=x_0^\mu+\lambda k^\mu$ be a complete affinely parametrized null line, $k^2=0$. In standard normalization the averaged null energy operator is formally

$$\mathcal E_k(x_\perp)= \int_{-\infty}^{\infty}d\lambda\, T_{\mu\nu}(x(\lambda))k^\mu k^\nu .$$

ANEC asserts $\langle\mathcal E_k\rangle_\psi\ge0$ for states and theories covered by a theorem. Rescaling $\lambda\mapsto a\lambda$ also rescales $k^\mu$; holding one fixed while changing the other produces an inconsistent numerical comparison. The meaningful sign is invariant under an orientation-preserving affine rescaling.

Proofs use different hypotheses. Modular-Hamiltonian monotonicity gives ANEC in broad relativistic QFT settings through deformed half-spaces, as developed by Faulkner et al. 2016, §§ 2–4. Causality and unitarity of correlation functions give another route in interacting Lorentz-invariant theories; see Hartman, Kundu, and Tajdini 2017, §§ II–IV. The assumptions of one proof should not be silently replaced by those of the other.

An excited-state check

For a normalized wavepacket excitation $|\psi\rangle=\mathcal O(f)|0\rangle/\|\mathcal O(f)|0\rangle\|$, evaluate a regulated integral

$$\mathcal E_k(L)=\int_{-L}^{L}d\lambda\, \langle\psi|T_{kk}(x_0+\lambda k)|\psi\rangle$$

and increase $L$. A numerical ANEC check needs a stable plateau, controlled transverse smearing if used, and a renormalization prescription held fixed. Positivity at one finite $L$ is not a proof; negativity at finite $L$ is not a counterexample unless the omitted tails are bounded.

ANEC shares the renormalized-stress input with QEIs but uses a complete null integral and its own theorem hypotheses. It should remain distinct from the local QNEC entropy variation. The diagram is schematic.

Completeness and boundaries

A segment $\int_{\lambda_1}^{\lambda_2}T_{kk}\,d\lambda$ can be negative. ANEC’s compensating positive contribution may lie outside that segment. Likewise, a null geodesic that terminates on a boundary or defect is not automatically covered by a theorem for complete Minkowski generators. Boundary terms and reflected energy flux must be treated explicitly.

Truncating the null line, using a nonaffine parameter without the correct weights, or importing a boundary-free theorem into a boundary problem changes the claim. The map is schematic.

Exercises

Under $\lambda'=a\lambda$ with $a>0$, determine how $k'^\mu=dx^\mu/d\lambda'$ and $\mathcal E_{k'}$ scale.

Solution

$k'^\mu=k^\mu/a$ and $d\lambda'=a\,d\lambda$, hence $\mathcal E_{k'}=\mathcal E_k/a$. Its sign is unchanged.

References

• Faulkner, Thomas, Robert G. Leigh, Onkar Parrikar, and Huajia Wang. “Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition.” Journal of High Energy Physics 2016, no. 9 (2016): 038. DOI.
• Hartman, Thomas, Sandipan Kundu, and Amirhossein Tajdini. “Averaged Null Energy Condition from Causality.” Physical Review D 95 (2017): 066008. DOI.