Averaged Null Energy Condition

The averaged null energy condition (ANEC) says that a renormalized null-null stress tensor has nonnegative expectation after integration along an entire affinely parametrized null generator. It is weaker than the pointwise null energy condition, but it is not a timelike quantum energy inequality with a narrow sampler. Completeness, affine normalization, state and operator domains, transverse smearing, stress-tensor prescription, spacetime geometry, and the chosen theorem all matter. This page states those hypotheses, sketches the modular proof in Minkowski QFT, and computes a fully normalized free-field example.

Required background. Conserved currents and the CFT stress tensor provide the null component; Hilbert positivity and unitary evolution provide the positivity notion; and shape deformations and modular perturbation theory supply the half-space argument.

Helpful background. Quantum energy inequalities give the contrasting timelike, smoothly weighted bounds and explain why a weighted null average is a different problem.

For the chapter-wide dictionary, see Enter this chapter; for the hypotheses attached to each claim, see the claim-validity summary; and for controls that invalidate tempting conclusions, see failure controls.

Complete affine null averages

Let

$$x^\mu(\lambda)=x_0^\mu+\lambda k^\mu, \qquad k^\mu k_\mu=0,$$

be a complete null line in Minkowski spacetime, with $k^\mu$ future directed and $\lambda\in\mathbb R$ affine. Formally, its averaged null energy is

$$\mathcal E_k(x_\perp) =\int_{-\infty}^{\infty}d\lambda\, T_{\mu\nu}(x_0+\lambda k)k^\mu k^\nu.$$

ANEC asserts

$$\langle\psi|\mathcal E_k(x_\perp)|\psi\rangle\ge0$$

for the theory and state domain covered by a particular theorem. The formula is formal because $T_{\mu\nu}$ is an operator-valued distribution. A careful construction first uses a longitudinal cutoff $h_R(\lambda)$ tending to one and, in dimensions above two, often smears in $x_\perp$. The limit may define a positive quadratic form on a dense domain rather than a bounded operator on the whole Hilbert space.

Affine rescaling changes the numerical normalization but not the sign. If

$$\lambda'=a\lambda+b, \qquad a>0,$$

then $k'=k/a$ and

$$\mathcal E_{k'} =\int d\lambda'\,T_{\mu\nu}k'^\mu k'^\nu =\frac{1}{a}\mathcal E_k.$$

Quoting an ANEC number without fixing $k$ or $\lambda$ is therefore meaningless; quoting its sign is invariant under future-oriented affine changes.

The modular monotonicity route

For the Minkowski vacuum and an undeformed half-space $A_0$, the full modular Hamiltonian is the difference between the modular Hamiltonians of the region and its complement. If a null deformation shrinks the region to $A\subset A_0$, monotonicity of relative entropy implies the quadratic-form inequality

$$\widehat K_{A_0}-\widehat K_A\ge0.$$

At first order in a future-horizon displacement $\varepsilon\zeta^+(x_\perp)$ with $\zeta^+\ge0$, modular perturbation theory gives

$$\widehat K_{A_0}-\widehat K_A =2\pi\varepsilon \int d^{d-2}x_\perp\,\zeta^+(x_\perp) \int_{-\infty}^{\infty}dx^+\,T_{++}(x^+,0,x_\perp) +O(\varepsilon^2).$$

Take the expectation in an admitted state, divide by $2\pi\varepsilon$, and send $\varepsilon\to0^+$. Because the nonnegative transverse test function is arbitrary,

$$\int d^{d-2}x_\perp\,\zeta^+(x_\perp) \langle\mathcal E_+(x_\perp)\rangle_\psi\ge0.$$

This is ANEC in its distributionally meaningful, transversely smeared form. The first-order modular formula and the positivity argument are derived in Faulkner et al. 2016, §§2.2 and 3.

The derivation is powerful but not assumption free. It uses relativistic QFT in Minkowski space, the vacuum half-space modular structure, an appropriate stress tensor, controlled shape perturbation, and states for which the modular-energy differences and null integrals exist. Another route uses microcausality, unitarity, Lorentz invariance, and the lightcone operator product expansion in interacting theories above two dimensions; its hypotheses and positive sum rule are given by Hartman, Kundu, and Tajdini 2017, §§2–4. The two arguments support overlapping conclusions, but their assumptions are not interchangeable.

Reproducible coherent-state benchmark

An elementary two-dimensional example fixes every normalization. Use $u=t-x$, $v=t+x$, so $ds^2=du\,dv$. Along the complete null line $v=v_0$, take $u$ itself as affine parameter. Then

$$k^\mu=\frac{dx^\mu}{du}=\left(\frac12,-\frac12\right), \qquad T_{kk}=T_{uu}.$$

For the derivative algebra of a free massless scalar, choose a coherent state whose classical right-moving profile is

$$\phi_{\rm cl}(u)=A\exp\!\left(-\frac{u^2}{2L^2}\right).$$

The profile solves the massless wave equation, has finite chiral energy, and its derivative has no zero mode. Vacuum normal ordering in a coherent state replaces the field by the classical solution, so

$$\left\langle{:}T_{uu}(u){:}\right\rangle =\left(\partial_u\phi_{\rm cl}\right)^2 =\frac{A^2u^2}{L^4}e^{-u^2/L^2}.$$

The complete integral is analytic:

$$\mathcal E_u =\frac{A^2}{L^4}\int_{-\infty}^{\infty}du\,u^2e^{-u^2/L^2} =\frac{A^2\sqrt\pi}{2L}>0.$$

For a reproducible cutoff study,

$$\begin{aligned} \mathcal E_u(R) &=\int_{-R}^{R}du\, \left\langle{:}T_{uu}(u){:}\right\rangle\\ &=\frac{A^2}{L} \left[ \frac{\sqrt\pi}{2}\operatorname{erf}(R/L) -\frac{R}{L}e^{-R^2/L^2} \right]. \end{aligned}$$

Set $A=0.4$ and $L=2$ in the chosen affine-length units. The exact complete value is

$$\mathcal E_u(\infty)=0.070898154.$$

At $R/L=1,2,3,4$, the cutoff values are respectively

$$0.03031557, \quad 0.06763600, \quad 0.07086697, \quad 0.07089812.$$

The $R=4L$ result differs from the limit by $3.7\times10^{-8}$. This example checks the $u$-coordinate tangent normalization and the complete-limit procedure. It does not prove ANEC for arbitrary states: positivity here is immediate because a coherent state’s normal-ordered stress tensor is classical. Free-field proofs covering much wider state domains are given by Klinkhammer 1991, §§II–III and, in two-dimensional curved spacetime under stated asymptotic conditions, Wald and Yurtsever 1991, §§IV–V.

Completeness is not decorative

ANEC does not assert positivity on every finite null segment. Quantum states may contain a negative pulse whose compensating positive energy lies farther along the generator. As a purely analytic control—not a claimed stress tensor of a particular state—consider

$$q(\lambda)=\frac{1}{L^2} \left[ e^{-(\lambda-2L)^2/L^2} -0.8e^{-(\lambda+2L)^2/L^2} \right].$$

Its complete integral is $0.2\sqrt\pi/L>0$, while a segment around $\lambda=-2L$ is negative. This demonstrates the logical point: a nonnegative complete integral supplies no sign constraint on each partial integral. Actual QFT states can also have negative weighted null averages; in four-dimensional Minkowski space there is no state-independent QEI for arbitrary smooth null weights, even though ANEC can hold Fewster and Roman 2003, §§II–IV.

A nonaffine reparametrization introduces precisely such a weight. For $\lambda'=f(\lambda)$ with $f'(\lambda)>0$,

$$\int d\lambda'\,T_{\mu\nu} \frac{dx^\mu}{d\lambda'}\frac{dx^\nu}{d\lambda'} =\int d\lambda\,\frac{T_{kk}(\lambda)}{f'(\lambda)}.$$

Only constant $f'$ gives the affine rescaling law. A variable $f'$ changes the observable to a weighted null average, so ANEC cannot be invoked to fix its sign.

Theorem boundaries

State and operator domain. The formal light-ray integral need not converge on every vector. State whether the claim is an expectation-value limit, a positive quadratic form, or a self-adjoint operator statement, and give the dense domain or preparation class.

Geometry. The modular proof above is a flat-space half-space result. Curvature can introduce focusing, anomalies, and renormalization terms, and ANEC is not true in arbitrary curved spacetime without further restrictions. Achronality, global structure, and asymptotic behavior can be decisive.

Boundaries and defects. A null geodesic that hits a boundary or whose causal relations are changed by a boundary is not automatically covered by a boundary-free theorem. There are positive results when a complete geodesic stays in a flat tubular neighborhood with suitable causal properties; see Fewster, Olum, and Pfenning 2007, theorem in §III. Those extra geometric hypotheses must be checked rather than presumed.

Stress-tensor improvements. If an improvement changes $T_{kk}$ by a total second derivative along the null line, its complete integral is unchanged only when the boundary term vanishes. On a finite segment—or for states with insufficient decay—the same improvement can change the result.

ANEC versus QNEC. ANEC is a complete light-ray integral. QNEC is a local inequality involving $T_{kk}$ and a second null shape variation of entropy. One may help prove the other under additional assumptions, but they are not definitions of the same quantity.

Failure controls

Truncate the line. Replacing $(-\infty,\infty)$ by $[-R,R]$ yields a cutoff diagnostic. Its sign is not protected; convergence requires a plateau and a tail bound, not positivity at one $R$.

Use a nonaffine parameter naively. If the tangent is recomputed correctly, a variable weight $1/f'$ appears. Omitting it changes both dimensions and the physical average.

Import a flat theorem into a boundary geometry. First verify completeness, achronality or the theorem’s causal condition, distance from the boundary, and the reference-state subtraction. Otherwise the theorem has not been tested.

Common pitfalls

Equating ANEC with pointwise NEC. A nonnegative complete integral permits negative local regions. The control profile makes this quantifier difference explicit.

Treating a formal integral as a bounded observable. Light-ray operators are typically distributional and unbounded. Positivity must be stated on the domain supplied by the theorem.

Inferring ANEC from a finite positive numerical window. Positive omitted tails are not the only possibility. Demonstrate convergence and bound both tails before reporting the complete average.

Exercises

1. Under $\lambda'=a\lambda+b$ with $a>0$, derive the scaling of $k$ and $\mathcal E_k$.

Solution

$k'=dx/d\lambda'=k/a$ and $d\lambda'=a\,d\lambda$. Therefore

$$\mathcal E_{k'}=\int a\,d\lambda\, T_{\mu\nu}\frac{k^\mu}{a}\frac{k^\nu}{a} =\frac{1}{a}\mathcal E_k.$$

The magnitude depends on affine normalization, while the sign does not.

2. Derive the coherent benchmark’s finite-cutoff formula.

Solution

Set $z=u/L$ and use

$$\int z^2e^{-z^2}dz =\frac{\sqrt\pi}{4}\operatorname{erf}z -\frac{z}{2}e^{-z^2}.$$

Evaluating symmetrically between $-R/L$ and $R/L$ and multiplying by $A^2/L$ gives the displayed expression. The exponential term vanishes and $\operatorname{erf}(R/L)\to1$ as $R\to\infty$.

3. The diagnostic profile $q(\lambda)$ has two Gaussian pulses. Compute its complete integral and explain why it does not prove a QFT theorem.

Solution

Each translated Gaussian integrates to $\sqrt\pi L$, so

$$\int q(\lambda)d\lambda =\frac{1-0.8}{L^2}\sqrt\pi L =\frac{0.2\sqrt\pi}{L}>0.$$

The calculation only exhibits a function with a positive complete integral and a negative subregion. No state was constructed whose renormalized stress tensor equals $q$, so the profile is a logical control, not evidence for or against ANEC in a particular theory.

4. Suppose $T_{kk}$ is improved to $T_{kk}+\partial_\lambda^2F$. When is the complete ANEC integral unchanged?

Solution

The change is

$$\int_{-\infty}^{\infty}d\lambda\,\partial_\lambda^2F =\partial_\lambda F(+\infty)-\partial_\lambda F(-\infty).$$

It vanishes when the derivative has equal limiting values, usually zero under sufficient decay. On a finite segment the endpoint derivatives remain, so improvement independence is not automatic.

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.
• Fewster, Christopher J., Ken D. Olum, and Michael J. Pfenning. “Averaged Null Energy Condition in Spacetimes with Boundaries.” Physical Review D 75 (2007): 025007. DOI.
• Fewster, Christopher J., and Thomas A. Roman. “Null Energy Conditions in Quantum Field Theory.” Physical Review D 67 (2003): 044003; erratum Physical Review D 80 (2009): 069903. DOI.
• Hartman, Thomas, Sandipan Kundu, and Amirhossein Tajdini. “Averaged Null Energy Condition from Causality.” Journal of High Energy Physics 2017, no. 7 (2017): 066. DOI.
• Klinkhammer, Gunnar. “Averaged Energy Conditions for Free Scalar Fields in Flat Space-Times.” Physical Review D 43 (1991): 2542–2548. DOI.
• Wald, Robert M., and Ulvi Yurtsever. “General Proof of the Averaged Null Energy Condition for a Massless Scalar Field in Two-Dimensional Curved Spacetime.” Physical Review D 44 (1991): 403–416. DOI.