Worldline, Spacetime-Volume, and Null-Energy Bounds

“Averaged energy” names several inequivalent observables. A timelike-worldline QEI, a spacetime-volume QEI, a null-contracted tensor sampled on a timelike curve, and a tensor sampled on a null geodesic have different pullback conditions and different lower-bound theorems. Dimension matters just as much: a sharp null bound exists for broad two-dimensional conformal theories, while a compactly weighted null-geodesic bound fails even for the free massless scalar in four-dimensional Minkowski space.

Required background. Propagation of singularities for hyperbolic fields supplies the characteristic covectors that govern restrictions to curves.

Helpful background. Absolute and difference QEIs supplies the timelike proof. Higher-point microlocal spectrum conditions controls composite observables. Relativistic KMS analyticity and spectrum gives a thermal comparison. The quantum null energy condition concerns an entropy variation, not the QEI below. Null-smeared stress observables, QNEC on curved backgrounds, and energy and entropy bounds keep these domains separate.

Four averaging domains

For a timelike curve $\gamma$ with unit tangent $u^a$, the energy density average is

$$\int g(\tau)^2\langle T_{ab}u^au^b\rangle(\gamma(\tau))\,\mathrm d\tau.$$

The Hadamard wavefront set can be pulled back to a timelike curve, and positivity yields a lower bound for free fields. One may instead contract with a smooth null vector field $\ell^a$ along the same timelike curve. The observable $T_{ab}\ell^a\ell^b$ can also obey a quantum null energy inequality (QNEI), but the support remains timelike. It is not a null-geodesic average.

A spacetime-volume average uses $F\in C_0^\infty(M)$ and integrates $F^2T_{ab}v^av^b$ over a region. There is no distributional pullback to a lower-dimensional submanifold, but the theorem still depends on the component, positivity decomposition, and field. Smearing only over a spacelike slice is a different operation and, in four dimensions, does not generally have a state-independent lower bound. Therefore “more smearing” is not ordered solely by the number of integration variables; the causal character is essential.

Finally, for an affinely parametrized null geodesic $\eta(\lambda)$ with tangent $k^a$, the direct null average is

$$\int f(\lambda)\langle T_{ab}k^ak^b\rangle(\eta(\lambda))\,\mathrm d\lambda.$$

Because null covectors can be normal to a null curve, the simple Hadamard pullback argument used on timelike curves fails. A theorem may still exist in a special dimension or theory, but it requires separate structure.

The two-dimensional null bound

In a unitary, positive-energy two-dimensional conformal field theory with central charge $c$, take one chiral stress component $T(v)$ on a complete null line. With the standard CFT normalization and real $g\in C_0^\infty(\mathbb R)$, the sharp bound has the form

$$\int_{\mathbb R}g(v)^2\langle T(v)\rangle_\omega\,\mathrm dv \geq-\frac{c}{12\pi}\int_{\mathbb R}|g'(v)|^2\,\mathrm dv.$$

For a positive weight $f=g^2$, the right side is $-c(48\pi)^{-1}\int f'^2/f$. The proof uses positive energy and the anomalous transformation law of the chiral stress tensor under an orientation-preserving diffeomorphism of the line. A diffeomorphism is chosen so that its derivative is related to the sampler; positivity controls the transformed generator, while the Schwarzian derivative produces the explicit lower term. Fewster and Hollands prove the result for a broad class of unitary positive-energy CFTs in Fewster and Hollands 2005, Theorem 4.1, pp. 591–595. Flanagan’s two-dimensional free-scalar energy-density inequality, which combines the two chiral sectors, is Flanagan 1997, Eqs. (1.7)–(1.9), pp. 4923–4925.

Under $g_L(v)=L^{-1/2}g(v/L)$, the averaged operator has a lower bound proportional to $L^{-2}$. This is the two-dimensional scaling expected because the stress tensor has engineering dimension two.

Why four-dimensional null sampling differs

For the massless minimally coupled scalar in four-dimensional Minkowski space, Fewster and Roman construct Hadamard states for which every fixed compactly weighted null-geodesic average is arbitrarily negative Fewster and Roman 2003, § II and Eqs. (2.10)–(2.20), pp. 044003-2–044003-6. Hence no state-independent lower bound of the preceding kind exists on that state class. The same paper proves bounds for null contraction along timelike curves, displaying exactly why the curve’s causal character cannot be omitted.

This negative result does not contradict ANEC. The constructed states can have a nonnegative complete unweighted null integral even though a compactly weighted portion is unbounded below. Nor does it contradict QNEC, which relates a local null stress expectation to a second null shape variation of entropy under quite different assumptions.

The comparison is developed physically on curved-spacetime quantum energy inequalities: the four-dimensional timelike bound scales with four derivatives of length, whereas the two-dimensional chiral null bound scales with two.

Failure boundary: an uncontrolled infinite boost

Let $u_\eta=(\cosh\eta,\sinh\eta,0,0)$ approach the null direction $k=(1,1,0,0)$. Although a rescaled contraction of $T_{ab}u_\eta^au_\eta^b$ approaches $T_{ab}k^ak^b$, a fixed proper-time sampler does not approach a fixed affine null sampler: its support and normalization change under the boost. The timelike QEI bound changes at the same time. Holding the wrong normalization fixed and interchanging the limit $\eta\to\infty$ with the infimum over states erases the divergence that prevents a four-dimensional null QEI.

The strongest valid conclusion is therefore dimension- and domain-specific: timelike sampled free-field bounds in four dimensions; a chiral null bound in two-dimensional positive-energy CFT; no general compactly weighted null-geodesic bound for the four-dimensional free massless scalar.

Exercises

Show that the two-dimensional chiral bound scales as $L^{-2}$ for an $L^2$-normalized sampler.

Solution

For $g_L(v)=L^{-1/2}g(v/L)$, one has $g_L'(v)=L^{-3/2}g'(v/L)$. Thus $\int|g_L'|^2\mathrm dv=L^{-3}L\int|g'|^2\mathrm ds=L^{-2}\int|g'|^2\mathrm ds$. The left side uses a weight with unchanged integral, so both the physical dimension and the bound are consistent.

References

• Fewster, Christopher J., and Stefan Hollands. “Quantum Energy Inequalities in Two-Dimensional Conformal Field Theory.” Reviews in Mathematical Physics 17 (2005): 577–612. DOI; Open PDF.
• Fewster, Christopher J., and Thomas A. Roman. “Null Energy Conditions in Quantum Field Theory.” Physical Review D 67 (2003): 044003; erratum 80 (2009): 069903. DOI; Open PDF.
• Flanagan, Éanna É. “Quantum Inequalities in Two-Dimensional Minkowski Spacetime.” Physical Review D 56 (1997): 4922–4926. DOI; Open PDF.