Absolute and Difference Quantum Energy Inequalities

A quantum energy inequality (QEI) bounds a specified spacetime average of a specified stress-tensor component from below over a specified state class. It is not a pointwise energy condition. A difference QEI measures energy relative to a Hadamard reference state; an absolute QEI replaces that reference dependence by local geometry. The distinction matters whenever the curvature, field coupling, sampler, or renormalization prescription changes.

Required background. Hadamard states and their wavefront characterization licenses restriction of differentiated two-point functions to timelike curves. The renormalized stress tensor fixes the observable and its finite curvature freedom.

Helpful background. Passivity, complete passivity, and ground states gives the thermodynamic comparison. Wavefront-set products, pullbacks, and pushforwards supplies the restriction criterion. Quantum energy inequalities, quantum interest, and quantum energy teleportation separate energetic from information-theoretic uses. Classical energy conditions and quantum violations, curved-spacetime QEIs, QEI sampling, scheme, and state domains, and negative energy and quantum interest provide physical applications.

Difference and absolute bounds

Let $\gamma:\mathbb R\to M$ be a smooth timelike worldline parametrized by proper time, $u^a=\dot\gamma^a$, and $g\in C_0^\infty(\mathbb R;\mathbb R)$. For Hadamard states $\omega$ and $\omega_0$, a difference QEI has the directional form

$$\int_{\mathbb R}g(\tau)^2 \bigl(\langle T_{ab}u^au^b\rangle_\omega -\langle T_{ab}u^au^b\rangle_{\omega_0}\bigr)(\gamma(\tau)) \,\mathrm d\tau\geq-Q_{\omega_0}[g,\gamma].$$

The right side is finite because the Hadamard wavefront set permits the worldline pullback and gives rapid decay in the relevant positive-frequency cone. Positivity of the two-point function supplies the inequality. The reference state is not presumed to minimize energy; it is a device that makes the normal-ordered stress tensor well defined.

An absolute QEI instead reads

$$\int g^2\langle T_{ab}u^au^b\rangle_\omega\,\mathrm d\tau \geq-Q_{\mathrm{abs}}[g,\gamma,g_{ab},m,\xi],$$

where the bound contains no chosen $\omega_0$. For the minimally coupled massive Klein–Gordon field on a four-dimensional globally hyperbolic spacetime, Fewster and Smith obtain such a local geometric bound using a finite Hadamard expansion and Sobolev wavefront estimates Fewster and Smith 2008, Theorems 3.1 and 4.1, pp. 438–450. This is not a universal theorem for arbitrary interactions, nonminimal coupling, boundaries, or all stress components.

The massive Minkowski spectral bound

Take the minimally coupled real scalar of mass $m\geq0$ on four-dimensional Minkowski space, the inertial curve $\gamma(t)=(t,\mathbf0)$, and vacuum normal ordering. With

$$\widehat g(\nu)=\int_{\mathbb R}e^{i\nu t}g(t)\,\mathrm dt, \qquad \omega_{\mathbf k}=\sqrt{|\mathbf k|^2+m^2},$$

the positive-frequency argument gives, for every Hadamard state in the domain of the averaged energy density,

$$\int_{\mathbb R}g(t)^2 \langle{:}T_{00}{:}\rangle_\omega(t,\mathbf0)\,\mathrm dt \geq-B_m[g],$$

with

$$B_m[g]=\frac{1}{\pi}\int_0^\infty\mathrm d\alpha \int_{\mathbb R^3}\frac{\mathrm d^3\mathbf k}{(2\pi)^3} \frac{\omega_{\mathbf k}}{2} \left|\widehat g(\alpha+\omega_{\mathbf k})\right|^2.$$

To see the mechanism, point-split $T_{00}$ and act on the vacuum two-point function. The time-derivative, spatial-gradient, and mass terms sum to $\omega_{\mathbf k}/2$ after the on-shell relation is used. Multiply by $g(t)g(t')$, Fourier transform in both times, and integrate the positive auxiliary frequency $\alpha$. The expectation in an arbitrary Hadamard state differs smoothly from the vacuum, while positivity discards a nonnegative quadratic form. This is the rigorous flat-space construction of Fewster and Eveson 1998, Theorem 2.1 and Eqs. (2.17), (4.6), pp. 084010-3–084010-7.

For $m=0$, the momentum and $\alpha$ integrals reduce by Parseval to the useful check

$$B_0[g]=\frac{1}{16\pi^2}\int_{\mathbb R}|g''(t)|^2\,\mathrm dt.$$

The bound is state independent after the normal-ordering prescription is fixed, but it depends sharply on the sampler. It constrains the average, not each value of $T_{00}$.

The detailed curved-spacetime handoff is QEI sampling, scheme, and state domains, where the field, state class, curve, normalization, and geometric terms are recorded together.

What the hypotheses control

The timelike character of the curve makes the pullback possible: no nonzero null covector normal to the curve lies in the relevant Hadamard wavefront relation. Smooth compact support gives rapid Fourier decay, so the positive-frequency integral converges. The Hadamard condition fixes the ultraviolet singularity uniformly over the state class. Positivity of the two-point function, together with the classical sum-of-squares form of the minimally coupled energy density, supplies the lower estimate. Remove any one of these inputs and this proof stops at an identifiable step.

The mass dependence also passes two elementary checks. First, $B_m[g]$ is nonnegative because its integrand is nonnegative. Second, increasing $m$ raises the lowest frequency entering $\widehat g(\alpha+\omega_{\mathbf k})$; for a fixed smooth sampler its rapid decay suppresses the bound at large mass. This is decoupling for the chosen average, not a statement that every massive-field stress fluctuation is small.

There are two common normalizations for sampling. Here the measured weight is $g^2$, so rescaling $g$ by a constant rescales both sides quadratically. A theorem written for a positive weight $f$ can be compared only after setting $f=g^2$ and checking that $\sqrt f$ is smooth. A nonnegative smooth $f$ may have zeros for which $\sqrt f$ is not smooth, so the replacement is not merely notation.

Finally, an absolute bound does not erase the finite stress-tensor ambiguity. It is absolute relative to a declared locally covariant prescription. Changing an allowed curvature counterterm shifts both the averaged stress tensor and the geometric expression in the bound. Comparing numbers across prescriptions without translating that shift creates a false disagreement.

Failure boundary: the delta-sequence limit

Let $g_\tau(t)=\tau^{-1/2}g(t/\tau)$, so $\int g_\tau^2=\int g^2$. In the massless check,

$$B_0[g_\tau]=\tau^{-4}B_0[g].$$

Thus the lower bound tends to $-\infty$ as the sampler narrows. The QEI remains true for every positive $\tau$, but it supplies no state-independent pointwise lower bound. Treating the divergent right side as a finite limit is the adversarial error.

The converse fails as well: a finite QEI does not imply that negative energy is small in every norm or that a chosen state is passive. It controls only the declared sampled observable. Different samplers, trajectories, dimensions, and field theories require new theorems.

Exercises

Verify the $\tau^{-4}$ scaling of the four-dimensional massless bound.

Solution

Two derivatives give $g_\tau''(t)=\tau^{-5/2}g''(t/\tau)$. Therefore $\int|g_\tau''(t)|^2\mathrm dt=\tau^{-5}\int|g''(t/\tau)|^2\mathrm dt=\tau^{-4}\int|g''(s)|^2\mathrm ds$ after $s=t/\tau$.

References

• Fewster, Christopher J., and Simon P. Eveson. “Bounds on Negative Energy Densities in Flat Spacetime.” Physical Review D 58 (1998): 084010. DOI; Open PDF.
• Fewster, Christopher J., and Calvin J. Smith. “Absolute Quantum Energy Inequalities in Curved Spacetime.” Annales Henri Poincaré 9 (2008): 425–455. DOI; Open PDF.
• Fewster, Christopher J. “Lectures on Quantum Energy Inequalities.” In Quantum Field Theory and Gravity, edited by Felix Finster et al., 97–156. Basel: Birkhäuser, 2012. DOI; Open PDF.