Quantum Energy Inequalities

Quantum fields admit negative local energy density, so no state-independent pointwise lower bound survives. A quantum energy inequality (QEI) instead bounds a smooth average of the renormalized stress tensor for a specified field, trajectory or region, and state class. Its sampling scale quantifies how negative energy is limited in magnitude and duration.

Required background. Stress tensors and charges define the observable, point splitting and Wick products define its renormalization in free fields, and the spectrum condition supplies positive-energy structure.

Helpful background. Contact terms, domains, and errors help distinguish a distributional stress tensor from a pointwise function.

A sampled lower bound

For a massless minimally coupled scalar field in four-dimensional Minkowski spacetime, normal ordered relative to the vacuum, a standard worldline inequality is

$$\int_{-\infty}^{\infty}dt\,g(t)^2 \langle {:}T_{00}(t,\mathbf x){:}\rangle_\psi \ge -\frac{1}{16\pi^2} \int_{-\infty}^{\infty}dt\,|g''(t)|^2 ,$$

for real smooth compactly supported $g$ and suitable states. Fewster and Eveson 1998, Eqs. (1.7)–(1.9) make the scale dependence explicit. If $g_\tau(t)=\tau^{-1/2}g(t/\tau)$, the lower bound scales as $-\tau^{-4}$, with the left side an averaged energy density. Shorter sampling permits a more negative average. The field, state, and sampling hypotheses across major QEI families are reviewed in Fewster 2012, §§ 2–4, pp. 6–20.

The coefficient, derivatives, and even the existence of a state-independent bound change with dimension, mass, curvature, boundary conditions, coupling, and sampled observable. A QEI is therefore a family of theorems, not the statement “negative energy must be small.”

Squeezed-state check

Choose a finite-volume mode regulator and a squeezed vacuum. Interference terms make the normal-ordered energy density negative during parts of an oscillation, while the total Hamiltonian remains nonnegative. For a declared $g$, compute both

$$E_g=\int dt\,g(t)^2\langle{:}T_{00}{:}\rangle \quad\text{and}\quad B_g=-\frac{1}{16\pi^2}\int dt\,|g''(t)|^2 .$$

The check is $E_g\ge B_g$. Then vary the box size, mode cutoff, squeezing, and sampling width separately. Approximating a delta function drives $B_g$ to negative infinity and cannot be used to recover a pointwise bound.

A QEI starts from a renormalized stress tensor and a declared sampling functional. It neither implies a pointwise energy condition nor supplies the distinct completeness hypotheses of ANEC. The diagram is schematic.

What the inequality licenses

A proved QEI can rule out a proposed negative-energy profile if a smooth sampling function violates the bound. It can also constrain duration–magnitude tradeoffs and, with additional dynamical arguments, quantum-interest behavior. It does not by itself calculate detector switching work, prove ANEC on an incomplete null segment, or establish an entropy bound.

Changing the sampling profile, boundary condition, state domain, or renormalization prescription changes the QEI problem. Inferring a pointwise restriction from a finite-width average is the characteristic failure. The map is schematic.

Common pitfalls

Sharpening the sampler without rescaling the bound. The derivative norm grows as the support narrows. Recompute the right-hand side for every profile.

Importing a flat-space coefficient. Boundaries and curvature modify two-point functions and renormalization. This page’s explicit coefficient belongs only to the stated Minkowski free-field example.

References

• Fewster, Christopher J. “Lectures on Quantum Energy Inequalities.” In Quantum Field Theory and Gravity, edited by Felix Finster et al., 2012. arXiv.
• Fewster, Christopher J., and Simon P. Eveson. “Bounds on Negative Energy Densities in Flat Spacetime.” Physical Review D 58 (1998): 084010. DOI.