Quantum Energy Inequalities

Quantum fields can have negative renormalized energy density at a point, with no state-independent pointwise lower bound. A quantum energy inequality (QEI) asks a different question: after averaging a specified stress-tensor component with a smooth sampler along a specified timelike curve or region, how negative can the result be within a declared state class? The answer depends on the field, dimension, trajectory, renormalization, coupling, geometry, boundary conditions, and sampler. This page derives one standard Minkowski worldline bound and checks it against an explicit squeezed wavepacket.

Required background. Stress tensors and charges define the sampled observable; free Wick products and point splitting define the vacuum-normal-ordered stress tensor used here; and the spectrum condition supplies the positive-frequency structure behind the bound.

Helpful background. Contact terms, operator domains, and perturbative errors help distinguish a distribution-valued stress tensor from a pointwise operator.

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.

A timelike worldline bound

Consider a free, real, massless, minimally coupled scalar field in four-dimensional Minkowski spacetime. Normal order with respect to the Minkowski vacuum and evaluate the energy density on the inertial worldline $x(t)=(t,\mathbf0)$. For a real smooth $g$ of rapid decrease—compact support is also allowed—and states in the finite-energy/Hadamard domain for which the average exists,

$$\int_{-\infty}^{\infty}dt\,g(t)^2 \left\langle{:}T_{00}(t,\mathbf0){:}\right\rangle_\psi \ge -\frac{1}{16\pi^2} \int_{-\infty}^{\infty}dt\,|g''(t)|^2.$$

This is the massless four-dimensional specialization of Fewster and Eveson 1998, §V.B. It is state independent but not asserted to be the optimal bound. Every hypothesis matters:

• $T_{00}$ is the energy density seen by this inertial observer, not a null contraction or an arbitrary stress component.
• Normal ordering fixes the Minkowski vacuum to have zero expectation. A different reference state or curved background changes the right-hand side.
• The square $g^2$ is the nonnegative sampling weight. The theorem is not a bound for a discontinuous window or a delta function.
• The state must lie in a domain on which the smeared Wick polynomial has a finite expectation. “All states” never includes arbitrary nonnormal or infinite-energy functionals.

The dimensions provide a quick check. If $g^2$ is normalized to unit integral, then $g$ has dimension $L^{-1/2}$. Consequently $\int|g''|^2dt$ has dimension $L^{-4}$, the dimension of energy density in four spacetime dimensions.

Where the derivative norm comes from

The field’s positive-frequency expansion writes each derivative of $\phi$ on the worldline as a sum of annihilation and creation parts. Multiply by $g(t)$, Fourier transform, and form a family of positive operators $A(u)^\dagger A(u)$ for $u\ge0$. Their expectation values are nonnegative. Expanding and integrating over $u$ produces two pieces: the normal-ordered sampled energy density and a state-independent vacuum commutator term. The general worldline construction is developed in Fewster 2000, §§ II–III. Moving that commutator term to the right gives a frequency-space bound proportional to

$$-\int_0^\infty du\,u^4|\widehat g(u)|^2.$$

Because $g$ is real, the integrand extends evenly to the full frequency axis. Parseval’s identity then gives

$$\int_{-\infty}^{\infty}du\,u^4|\widehat g(u)|^2 \longleftrightarrow \int_{-\infty}^{\infty}dt\,|g''(t)|^2,$$

with the displayed coefficient fixed by the Fourier and field normalizations. This proof mechanism explains both the lower bound and its sensitivity to rapid switching: high frequencies are weighted by $u^4$.

Rescale a fixed profile by

$$g_\tau(t)=\tau^{-1/2}g(t/\tau).$$

Then $\int g_\tau^2dt=\int g^2dt$ while

$$\int|g_\tau''(t)|^2dt =\tau^{-4}\int|g''(s)|^2ds.$$

Thus shorter sampling permits a lower average of order $-\tau^{-4}$. In the formal point-sampling limit $\tau\to0$, the lower bound tends to $-\infty$; a QEI does not recover pointwise positivity.

Gaussian sampling benchmark

Choose the normalized rapidly decreasing profile

$$g_\tau(t)=\frac{1}{(\pi\tau^2)^{1/4}} \exp\!\left(-\frac{t^2}{2\tau^2}\right), \qquad \int_{-\infty}^{\infty}g_\tau(t)^2dt=1.$$

Its second derivative is

$$g_\tau''(t)= \left(\frac{t^2}{\tau^4}-\frac{1}{\tau^2}\right)g_\tau(t).$$

With respect to the probability density $g_\tau^2$, one has $\langle t^2\rangle=\tau^2/2$ and $\langle t^4\rangle=3\tau^4/4$. Therefore

$$\int_{-\infty}^{\infty}|g_\tau''(t)|^2dt =\frac{3}{4\tau^4},$$

and the QEI becomes

$$E_{g_\tau} \equiv\int dt\,g_\tau(t)^2 \left\langle{:}T_{00}(t,\mathbf0){:}\right\rangle \ge -\frac{3}{64\pi^2\tau^4}.$$

This analytic number is useful for testing code: at $\tau=1$, the bound is $-0.0047494305$ in the corresponding inverse-length units.

An explicit negative-energy squeezed wavepacket

We now construct a normalizable state rather than merely assert that negative energy occurs. Use

$$\phi(t,\mathbf x)= \int\frac{d^3\mathbf k}{(2\pi)^{3/2}\sqrt{2k}} \left[a(\mathbf k)e^{-ikt+i\mathbf k\cdot\mathbf x} +a^\dagger(\mathbf k)e^{ikt-i\mathbf k\cdot\mathbf x}\right],$$

with $[a(\mathbf k),a^\dagger(\mathbf k')]=\delta^3(\mathbf k-\mathbf k')$ and $k=|\mathbf k|$. Define a normalized radial mode

$$f_\ell(\mathbf k)= \frac{\ell}{\sqrt\pi\,k^{1/2}}e^{-\ell k}, \qquad b=\int d^3\mathbf k\,f_\ell(\mathbf k)^*a(\mathbf k).$$

Indeed,

$$\int d^3\mathbf k\,|f_\ell(\mathbf k)|^2 =4\ell^2\int_0^\infty dk\,k e^{-2\ell k}=1,$$

so $[b,b^\dagger]=1$. Squeeze only this mode with

$$S(\zeta)=\exp\!\left[\frac12(\zeta^*b^2-\zeta b^{\dagger2})\right], \qquad \zeta=re^{i\theta}.$$

In $S(\zeta)|0\rangle$,

$$\langle b^\dagger b\rangle=\sinh^2r, \qquad \langle bb\rangle=-e^{i\theta}\sinh r\cosh r.$$

Let $u_f$ be the positive-frequency wavepacket multiplying $b$. Spherical symmetry gives $\boldsymbol\nabla u_f(t,\mathbf0)=0$, while direct radial integration gives

$$\dot u_f(t,\mathbf0) =-\frac{2i\ell}{\pi(\ell+it)^3}.$$

For $T_{00}=\tfrac12[(\partial_t\phi)^2+|\boldsymbol\nabla\phi|^2]$, vacuum normal ordering then yields

$$\begin{aligned} \left\langle{:}T_{00}(t,\mathbf0){:}\right\rangle =\frac{4\ell^2}{\pi^2} \bigg[ &\frac{\sinh^2r}{(\ell^2+t^2)^3}\\ &+\sinh r\cosh r\, \operatorname{Re}\frac{e^{i\theta}}{(\ell+it)^6} \bigg]. \end{aligned}$$

Choose $r=0.10$ and $\theta=\pi$. At the sampling center,

$$\left\langle{:}T_{00}(0,\mathbf0){:}\right\rangle =\frac{4}{\pi^2\ell^4} \left(\sinh^2r-\sinh r\cosh r\right) =-\frac{0.03673283}{\ell^4}.$$

The energy is genuinely negative locally. Now sample with the preceding Gaussian at $\tau=\ell/2$. Simpson integration of the displayed function over $-8\ell\le t\le8\ell$ with step $10^{-3}\ell$ gives

$$E_{g_{\ell/2}}=-\frac{0.005339454}{\ell^4}, \qquad B_{g_{\ell/2}}=-\frac{0.075990888}{\ell^4}.$$

The safety margin is $E_g-B_g=0.07065143/\ell^4>0$. Halving the step changes the quoted sampled value by less than $10^{-10}/\ell^4$. This state is far from saturating the inequality: the purpose is to reproduce a negative average and verify the correct side of the bound, not to optimize squeezing.

What changes outside this theorem

Mass and field content. A massive scalar has a frequency-space bound containing a mass-dependent spectral factor. Dirac, Maxwell, interacting, and nonminimally coupled fields require their own results; neither the coefficient nor the derivative order can be copied by analogy.

Curvature and boundaries. Curved-spacetime QEIs may be difference bounds relative to a reference Hadamard state or absolute bounds containing local geometric terms. Reflecting boundaries change the two-point function and can add distance-to-boundary dependence. The flat-space number above is not a universal short formula for those settings; an absolute curved-spacetime construction is given by Fewster and Smith 2008, Theorem 3.1 and §4.

Null sampling. A timelike QEI cannot be turned into a null QEI by informally taking an observer’s speed to light speed. For the four-dimensional massless scalar, smooth weighted averages of $T_{kk}$ along a null geodesic are unbounded below on Hadamard states, even though the complete unweighted ANEC integral can be nonnegative. This sharp distinction is proved in Fewster and Roman 2003, §§II–IV.

Renormalization. In Minkowski space, vacuum normal ordering fixes the zero used above. In curved spacetime, allowed finite local curvature counterterms shift the stress tensor. A theorem must state how those ambiguities are fixed before its numerical bound can be compared with data.

Failure controls

Sharpen the sampler. Replacing $\tau$ by $\tau/2$ multiplies the Gaussian lower-bound magnitude by $16$. If a calculation keeps the old right-hand side, it has silently changed the theorem.

Replace the Gaussian by a top-hat. A discontinuous square root has distributional derivatives whose squared norm is not finite. Smoothing an edge over width $\varepsilon$ makes the derivative contribution grow as $\varepsilon\to0$; the top-hat is not obtained with a finite unchanged bound.

Insert a boundary without recomputing the vacuum term. The image contribution alters the reference two-point function. A violation of the Minkowski coefficient in that setting tests the wrong inequality, not the QEI theorem.

Common pitfalls

Reading an average as a pointwise statement. Negative density at the center of the squeezed benchmark is allowed. Only the declared smooth average is bounded.

Suppressing the square root of the sampler. The theorem is naturally written with weight $g^2$ and bound $\|g''\|_2^2$. Writing a generic weight $f$ without tracking $g=\sqrt f$ changes the derivative expression.

Calling every lower bound “absolute.” Difference and absolute QEIs have different reference-state and renormalization content. State which one is being used.

Exercises

1. Derive $\int|g_\tau''|^2dt=3/(4\tau^4)$ for the normalized Gaussian.

Solution

Square $g_\tau''=(t^2/\tau^4-1/\tau^2)g_\tau$ and average the polynomial with probability density $g_\tau^2$. Using $\langle t^2\rangle=\tau^2/2$ and $\langle t^4\rangle=3\tau^4/4$ gives

$$\frac{3}{4\tau^4}-\frac{1}{\tau^4}+\frac{1}{\tau^4} =\frac{3}{4\tau^4}.$$

2. If a sampler is narrowed from $\tau$ to $\tau/3$, by what factor does the four-dimensional massless lower-bound magnitude change?

Solution

The derivative norm scales as $\tau^{-4}$. Therefore the magnitude grows by $3^4=81$. This is why a fixed bound cannot accompany a sharpened profile.

3. Verify the normalization of $f_\ell$ and the commutator of $b$.

Solution

In spherical momentum coordinates,

$$\int d^3\mathbf k\,|f_\ell|^2 =4\pi\frac{\ell^2}{\pi} \int_0^\infty dk\,k e^{-2\ell k} =4\ell^2\frac{1}{(2\ell)^2}=1.$$

Inserting the canonical commutator into $[b,b^\dagger]$ gives exactly this integral, hence $[b,b^\dagger]=1$.

4. Why does the squeezed benchmark not prove that the QEI is optimal?

Solution

It verifies one state and one sampler, with a large positive gap $E_g-B_g$. Optimality would require finding the infimum over the full admitted state class and proving that the bound is attained or approached. A successful inequality check supplies neither step.

References

• Fewster, Christopher J. “A General Worldline Quantum Inequality.” Classical and Quantum Gravity 17, no. 9 (2000): 1897–1911. DOI.
• Fewster, Christopher J., and Simon P. Eveson. “Bounds on Negative Energy Densities in Flat Spacetime.” Physical Review D 58 (1998): 084010. 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.
• Fewster, Christopher J., and Calvin J. Smith. “Absolute Quantum Energy Inequalities in Curved Spacetime.” Annales Henri Poincaré 9 (2008): 425–455. DOI.