Negative Energy in Quantum Field Theory

Quantum fields can have negative renormalized local energy density even though their total Hamiltonian is bounded below. The distinction is between a local, vacuum-subtracted observable—whose pair-creation terms interfere with the vacuum—and the global generator of time translations, whose spectrum is nonnegative in the vacuum representation. Negative energy is therefore real but constrained, not an instability of the ground state.

Required background. Congruences and Optical Scalars supplies the observer field used to define energy density, and Quantum Information supplies density matrices, variances, and entropy language.

Helpful background. Quantum Fields in Curved Spacetime supplies Hadamard states and stress renormalization, Einstein Equations supplies gravitational sourcing, and Black-Hole Geometry and Mechanics supplies later focusing applications.

Local energy as an indefinite observable

Work first in four-dimensional Minkowski spacetime with a free real scalar field,

$$S=-\frac12\int d^4x\, \left(\partial_a\phi\,\partial^a\phi+m^2\phi^2\right), \qquad m\ge0.$$

Use the Minkowski vacuum $|0\rangle$ as reference and normal order the stress tensor. For an inertial observer $u^a$, smear the measured density with a real nonnegative compactly supported function $f$:

$$A_f=\int d^4x\,f(x)\,{:}T_{ab}(x){:}\,u^au^b.$$

Vacuum subtraction gives $\langle0|A_f|0\rangle=0$. Yet $A_f|0\rangle$ is generally nonzero: the local stress tensor contains two-creation terms, and finite smearing does not impose the momentum cancellation that removes them from the total Hamiltonian.

Normalize $|2_f\rangle=A_f|0\rangle/\lVert A_f|0\rangle\rVert$ and define

$$|\psi_\alpha\rangle =\cos\alpha\,|0\rangle+\sin\alpha\,|2_f\rangle.$$

Writing $a=\lVert A_f|0\rangle\rVert\gt 0$ and $b=\langle2_f|A_f|2_f\rangle$, direct expansion gives

$$\langle\psi_\alpha|A_f|\psi_\alpha\rangle =a\sin(2\alpha)+b\sin^2\alpha =2a\alpha+O(\alpha^2).$$

It is negative for sufficiently small $\alpha\lt 0$. This is an interference effect between the zero- and two-particle sectors, not a negative occupation number. Fewster gives the corresponding Wick-square argument, its local-observable interpretation, and stress-energy examples in Fewster 2012, § 1.2, pp. 7–10, PDF.

Why the Hamiltonian remains positive

The normal-ordered Hamiltonian is the integral over a complete Cauchy surface,

$$H=\int d^3x\,{:}T_{00}{:} =\int\frac{d^3k}{(2\pi)^3}\, \omega_{\boldsymbol k}\, a^\dagger(\boldsymbol k)a(\boldsymbol k), \qquad \omega_{\boldsymbol k}=\sqrt{\boldsymbol k^2+m^2}.$$

Thus $H\ge0$ and $H|0\rangle=0$. The spatial integral and field equation cancel the pair-creation terms that survive in a local sampling. Equivalently, the Hamiltonian is a nonlocal observable relative to every bounded region, so the local spectral argument does not apply to it.

This distinction is structural. If a nonzero local observable $A$ had $\langle0|A|0\rangle=0$ and were positive, then

$$\lVert A^{1/2}|0\rangle\rVert^2=0$$

would imply $A|0\rangle=0$. Vacuum cyclicity and locality then force $A$ to be trivial under the standard hypotheses. Epstein, Glaser, and Jaffe established this nonpositivity result for local energy density in Epstein, Glaser, and Jaffe 1965, pp. 1016–1022.

Casimir and squeezed realizations

Between ideal parallel conducting plates separated by $L$, symmetry, conservation, tracelessness, and the electromagnetic calculation give

$$\langle T_{\mu\nu}\rangle_{\mathrm{ren}} =\frac{\pi^2}{720L^4} \operatorname{diag}(-1,1,1,-3).$$

The energy density between the plates is negative relative to the empty-space vacuum, while stresses on the plates and positive energy in the apparatus complete the physical system. The $L^{-4}$ scaling is fixed by dimensions; the sign is consistent with the attractive force. Fewster derives this interpretation in Fewster 2012, § 1.2, pp. 8–9, PDF.

A squeezed state contains coherent superpositions of even particle numbers. Its anomalous correlators $\langle aa\rangle$ and $\langle a^\dagger a^\dagger\rangle$ oscillate relative to the positive number term, producing subvacuum intervals in a local quadrature or energy density. Positive and negative fringes alternate, so a timelike observer cannot remain in an arbitrarily deep negative trough indefinitely. The next page converts that intuition into a sampled lower bound.

Regime and checks

The derivation is fixed-background free-field QFT at leading matter-loop order. The state, reference vacuum, smearing, and normal-ordering prescription are explicit. In curved spacetime one replaces normal ordering by a locally covariant renormalized stress tensor in a Hadamard state; finite curvature counterterms and reference choices must then be declared.

Dimensions check the examples: $T_{ab}$ has mass dimension four in four spacetime dimensions, so the Casimir density scales as $L^{-4}$. Conservation holds away from the plates. The flat-space limit of the curved prescription recovers vacuum normal ordering. The calculation fails as a self-consistent spacetime prediction when $G\lvert\langle T_{ab}\rangle\rvert$ is no longer a small curvature source, when idealized boundaries probe their material cutoff, or when an unsmeared coincidence observable is used.

Exercise

Let a Hermitian local observable have matrix elements $\langle0|A|0\rangle=0$, $\langle0|A|2\rangle=a\gt 0$, and $\langle2|A|2\rangle=b$. For the normalized state $|\psi_\alpha\rangle=\cos\alpha|0\rangle+\sin\alpha|2\rangle$, show explicitly that negative expectation values coexist with a positive Hamiltonian if $H|0\rangle=0$ and $H|2\rangle=E|2\rangle$ with $E\gt 0$.

Solution

Hermiticity makes the off-diagonal elements equal and real after a phase choice, so

$$\langle A\rangle_{\psi_\alpha} =a\sin(2\alpha)+b\sin^2\alpha.$$

For $\alpha=-\varepsilon$ with $\varepsilon\gt 0$ small, this is $-2a\varepsilon+O(\varepsilon^2)\lt 0$. By contrast,

$$\langle H\rangle_{\psi_\alpha} =E\sin^2\alpha =E\varepsilon^2+O(\varepsilon^4)\ge0.$$

The negative local expectation appears at first order in the interference amplitude, whereas the positive total excitation energy appears at second order. There is no contradiction because $A$ and $H$ are different operators with different spectra; a lower bound on the global generator does not impose pointwise positivity on its density.

References

• Epstein, H., Glaser, V., and Jaffe, A. 1965. “Nonpositivity of the Energy Density in Quantized Field Theories.” Il Nuovo Cimento 36, 1016–1022. DOI.
• Fewster, C. J. 2012. “Lectures on Quantum Energy Inequalities.” arXiv:1208.5399. Open PDF.