13.09.11 · gr-cosmology / microlocal-qft-curved-spacetimes

Quantum energy inequalities (Fewster)

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Anchor (Master): Ford, *Proc. Roy. Soc. A* 364 (1978) 227; Ford and Roman, *Phys. Rev. D* 51 (1995) 4277; Fewster and Eveson, *Phys. Rev. D* 58 (1998) 084010; Fewster, *Class. Quantum Grav.* 17 (2000) 1897; Fewster and Smith, *Ann. Henri Poincaré* 9 (2008) 425; Fewster, *Lectures on Quantum Energy Inequalities* (arXiv:1208.5399, 2012)

Intuition Beginner

In everyday physics, energy density is never negative. A box of gas, a stretched spring, a beam of light — each carries a positive amount of energy in every little region. This rule, that the energy density seen by any observer is at least zero, is called the weak energy condition, and it is one of the assumptions behind the theorems that say spacetime must contain black holes and a Big Bang.

Quantum fields break this rule. In quantum theory you can prepare a field in a state where, at a chosen point and moment, the energy density is below zero. The vacuum has zero energy density on average, and a clever quantum state can dip below the vacuum here while rising above it there. These negative dips are real and have been measured indirectly, for example in the tiny attractive force between two close metal plates.

If negative energy were unlimited, the universe would be in trouble. You could imagine building a wormhole to walk through, or a warp bubble to outrun light, or a machine that drains heat from a cold body and dumps it into a hot one, breaking the second law. So a natural question follows: just how negative can the energy density get, and for how long?

The answer is a quantum energy inequality. It says you cannot have a large negative energy density for a long time. If you measure the energy density along your path and average it with a smooth weighting that lasts a time $T$ , the average can be negative, but only down to a floor of roughly minus-one-over- $T$ -to-the-fourth-power. A deep negative dip must be brief; a long-lasting dip must be shallow. Nature lets you borrow negative energy, but the loan is small and short, and it must be paid back. The remarkable part is that this floor does not depend on which quantum state you chose — it depends only on how you did the averaging.

Visual Beginner

Picture the energy density a detector measures as it moves along its worldline, plotted against its own clock. The curve wiggles above and below the zero line; the regions below zero are the negative-energy intervals that quantum theory allows.

Three things in the picture carry the meaning. First, the curve does cross below zero — negative energy density is genuinely allowed, unlike in classical physics. Second, the bell-shaped weighting of width $T$ is how we average: we are not asking about a single instant but about a smeared interval. Third, the boxed inequality is the floor. The deeper the dip, the narrower it must be; the area of negative energy you can scoop up shrinks as you spread the averaging over a longer time. The crossed-out grey region is the combination nature refuses: large and negative and long-lasting all at once.

Worked example Beginner

Estimate how negative the averaged energy density of a quantum field can be when you average over one second, using the Ford-Roman bound, and see why this is far too tiny to ever build a wormhole.

Step 1. Write the bound in the simple form. For averaging with a smooth bump of duration $T$ , the time-averaged energy density of the field has a floor of about $$ \text{average energy density} \geq -\frac{C}{T^4}, $$ where $C$ is a pure number close to one. This is the four-dimensional Ford-Roman bound. The key feature is the fourth power of the time.

Step 2. Choose a time and put in numbers, in natural units where the speed of light and Planck's constant are one. Take $T = 1$ second. In natural units one second is a very large number of natural time units, about $1.5 \times 1 0^{43}$ . So the floor is $$ -\frac{1}{T^4} \approx -\frac{1}{(1.5 \times 10^{43})^4} \approx -2 \times 10^{-172}. $$ That is an almost unimaginably small negative number.

Step 3. Compare with something real. The energy density of ordinary water is, in these same natural units, a number of order $1 0^{- 94}$ . So the most negative energy density you can sustain for a full second is smaller in size than the energy density of water by a factor of about $1 0^{78}$ — utterly negligible.

Step 4. See the trade-off the other way. To get a sizeable negative energy density, you must shrink the averaging time $T$ . Because of the fourth power, making the negative energy a billion times bigger costs you a factor of about $178$ in shorter time. Strong negative energy is possible only on extremely short timescales and over extremely short distances.

What this tells us: negative energy is real but feeble and fleeting. The fourth-power law means that any plan needing a lot of negative energy for a usable length of time runs into a wall set by the inequality. The same arithmetic is why proposals for traversable wormholes and warp drives, which need large amounts of negative energy spread over macroscopic regions, are squeezed into geometries so thin they are physically implausible.

Check your understanding Beginner

Exercise (easy, multiple choice).

A quantum energy inequality says the time-averaged energy density along a worldline is bounded below by $- C / T^{4}$ , where $T$ is the averaging time. If you double the averaging time $T$ , how does the floor change?

A. It becomes twice as deep (twice as negative) B. It stays the same C. It becomes sixteen times shallower (sixteen times closer to zero) D. It becomes four times deeper

Hint

The floor is proportional to $1/ T^{4}$ . Doubling $T$ multiplies $T^{4}$ by $2^{4} = 16$ , which divides the size of the floor by sixteen.

Answer

C. It becomes sixteen times shallower.

The floor is $- C / T^{4}$ . Replacing $T$ by $2 T$ replaces $T^{4}$ by $16 T^{4}$ , so the floor becomes $- C / (16 T^{4})$ , which is sixteen times closer to zero. Longer averaging forces a shallower allowed dip; this is the magnitude-duration trade-off at the heart of every quantum energy inequality. Option A and option D have the dependence backwards, and option B ignores the time dependence entirely.

Exercise (easy, true-false).

True or false: the lower bound in a quantum energy inequality depends on which quantum state the field is in, so you must know the state before you can state the bound.

Hint

The whole power of these inequalities is that one number works for every admissible state. What does the bound depend on instead?

Answer

False.

The defining feature of a quantum energy inequality is that the lower bound is independent of the state: it depends only on the sampling function used to average and on the spacetime, but not on which admissible (Hadamard) state the field occupies. That is exactly what makes the bound useful — it constrains every physically reasonable state at once, including states nobody has written down, so it can rule out wormhole and warp-drive geometries without needing to know the matter state in detail.

Formal definition Intermediate+

Throughout, $(M, g)$ is a globally hyperbolic Lorentzian manifold in the sense of 13.09.01 with signature $(-, +, +, +)$ , and $ϕ$ is the real Klein-Gordon field with operator $P = □_{g} + m^{2} + ξ R$ studied in 13.09.02. A state $ω$ is Hadamard when its two-point function $Λ (x, y) = ω (ϕ (x) ϕ (y))$ satisfies the microlocal spectrum condition of 13.09.03. The Hadamard parametrix is the locally constructed bidistribution $H (x, y)$ with the universal short-distance singularity $H = (8 π^{2})^{- 1} (U / σ_{ϵ}) + V lo g σ_{ϵ} + W$ , where $σ$ is Synge's world function. For any Hadamard $ω$ the difference $Λ - H$ is smooth.

The renormalised normal-ordered energy density along a smooth timelike worldline $γ$ with unit tangent $u^{a}$ is obtained by point-splitting the classical density $T_{ab} u^{a} u^{b}$ and subtracting the parametrix: $$ \langle :!\rho!:\rangle_\omega(\tau) ;=; \lim_{y \to \gamma(\tau)} \mathcal{D}{xy}\big(\Lambda(x, y) - H(x, y)\big)\Big|{x = \gamma(\tau)}, $$ where $D_{x y}$ is the symmetric bidifferential operator implementing $T_{ab} u^{a} u^{b}$ on the two arguments (Wald 1977; 13.09.06). The coincidence limit exists because $Λ - H$ is smooth. For two Hadamard states $ω$ , $ω_{0}$ the difference of densities is unambiguous and parametrix-independent: $$ \langle :!\rho!:\rangle_\omega(\tau) - \langle :!\rho!:\rangle_{\omega_0}(\tau) = \lim_{y \to \gamma(\tau)} \mathcal{D}_{xy}\big(\Lambda(x, y) - \Lambda_0(x, y)\big), $$ since the parametrix cancels.

A sampling function is a real-valued $g \in C_{c}^{\infty} (R)$ , regarded as a weight along $γ$ via the pullback $g (τ)$ . One normalises $\int g (τ)^{2} d τ = 1$ .

Definition (worldline quantum energy inequality). A difference quantum energy inequality for the class of Hadamard states relative to a reference $ω_{0}$ is a bound, holding for every Hadamard $ω$ , $$ \int_{\mathbb{R}} \Big(\langle :!\rho!:\rangle_\omega(\tau) - \langle :!\rho!:\rangle_{\omega_0}(\tau)\Big),g(\tau)^2,d\tau ;\geq; -\mathcal{Q}[g], $$ where the bound $Q [g] \geq 0$ depends on $g$ , on $γ$ , and on the geometry, but not on the state $ω$ . An absolute quantum energy inequality is the same bound with $⟨ : ρ : ⟩_{ω_{0}}$ absent — the Hadamard-subtracted density itself bounded below.

The classical weak energy condition $T_{ab} u^{a} u^{b} \geq 0$ does not survive quantisation: the energy density operator of a Wightman field is not bounded below ^{[Epstein-Glaser-Jaffe 1965]}. The quantum energy inequality is the surviving averaged remnant.

Counterexamples to common slips

The pointwise bound $⟨ : ρ : ⟩_{ω} (τ) \geq - C$ is false for every constant $C$ . Given any $C$ one constructs a Hadamard state — a suitable superposition of the vacuum and a two-mode squeezed state — with $⟨ : ρ : ⟩_{ω} (0) < - C$ . Only the time-averaged statement survives. The averaging is not a technical convenience; it is the precise content of what is true.
The sampling function enters as $g^{2}$ , not $g$ , and the normalisation is $\int g^{2} = 1$ . Writing the smearing as $\int ⟨ ρ ⟩ g d τ$ with $g$ of one sign is a different and weaker object; the QEI controls the quadratic smearing because the proof rests on a positivity statement quadratic in the field.
"State-independent" means independent of the choice of Hadamard state, not independent of the worldline or the geometry. The bound $Q [g]$ contains the sampling function and, in the curved case, curvature and the reference state's smooth remainder. On a fixed worldline in fixed geometry it is one fixed number per sampling function.
A QEI does not say the total energy is positive. It bounds a local weighted average. The averaged null energy condition and the averaged weak energy condition along complete geodesics are stronger, separate statements; a worldline QEI implies neither directly, though it is the technical input to singularity theorems that play the same role.

Key theorem with proof Intermediate+

Theorem (Fewster-Eveson worldline QEI for the massless scalar in four-dimensional Minkowski space). Let $ϕ$ be the free massless scalar field in Minkowski space $R^{1, 3}$ , $γ (τ) = (τ, 0)$ an inertial worldline, and $: ρ := \frac{1}{2} (: \dot{ϕ}^{2} : + : ∣\nabla ϕ ∣^{2} :)$ the normal-ordered energy density relative to the Minkowski vacuum. For every Hadamard state $ω$ and every real $g \in C_{c}^{\infty} (R)$ , $$ \int_{\mathbb{R}} \langle :!\rho!:\rangle_\omega(\tau),g(\tau)^2,d\tau ;\geq; -\frac{1}{16\pi^2}\int_{\mathbb{R}} g''(\tau)^2,d\tau. $$ The bound is independent of $ω$ ^{[Fewster-Eveson 1998]}.

Proof. The argument rests on a single positivity statement and the Parseval identity. Write the point-split two-point function of any state as $Λ (x, y) = ω (ϕ (x) ϕ (y))$ ; for a Hadamard state $Λ - Λ_{vac}$ is smooth, where $Λ_{vac}$ is the vacuum two-point function. The normal-ordered density is the coincidence limit of a differential operator applied to $Λ - Λ_{vac}$ .

Step 1: positivity of the point-split sampled field. For each frequency $λ \in R$ define the smeared, frequency-shifted field operator $$ \Phi_\lambda ;=; \int_{\mathbb{R}} g(\tau),e^{i\lambda\tau},(\partial_\tau\phi)(\gamma(\tau)),d\tau . $$ For any state $ω$ , the operator $Φ_{λ}^{†} Φ_{λ}$ has non-negative expectation: $$ \omega(\Phi_\lambda^\dagger\Phi_\lambda) ;=; \iint \overline{g(\tau)},e^{-i\lambda\tau},g(\tau'),e^{i\lambda\tau'},\partial_\tau\partial_{\tau'}\Lambda(\gamma(\tau),\gamma(\tau')),d\tau,d\tau' ;\geq; 0, $$ since $ω (A^{†} A) \geq 0$ for any operator $A$ and any state. This is the only positivity used.

Step 2: extract the sampled energy density. The sampled normal-ordered energy density along $γ$ is $$ \int \langle :!\rho!:\rangle_\omega(\tau),g(\tau)^2,d\tau = \int g(\tau)^2 \lim_{\tau' \to \tau} \partial_\tau\partial_{\tau'}\big(\Lambda - \Lambda_{\mathrm{vac}}\big)(\gamma(\tau),\gamma(\tau')),d\tau, $$ restricting to the $\dot{ϕ}^{2}$ part of $ρ$ ; the spatial-gradient part is handled identically and contributes the same final form by the vacuum mass-shell relation $∣ k ∣^{2} = ω_{k}^{2}$ for the massless field. Insert the spectral representation of $\partial_{τ} \partial_{τ^{'}} Λ$ pulled back to $γ$ . Both $Λ$ and $Λ_{vac}$ are, on the worldline, distributions of the time difference plus a smooth state-dependent piece; their derivative two-point function has a representation as a positive-frequency integral $$ \partial_\tau\partial_{\tau'}\Lambda(\gamma(\tau),\gamma(\tau')) = \int_0^\infty \mu_\omega(d\lambda),e^{-i\lambda(\tau - \tau')} + (\text{c.c. of vacuum reference}), $$ with $μ_{ω}$ a positive measure controlled by the state.

Step 3: average over the frequency shift and apply Parseval. Integrate the Step 1 positivity over $λ \in (0, \infty)$ against the vacuum spectral weight. The key identity is that the sampled vacuum-subtracted energy density equals a frequency integral of the manifestly non-negative quantity $ω (Φ_{λ}^{†} Φ_{λ})$ minus a fixed $ω$ -independent term: $$ \int \langle :!\rho!:\rangle_\omega,g^2,d\tau = \frac{1}{\pi}\int_0^\infty \omega(\Phi_\lambda^\dagger\Phi_\lambda),d\lambda ;-; \frac{1}{\pi}\int_0^\infty !!\int |\widehat{g}(\lambda + u)|^2,u,\frac{du}{2\pi},d\lambda, $$ where $g$ is the Fourier transform of $g$ . The first term is non-negative by Step 1, so the sampled density is bounded below by minus the second term, which is independent of $ω$ . Evaluating the $ω$ -independent term for the massless field in four dimensions, the $u$ -weight is $u$ (one power from each derivative and the measure), and the double integral collapses by Parseval to $$ \frac{1}{16\pi^2}\int_{\mathbb{R}} \lambda^4,|\widehat{g}(\lambda)|^2,\frac{d\lambda}{2\pi} = \frac{1}{16\pi^2}\int_{\mathbb{R}} g''(\tau)^2,d\tau, $$ using $g^{''} (λ) = - λ^{2} g (λ)$ and the Parseval identity $\int ∣ g^{''} ∣^{2} d λ /2 π = \int (g^{''})^{2} d τ$ . Therefore $$ \int \langle :!\rho!:\rangle_\omega,g^2,d\tau ;\geq; -\frac{1}{16\pi^2}\int g''(\tau)^2,d\tau, $$ with the right-hand side independent of $ω$ . $□$

Bridge. This theorem builds toward the curved-spacetime difference inequality and appears again in 13.09.10, where the same positivity-plus-Hadamard-subtraction logic bounds how negative the energy density of an admissible black-hole state can be near the horizon. The foundational reason the bound is state-independent is that only the vacuum reference and the sampling function survive the subtraction: the state-dependent piece enters solely through the manifestly non-negative term $ω (Φ_{λ}^{†} Φ_{λ})$ , so dropping it can only lower the right-hand side, and what remains is the fixed Sobolev seminorm $\int (g^{''})^{2}$ . This is exactly the structure that, in the curved case, becomes the difference QEI: the vacuum reference is replaced by an arbitrary Hadamard reference state $ω_{0}$ , the difference $Λ - Λ_{0}$ is smooth by the microlocal spectrum condition of 13.09.03, and the same positivity gives a finite bound. The construction here generalises from the single vacuum to the whole folium of Hadamard states, and the bridge is the observation that the Hadamard parametrix is the universal local reference: the central insight is that a QEI is a positivity statement about a smeared field operator, dressed by the geometry only through the sampling tube. Putting these together, the worldline QEI, the difference QEI, and the absolute QEI are one inequality at three levels of state-independence.

Exercises Intermediate+

Exercise 1 (easy, numeric).

Take the Gaussian sampling function $g (τ) = (2 π T^{2})^{- 1/4} e^{- τ^{2} / (4 T^{2})}$ , normalised so $\int g^{2} d τ = 1$ . The Fewster-Eveson bound involves $\int (g^{''})^{2} d τ$ , which for this Gaussian scales as $a / T^{4}$ for a pure number $a$ . By dimensional analysis alone, what power of $T$ does the bound $\frac{1}{16 π ^{2}} \int (g^{''})^{2}$ scale as?

Hint

Each derivative $\partial_{τ}$ brings down a factor of $1/ T$ in magnitude, and the integral $\int d τ$ contributes a factor of $T$ ; count the powers in $\int (g^{''})^{2} d τ$ given $\int g^{2} = 1$ .

Answer

$1/ T^{4}$ . Two derivatives in $g^{''}$ give $1/ T^{2}$ each in amplitude, so $(g^{''})^{2}$ scales as $g^{2} / T^{4}$ , and integrating against $d τ$ with $\int g^{2} = 1$ leaves $1/ T^{4}$ . This reproduces the Ford-Roman fourth-power law: the floor on averaged energy density goes as $- 1/ T^{4}$ in four dimensions. Rubric: full credit for $1/ T^{4}$ with the derivative-counting justification.

Exercise 5 (medium, short-answer).

Explain why the QEI is formulated as a bound on $\int ⟨ : ρ : ⟩_{ω} g^{2} d τ$ with $g^{2} \geq 0$ , rather than as a bound on $\int ⟨ : ρ : ⟩_{ω} h d τ$ for a sign-changing test function $h$ .

Hint

Look at where positivity enters the proof. The non-negative quantity is $ω (Φ^{†} Φ)$ , a square.

Answer

The proof's only positivity input is $ω (Φ_{λ}^{†} Φ_{λ}) \geq 0$ , which is quadratic in the smeared field and therefore quadratic in the sampling weight. Writing the smearing as $g^{2}$ makes the weight a perfect square, matching the quadratic structure of the field positivity: the sampled density is then a frequency integral of $ω (Φ^{†} Φ)$ minus an $ω$ -independent term. A linear smearing $\int ⟨ ρ ⟩ h d τ$ with sign-changing $h$ cannot be tied to a single square and admits no state-independent lower bound — by choosing $ω$ one can make a linearly smeared density arbitrarily negative wherever $h > 0$ . The quadratic weighting is what the positivity argument demands; it is not a cosmetic choice. Rubric: full credit for tying $g^{2}$ to the squared-operator positivity and noting linear smearings have no state-independent bound.

Exercise 6 (hard, short-answer).

A difference QEI on a curved spacetime bounds $\int (⟨ : ρ : ⟩_{ω} - ⟨ : ρ : ⟩_{ω_{0}}) g^{2} d τ$ from below by a quantity finite for every Hadamard $ω$ . Identify precisely which property of Hadamard states makes the bound finite, and explain why a non-Hadamard reference $ω_{0}$ would break the construction.

Hint

The difference $Λ - Λ_{0}$ of two two-point functions appears under a coincidence-limit differential operator. What does the microlocal spectrum condition of 13.09.03 say about this difference?

Answer

The microlocal spectrum condition of 13.09.03 fixes the wave-front set of every Hadamard two-point function to the same set — the future-pointing half of the null-bicharacteristic relation. Two distributions with identical wave-front sets have a difference whose singular directions cancel, so $Λ - Λ_{0}$ is a smooth function, and applying the coincidence-limit differential operator $D_{x y}$ to a smooth function yields a finite, well-defined density difference. This is what makes $\int (⟨ : ρ : ⟩_{ω} - ⟨ : ρ : ⟩_{ω_{0}}) g^{2}$ finite without any parametrix subtraction: the subtraction is performed by the reference state itself. If $ω_{0}$ were non-Hadamard, its wave-front set would differ from that of $ω$ , the difference $Λ - Λ_{0}$ would carry singularities, the coincidence limit would diverge, and there would be no finite lower bound — the construction collapses. The state-independence of the right-hand side survives because the same positivity argument as in flat space applies to the smeared field of $ω$ relative to the smooth reference. Rubric: full credit for naming the equal-wave-front-set / smooth-difference property and explaining the non-Hadamard divergence.

Exercise 7 (hard, short-answer).

The quantum-interest conjecture of Ford and Roman states that a pulse of negative energy must be over-compensated by a nearby positive pulse, with the "interest" growing as the separation grows. Using the worldline QEI with a sampling function that resolves two pulses separated by a time $Δ$ , argue heuristically why a lone negative pulse of fixed magnitude and width cannot persist in isolation.

Hint

Apply the QEI with a sampling function broad enough to cover both the negative pulse and the region after it. The averaged density over that window is bounded below.

Answer

Choose a sampling function $g^{2}$ broad enough to cover a window containing the negative pulse and the interval of duration $Δ$ after it. The QEI bounds the $g^{2}$ -weighted average of the energy density over that whole window below by $- \frac{1}{16 π ^{2}} \int (g^{''})^{2}$ , a finite number controlled by the window width. If the window held only the negative pulse and nothing else, the weighted average would be the (negative) pulse value times its fractional weight; as the window widens to include more empty space the bound on the average tightens toward zero, forcing the total integrated negative energy in any fixed window to be bounded. A lone negative pulse with no compensating positive energy would, sampled by a sufficiently broad and smooth $g$ , give an averaged density more negative than the QEI floor for that $g$ , a contradiction. Hence a positive pulse must accompany the negative one within a time set by the magnitude-duration trade-off, and broadening the separation $Δ$ forces the sampling function to be smoother over the gap, raising the required positive "interest". This is the QEI origin of quantum interest. Rubric: full credit for the broad-window argument plus the conclusion that isolated negative pulses violate the floor.

Advanced results Master

The flat-space worldline inequality is the first rung of a ladder that reaches arbitrary globally hyperbolic spacetimes and eliminates the reference state entirely. The following sharpen the basic theorem.

Theorem (difference QEI on a curved spacetime). Let $(M, g)$ be globally hyperbolic, $ϕ$ a Klein-Gordon field with a Hadamard reference state $ω_{0}$ , and $γ$ a smooth timelike worldline with proper-time sampling $g \in C_{c}^{\infty}$ . For every Hadamard state $ω$ , $$ \int \Big(\langle :!\rho!:\rangle_\omega - \langle :!\rho!:\rangle_{\omega_0}\Big)(\tau),g(\tau)^2,d\tau ;\geq; -\int_0^\infty !\big[,\widehat{g \otimes g,\cdot,(\partial\partial T_{\omega_0})},\big](-\lambda, \lambda),\frac{d\lambda}{\pi}, $$ where $T_{ω_{0}} = Λ - Λ_{0}$ restricted to $γ \times γ$ is smooth and the right-hand side is finite and independent of $ω$ ^{[Fewster 2000]}. The proof is the curved-space version of the flat argument: the positivity $ω (Φ_{λ}^{†} Φ_{λ}) \geq 0$ holds verbatim; the only change is that the vacuum reference is replaced by $ω_{0}$ and the subtraction is performed by $ω_{0}$ rather than by the Hadamard parametrix. Finiteness is the smoothness of $Λ - Λ_{0}$ guaranteed by the microlocal spectrum condition of 13.09.03. The bound depends on the reference state only through its smooth coincidence-limit data along $γ$ , never on $ω$ .

Theorem (absolute QEI). In the locally covariant framework there is a reference-free bound: for every Hadamard $ω$ , $$ \int \langle :!\rho!:\rangle_\omega(\tau),g(\tau)^2,d\tau ;\geq; -\mathcal{Q}{\mathrm{abs}}[g; \gamma, g{ab}], $$ where $Q_{abs}$ is built from the Hadamard parametrix coefficients $U, V$ and the geometry of the sampling tube, with no reference state appearing ^{[Fewster-Smith 2008]}. The construction subtracts the parametrix $H$ in place of a reference two-point function and controls the resulting Hadamard-subtracted density directly; the absolute bound reduces to the difference bound plus the finite, geometry-determined difference $⟨ : ρ : ⟩_{ω_{0}} - (parametrix coincidence terms)$ , which is computable from curvature invariants and the worldline acceleration.

Theorem (sharp constant in two dimensions). For the massless scalar in two-dimensional Minkowski space the optimal worldline bound is $$ \int \langle :!T_{00}!:\rangle_\omega,g^2,d\tau ;\geq; -\frac{1}{24\pi}\int \frac{(g'(\tau))^2}{g(\tau)},d\tau \qquad (g \geq 0), $$ and the constant $1/24 π$ cannot be improved: there is a sequence of Hadamard states saturating it ^{[Flanagan 1997]}. The optimal constant follows from a calculus-of-variations argument: substituting $g = h^{2}$ turns the right-hand side into $- \frac{1}{6 π} \int (h^{'})^{2}$ , and the extremising state is the one whose mode content matches the minimiser of the associated quadratic form. The four-dimensional Fewster-Eveson constant $1/16 π^{2}$ , by contrast, is not known to be optimal; the QEI there is a true bound but not provably the best one.

Theorem (QEIs and singularity theorems). A Hawking-type singularity theorem holds under a smeared energy condition of the form supplied by a QEI: if the Ricci-tidal contraction $R_{ab} u^{a} u^{b}$ along every timelike geodesic satisfies a worldline averaged bound of the QEI type, then a globally hyperbolic spacetime with a contracting Cauchy surface is future timelike geodesically incomplete ^{[Fewster-Galloway 2011]}. The classical Hawking theorem assumes the pointwise strong energy condition $R_{ab} u^{a} u^{b} \geq 0$ ; quantum fields violate it, but the semiclassical Einstein equation ties $R_{ab} u^{a} u^{b}$ to $⟨ T_{ab} ⟩ u^{a} u^{b}$ , which a QEI bounds below in the averaged sense. The averaged condition is enough to run the Riccati/focusing comparison argument and force a conjugate point, hence incompleteness.

Synthesis. The quantum energy inequality is the foundational reason the singularity theorems and the topological-censorship results survive the failure of the classical energy conditions: the pointwise weak and strong energy conditions are false for quantum matter, but their averaged QEI remnants are true, state-independent, and strong enough to drive the focusing arguments. The central insight is that a QEI is a positivity statement about a smeared field operator, $ω (Φ^{†} Φ) \geq 0$ , dressed by the geometry only through the sampling tube and the Hadamard subtraction; everything state-dependent enters through a manifestly non-negative term that can only help the bound. This is exactly the structure that connects the flat worldline inequality, the curved difference inequality, and the reference-free absolute inequality: putting these together, they are one positivity argument at three levels of state-independence, with the microlocal spectrum condition of 13.09.03 supplying the smoothness that makes each finite. The construction generalises the single Minkowski vacuum to the entire folium of Hadamard states, and identifies the energy-condition content of general relativity with a Sobolev-type seminorm of the sampling function. The same analytic continuation and modular structure that gives the Unruh temperature in 13.09.09 reappears here as the positive-frequency decomposition behind $Φ_{λ}$ , and the Boulware-state divergence of 13.09.10 is the sharpest physical witness that the pointwise bound it repairs is genuinely false.

Full proof set Master

Proposition (failure of the pointwise weak energy condition). For the free scalar field in Minkowski space there is no constant $C$ with $⟨ : ρ : ⟩_{ω} (0) \geq - C$ for all Hadamard $ω$ .

Proof. Consider the normalised superposition $∣ ψ ⟩ = (1 + ∣ c ∣^{2})^{- 1/2} (∣0 ⟩ + c ∣ 2_{k} ⟩)$ of the vacuum and a two-particle state with both quanta in mode $k$ , $c \in C$ . This is a finite-particle vector, hence Hadamard (it differs from the vacuum by a smooth two-point function). The normal-ordered energy density at the origin has expectation $$ \langle :!\rho!:\rangle_\psi(0) = \frac{1}{1 + |c|^2}\Big( |c|^2,\langle 2_{\mathbf{k}}|:!\rho!:|2_{\mathbf{k}}\rangle + 2,\mathrm{Re}\big[\bar c,\langle 0|:!\rho!:|2_{\mathbf{k}}\rangle\big]\Big). $$ The diagonal term is non-negative and of order $∣ c ∣^{2}$ for large $∣ c ∣$ ; the cross term $⟨ 0∣ : ρ : ∣ 2_{k} ⟩$ is a nonzero complex number $A$ of order $∣ c ∣$ in the prefactor-weighted expression, and by choosing the phase of $c$ so that $\overset{c}{ˉ} A$ is real and negative, the cross term is $- 2∣ c ∣∣ A ∣/ (1 + ∣ c ∣^{2})$ . For small $∣ c ∣$ the cross term dominates the diagonal term (linear versus quadratic in $∣ c ∣$ ), giving $⟨ : ρ : ⟩_{ψ} (0) < 0$ . Scaling: replace $k$ by a wave packet peaked at large $∣ k ∣$ and centred so the cross term scales with the energy of the mode; the achievable negative value grows without bound as the packet is pushed to higher frequency. Hence for any $C$ a choice of $c$ and packet gives $⟨ : ρ : ⟩_{ψ} (0) < - C$ . No pointwise lower bound exists ^{[Epstein-Glaser-Jaffe 1965]}. $□$

Proposition (the positivity at the heart of the QEI). For any state $ω$ , any $g \in C_{c}^{\infty} (R)$ , and any $λ \in R$ , the smeared operator $Φ_{λ} = \int g (τ) e^{iλ τ} (\partial_{τ} ϕ) (γ (τ)) d τ$ satisfies $ω (Φ_{λ}^{†} Φ_{λ}) \geq 0$ , with $$ \omega(\Phi_\lambda^\dagger\Phi_\lambda) = \iint \overline{g(\tau)},g(\tau'),e^{-i\lambda(\tau - \tau')},\partial_\tau\partial_{\tau'}\Lambda(\gamma(\tau),\gamma(\tau')),d\tau,d\tau'. $$

Proof. For any operator $A$ on the GNS Hilbert space of $ω$ with cyclic vector $Ω_{ω}$ , $ω (A^{†} A) = ∥ A Ω_{ω} ∥^{2} \geq 0$ . Take $A = Φ_{λ}$ . Writing $Φ_{λ} = \int g (τ) e^{iλ τ} π (τ) d τ$ with $π (τ) = (\partial_{τ} ϕ) (γ (τ))$ , the adjoint is $Φ_{λ}^{†} = \int \overline{g (τ)} e^{- iλ τ} π (τ) d τ$ since $π$ is the smearing of a real field against a real density and is self-adjoint. Then $$ \omega(\Phi_\lambda^\dagger\Phi_\lambda) = \iint \overline{g(\tau)}e^{-i\lambda\tau},g(\tau')e^{i\lambda\tau'},\omega(\pi(\tau)\pi(\tau')),d\tau,d\tau', $$ and $ω (π (τ) π (τ^{'})) = \partial_{τ} \partial_{τ^{'}} Λ (γ (τ), γ (τ^{'}))$ by definition of the two-point function. The double integral converges because $g$ is compactly supported and smooth and $\partial\partial Λ$ is a distribution of finite order paired against a Schwartz-class kernel on the worldline. Non-negativity is the norm-square statement. $□$

Proposition (vacuum subtraction lowers the bound). Let $S_{ω} [g] = \int ⟨ : ρ : ⟩_{ω} g^{2} d τ$ . Then $S_{ω} [g] = P_{ω} [g] - Q [g]$ where $P_{ω} [g] = \frac{1}{π} \int_{0}^{\infty} ω (Φ_{λ}^{†} Φ_{λ}) d λ \geq 0$ and $Q [g] = \frac{1}{16 π ^{2}} \int (g^{''})^{2} d τ$ is independent of $ω$ . Consequently $S_{ω} [g] \geq - Q [g]$ .

Proof. The normal-ordered density is $⟨ : ρ : ⟩_{ω} = lim_{τ^{'} \to τ} \partial_{τ} \partial_{τ^{'}} (Λ - Λ_{vac})$ , restricting to the $\dot{ϕ}^{2}$ contribution; the gradient contribution doubles the coefficient and is absorbed in the constant. Insert the integral representation $δ (τ - τ^{'}) = \frac{1}{2 π} \int_{R} e^{- iλ (τ - τ^{'})} d λ$ to resolve the coincidence limit, splitting into positive and negative frequency. The positive-frequency half assembles into $\frac{1}{π} \int_{0}^{\infty} ω (Φ_{λ}^{†} Φ_{λ}) d λ$ by Proposition 2; the negative-frequency half, together with the vacuum subtraction, assembles into the $ω$ -independent integral. Evaluating the latter for the massless field in four dimensions: the vacuum derivative two-point function on the inertial worldline is $\partial_{τ} \partial_{τ^{'}} Λ_{vac} = - \frac{1}{4 π ^{2}} \partial_{τ} \partial_{τ^{'}} \frac{1}{( τ - τ ^{'} - i 0 ) ^{2}}$ up to the standard $i 0$ prescription, whose sampled second moment is $\frac{1}{16 π ^{2}} \int λ^{4} ∣ g (λ) ∣^{2} \frac{d λ}{2 π}$ . By the Parseval identity (Exercise 3) this equals $\frac{1}{16 π ^{2}} \int (g^{''})^{2} d τ =: Q [g]$ . Since $P_{ω} [g] \geq 0$ by Proposition 2, $S_{ω} [g] = P_{ω} [g] - Q [g] \geq - Q [g]$ , with $Q [g]$ independent of the state. $□$

Proposition (scaling form / Ford-Roman bound). For the Gaussian sampling $g (τ) = (2 π T^{2})^{- 1/4} e^{- τ^{2} / (4 T^{2})}$ with $\int g^{2} = 1$ , the Fewster-Eveson bound is $- Q [g] = - \frac{3}{128 π ^{2}} \frac{1}{T ^{4}}$ .

Proof. Compute $g^{''}$ . With $g = N e^{- τ^{2} / (4 T^{2})}$ , $N = (2 π T^{2})^{- 1/4}$ , one has $g^{'} = - \frac{τ}{2 T ^{2}} g$ and $g^{''} = (\frac{τ ^{2}}{4 T ^{4}} - \frac{1}{2 T ^{2}}) g$ . Then $$ \int (g'')^2,d\tau = \int \Big(\frac{\tau^2}{4T^4} - \frac{1}{2T^2}\Big)^2 g^2,d\tau = \frac{1}{16T^8}\langle\tau^4\rangle - \frac{1}{4T^6}\langle\tau^2\rangle + \frac{1}{4T^4}, $$ where $⟨ τ^{2 n} ⟩ = \int τ^{2 n} g^{2} d τ$ are the moments of the normalised Gaussian $g^{2}$ of variance $T^{2}$ : $⟨ τ^{2} ⟩ = T^{2}$ , $⟨ τ^{4} ⟩ = 3 T^{4}$ . Substituting, $\int (g^{''})^{2} = \frac{3 T ^{4}}{16 T ^{8}} - \frac{T ^{2}}{4 T ^{6}} + \frac{1}{4 T ^{4}} = \frac{3}{16 T ^{4}} - \frac{1}{4 T ^{4}} + \frac{1}{4 T ^{4}} = \frac{3}{16 T ^{4}}$ . Therefore $Q [g] = \frac{1}{16 π ^{2}} \cdot \frac{3}{16 T ^{4}} = \frac{3}{256 π ^{2} T ^{4}}$ . Hence the bound is $- \frac{3}{256 π ^{2}} T^{- 4}$ , the Ford-Roman $1/ T^{4}$ law with an explicit constant. $□$

Connections Master

Hadamard states via the wave-front-set criterion 13.09.03. The finiteness of every quantum energy inequality is the smoothness of the difference of Hadamard two-point functions, which is the microlocal spectrum condition. The difference QEI subtracts a reference Hadamard state and is finite because $Λ - Λ_{0}$ is smooth; the absolute QEI subtracts the Hadamard parametrix and is finite because $Λ - H$ is smooth. In both cases the wave-front-set criterion of 13.09.03 is the exact instrument that turns a formally divergent coincidence limit into a finite, state-independent bound, so the QEI is the energy-positivity shadow of the Hadamard regularity studied there.
Hartle-Hawking and Unruh states on Schwarzschild 13.09.10. The Boulware state of 13.09.10 supplies the sharpest physical example of unbounded-below energy density: its renormalised stress tensor diverges to negative values in a freely falling frame as the horizon is approached, the most vivid witness that the pointwise weak energy condition fails. The Hartle-Hawking and Unruh states are the Hadamard reference states against which difference QEIs near a black hole are formulated, so this unit provides the inequality bounding how negative the averaged energy density of any admissible black-hole state can be, and the QEI constrains the negative-energy component of the Hawking flux that the hole absorbs as it evaporates.
Unruh effect via the Bisognano-Wichmann theorem 13.09.09. The positive-frequency decomposition behind the smeared operator $Φ_{λ}$ is the same analytic structure that, in 13.09.09, identifies the modular flow of the Rindler wedge with the boost and gives the Unruh temperature. A uniformly accelerated detector in the Minkowski vacuum registers the Unruh bath, and the worldline QEI bounds how negative the time-averaged energy density along its accelerated worldline can be; the averaging timescale in the QEI and the inverse Unruh temperature are tied through the same positive-frequency splitting, so the modular and energy-inequality structures share an analytic core.
Wick polynomials and the renormalised stress tensor 13.09.06. The object the QEI bounds, $⟨ : ρ : ⟩_{ω}$ , is the coincidence limit of the point-split, Hadamard-subtracted stress tensor constructed in 13.09.06. The QEI is therefore an inequality about the very quantity that the Hadamard-subtraction renormalisation programme defines: it takes the renormalised energy density of 13.09.06 and shows that, although it is unbounded below pointwise, it is bounded below in every weighted time-average, closing the loop between the definition of the stress tensor and its positivity properties.

Historical & philosophical context Master

Quantum energy inequalities grew from a thermodynamic worry. In 1965 Epstein, Glaser, and Jaffe ^{[Epstein-Glaser-Jaffe 1965]} proved that the energy density of a Wightman quantum field cannot be a positive operator, so the pointwise weak energy condition that underlies the classical singularity theorems is simply false for quantum matter. Larry Ford ^{[Ford 1978]} asked what survives: he argued that a sustained negative energy flux would let a beam carry away more entropy than energy and violate the second law of thermodynamics, and from that constraint he extracted the first quantum inequality bounding the magnitude and duration of negative energy. Ford and Thomas Roman ^{[Ford-Roman 1995]} developed the worldline form with Lorentzian sampling and connected it to the averaged weak energy condition, and in a parallel line Éanna Flanagan ^{[Flanagan 1997]} found the sharp constant in two dimensions by a variational argument, the first optimal bound.

The decisive technical advance was the recognition that the inequality is a microlocal statement. Christopher Fewster and Simon Eveson ^{[Fewster-Eveson 1998]} gave the clean derivation: a positivity statement about a frequency-shifted smeared field, combined with the Parseval identity, yields the four-dimensional scalar bound $- \frac{1}{16 π ^{2}} \int (g^{''})^{2}$ with no model-dependent input. Fewster ^{[Fewster 2000]} then generalised the derivation to arbitrary globally hyperbolic spacetimes, replacing the Minkowski vacuum by an arbitrary Hadamard reference state and using the smoothness of the difference of Hadamard two-point functions — the microlocal spectrum condition of Radzikowski (1996) underlying 13.09.03 — to guarantee finiteness, producing the difference QEI. Fewster and Calvin Smith ^{[Fewster-Smith 2008]} removed the reference state altogether in the locally covariant framework, giving absolute QEIs bounded by geometric data alone. The applications followed: Ford and Roman bounded traversable wormholes and the Alcubierre warp drive, and Fewster and Gregory Galloway ^{[Fewster-Galloway 2011]} proved a Hawking-type singularity theorem under a smeared energy condition of QEI type, recovering geodesic incompleteness from the averaged remnant of the discredited pointwise condition. The synthesis is collected in Fewster's lecture notes ^{[Fewster 2012]}.

Bibliography Master

@article{EpsteinGlaserJaffe1965,
  author  = {Epstein, Henri and Glaser, Vladimir and Jaffe, Arthur},
  title   = {Nonpositivity of the energy density in quantized field theories},
  journal = {Nuovo Cimento},
  volume  = {36},
  year    = {1965},
  pages   = {1016--1022}
}

@article{Ford1978,
  author  = {Ford, Lawrence H.},
  title   = {Quantum coherence effects and the second law of thermodynamics},
  journal = {Proc. Roy. Soc. A},
  volume  = {364},
  year    = {1978},
  pages   = {227--236}
}

@article{FordRoman1995,
  author  = {Ford, Lawrence H. and Roman, Thomas A.},
  title   = {Averaged energy conditions and quantum inequalities},
  journal = {Phys. Rev. D},
  volume  = {51},
  year    = {1995},
  pages   = {4277--4286}
}

@article{Flanagan1997,
  author  = {Flanagan, {\'E}anna {\'E}.},
  title   = {Quantum inequalities in two-dimensional {M}inkowski spacetime},
  journal = {Phys. Rev. D},
  volume  = {56},
  year    = {1997},
  pages   = {4922--4926}
}

@article{FewsterEveson1998,
  author  = {Fewster, Christopher J. and Eveson, Simon P.},
  title   = {Bounds on negative energy densities in flat spacetime},
  journal = {Phys. Rev. D},
  volume  = {58},
  year    = {1998},
  pages   = {084010}
}

@article{Fewster2000,
  author  = {Fewster, Christopher J.},
  title   = {A general worldline quantum inequality},
  journal = {Class. Quantum Grav.},
  volume  = {17},
  year    = {2000},
  pages   = {1897--1911}
}

@article{FewsterSmith2008,
  author  = {Fewster, Christopher J. and Smith, Calvin J.},
  title   = {Absolute quantum energy inequalities in curved spacetime},
  journal = {Ann. Henri Poincar\'e},
  volume  = {9},
  year    = {2008},
  pages   = {425--455}
}

@article{FordRoman1996,
  author  = {Ford, Lawrence H. and Roman, Thomas A.},
  title   = {Quantum field theory constrains traversable wormhole geometries},
  journal = {Phys. Rev. D},
  volume  = {53},
  year    = {1996},
  pages   = {5496--5507}
}

@article{FewsterGalloway2011,
  author  = {Fewster, Christopher J. and Galloway, Gregory J.},
  title   = {Singularity theorems from weakened energy conditions},
  journal = {Class. Quantum Grav.},
  volume  = {28},
  year    = {2011},
  pages   = {125009}
}

@misc{Fewster2012,
  author = {Fewster, Christopher J.},
  title  = {Lectures on quantum energy inequalities},
  year   = {2012},
  note   = {arXiv:1208.5399 [math-ph]}
}

@article{Radzikowski1996,
  author  = {Radzikowski, Marek J.},
  title   = {Micro-local approach to the {H}adamard condition in quantum field theory on curved space-time},
  journal = {Comm. Math. Phys.},
  volume  = {179},
  year    = {1996},
  pages   = {529--553}
}

Prerequisites

13.09.03
13.09.09
13.09.10

Tier anchors

beginner: Ford and Roman, 'Negative Energy, Wormholes and Warp Drive', *Scientific American* 282 (2000) 46; Fewster, *Lectures on Quantum Energy Inequalities* (arXiv:1208.5399, 2012), §1 (negative energy density and the averaging idea)
intermediate: Fewster and Eveson, *Phys. Rev. D* 58 (1998) 084010 (the microlocal/Fourier derivation of the scalar-field bound); Flanagan, *Phys. Rev. D* 56 (1997) 4922 (the sharp two-dimensional constant); Fewster, *Class. Quantum Grav.* 17 (2000) 1897 (the curved-spacetime difference QEI)
master: Ford, *Proc. Roy. Soc. A* 364 (1978) 227; Ford and Roman, *Phys. Rev. D* 51 (1995) 4277; Fewster and Eveson, *Phys. Rev. D* 58 (1998) 084010; Fewster, *Class. Quantum Grav.* 17 (2000) 1897; Fewster and Smith, *Ann. Henri Poincaré* 9 (2008) 425; Fewster, *Lectures on Quantum Energy Inequalities* (arXiv:1208.5399, 2012)

References

Ford, L. H. — 'Quantum coherence effects and the second law of thermodynamics', Proc. Roy. Soc. A 364 (1978) 227 · The first quantum inequality: a negative energy flux sustained over a time T is bounded, |F| T^2 <~ 1, derived from the requirement that a beam carrying negative energy not violate the second law of thermodynamics; the original motivation for averaged energy bounds.
Ford, L. H. & Roman, T. A. — 'Averaged energy conditions and quantum inequalities', Phys. Rev. D 51 (1995) 4277 · The worldline quantum inequality in Lorentzian-sampling form, integral of <T_00> against a Lorentzian peak of width tau_0 bounded below by -C/tau_0^4 in four dimensions; the bound governing the magnitude-duration trade-off for negative energy.
Flanagan, E. E. — 'Quantum inequalities in two-dimensional Minkowski spacetime', Phys. Rev. D 56 (1997) 4922 · The sharp (optimal) constant in two dimensions: integral of <T_00> g^2 dt >= -(1/24 pi) integral (g')^2 / g^2 dt, obtained by a variational argument and saturated by an explicit one-mode state, fixing the best possible bound.
Fewster, C. J. & Eveson, S. P. — 'Bounds on negative energy densities in flat spacetime', Phys. Rev. D 58 (1998) 084010 · The clean Fourier / point-split derivation of the worldline QEI for the free scalar field in four-dimensional Minkowski space, integral of <:T_00:> g^2 dt >= -(1/16 pi^2) integral (g'')^2 dt for the massless field, with the positivity of the point-split two-point function supplying the bound.
Fewster, C. J. — 'A general worldline quantum inequality', Class. Quantum Grav. 17 (2000) 1897 · The generalisation to arbitrary globally hyperbolic spacetimes: the difference quantum energy inequality bounding <T_00>_omega - <T_00>_omega0 against a reference Hadamard state omega0, finite because the difference of Hadamard two-point functions is smooth.
Fewster, C. J. & Smith, C. J. — 'Absolute quantum energy inequalities in curved spacetime', Ann. Henri Poincaré 9 (2008) 425 · Absolute (state-independent, reference-free) QEIs in curved spacetime, obtained in the locally covariant framework by bounding the Hadamard-subtracted energy density directly against geometric quantities of the sampling tube rather than a reference state.
Epstein, H., Glaser, V. & Jaffe, A. — 'Nonpositivity of the energy density in quantized field theories', Nuovo Cimento 36 (1965) 1016 · The proof that the renormalised energy density operator of a Wightman quantum field is not bounded below and cannot be a positive operator, the failure of the pointwise weak energy condition that QEIs are designed to repair.
Fewster, C. J. — 'Lectures on quantum energy inequalities', arXiv:1208.5399 (2012) · The synthesis lecture notes: the modern statement of worldline, difference, and absolute QEIs, the microlocal proof strategy, the sharp-constant question, and the applications to wormholes, warp drives, and singularity theorems.
Fewster, C. J. & Galloway, G. J. — 'Singularity theorems from weakened energy conditions', Class. Quantum Grav. 28 (2011) 125009 · A Hawking-type singularity theorem proved under a smeared (averaged) energy condition of the kind a QEI supplies, replacing the pointwise strong/weak energy condition; the bridge from QEIs to geodesic incompleteness.

Estimated time

beginner: 18m
intermediate: 46m
master: 76m