Modular Energy and Quantum Energy Inequalities

Relative entropy turns a comparison with a reference state into an inequality between modular energy and entropy. When a modular Hamiltonian is local—as for the Minkowski vacuum on a half-space—nested-region monotonicity can further constrain null energy flux. Curved-spacetime quantum energy inequalities have a different origin and scope: they bound suitably sampled renormalized stress energy for specified fields and Hadamard states. Keeping these statements separate prevents a flat-space ANEC theorem from being misreported as a generic curved-space result.

Required background. Quantum Information supplies relative entropy, data processing, and density-matrix modular Hamiltonians.

Helpful background. Flat-Spacetime Quantum Field Theory provides stress tensors and Rindler wedges; Partial Differential Equations and Green Functions defines Hadamard parametrices and sampling domains; Causality and Global Structure establishes timelike and null curves; Quantum Fields in Curved Spacetime provides Hadamard states and renormalized stress–energy expectation values.

The finite-dimensional bridge is the Gibbs form $\sigma=e^{-K}/Z$. Here $K$ is dimensionless and defined up to an additive constant. In the continuum, all state, region, algebra, stress-tensor subtraction, and sampling choices must be fixed before either “modular energy” or “energy bound” has a definite meaning.

Relative entropy gives the modular inequality

Let $\rho$ be a state and let the faithful reference state be $\sigma=e^{-K}/Z$ on a finite factor. Define

$$\Delta S=S_{\mathrm{vN}}(\rho)-S_{\mathrm{vN}}(\sigma), \qquad \Delta\langle K\rangle=\operatorname{tr}[(\rho-\sigma)K].$$

Using $\log\sigma=-K-\log Z$ gives

$$\begin{aligned} S(\rho\Vert\sigma) &=\operatorname{tr}\rho(\log\rho-\log\sigma)\\ &=-S_{\mathrm{vN}}(\rho)+\langle K\rangle_\rho+\log Z\\ &=\Delta\langle K\rangle-\Delta S. \end{aligned}$$

Positivity therefore yields

$$\Delta S\le\Delta\langle K\rangle.$$

This is the precise energy–information statement: the “energy” is expectation of the reference modular Hamiltonian, not necessarily the physical Hamiltonian. Casini uses this identity to formulate a finite local version of the Bekenstein bound and explains its continuum relative-entropy meaning in Casini 2008, §§2–4. Araki’s algebraic entropy makes the same comparison without assuming a local trace Araki 1976, pp. 809–833.

There is a useful infinitesimal check. Put $\rho=\sigma+\epsilon\delta\rho$ with $\operatorname{tr}\delta\rho=0$ and preserve positivity. Differentiating the entropy at $\epsilon=0$ gives $\delta S=-\operatorname{tr}(\delta\rho\log\sigma)=\operatorname{tr}(\delta\rho K)=\delta\langle K\rangle$. Thus relative entropy starts at second order. This “first law of entanglement” is a linearized identity around the reference state; a finite change obeys the inequality, not the linearized equality.

For an inclusion $\mathcal A(V)\subset\mathcal A(W)$, monotonicity reads

$$S_V(\rho\Vert\sigma)\le S_W(\rho\Vert\sigma).$$

Substituting the modular identity relates changes in modular energy to changes in vacuum-subtracted entropy. Turning this into a stress-tensor inequality additionally requires an explicit local expression for $K$; monotonicity alone does not manufacture one.

Local modular Hamiltonians are exceptional

For the Minkowski vacuum and the right half-space $x^1>0$ at $t=0$, the modular Hamiltonian, up to an irrelevant constant, is

$$K_{\mathrm{half}} =2\pi\int_{x^1>0}d^{d-1}x\;x^1T_{00}(0,\mathbf x).$$

The displayed expression is the one-sided modular-energy form. The standard-pair operator is the full boost generator, region minus complement; its commutant part acts trivially on observables in the right wedge. With $\sigma_t(A)=e^{-itK}Ae^{itK}$, positive $t$ corresponds to boost rapidity $-2\pi t$. In a conformal field theory a ball also has a local modular generator with a conformal weight. Generic states and shapes need not; in particular, the vacuum half-space remains geometric even for a massive or interacting relativistic theory satisfying the wedge theorem Casini 2008, §§3–4.

For an infinitesimal deformation of a Minkowski half-space, the first-order modular Hamiltonian contains integrals of null stress along the future and past Rindler horizons, together with a commutator contribution. Monotonicity of the full modular Hamiltonian—region minus complement—cancels the commutator term and gives the averaged null energy condition along complete Minkowski null generators,

$$\int_{-\infty}^{\infty}d\lambda\, \langle T_{ab}k^ak^b\rangle\ge0.$$

The vacuum supplies the reference modular Hamiltonian; the expectation value in the inequality is taken in the excited-state class for which the construction and null integral are defined. The perturbative parameter is the shape deformation, not necessarily the size of the state excitation. The calculation is developed in Faulkner et al. 2016, §§2–3 and Appendix A, PDF. Their §5 identifies the remaining rigorous step of matching the removed brick-wall regulator to continuum relative entropy. This qualification concerns the proof given there; the calculation does not establish nonnegativity of arbitrary null averages in curved spacetime.

Absolute QEIs on curved spacetime

Now consider a real, minimally coupled Klein–Gordon field of mass $m\ge0$ on a smooth four-dimensional globally hyperbolic spacetime. Fewster and Smith use the opposite metric signature; translated to the site’s $(-,+,+,+)$ convention, a unit timelike tangent obeys $u^au_a=-1$ and the measured energy density is $T_{ab}u^au^b$.

Their general result starts with an $n$-dimensional small timelike sampling domain $\Sigma$, $1\le n\le4$, lying in a globally hyperbolic convex normal neighborhood and carrying a hyperbolic coordinate chart. If $D$ is a real differential operator of order at most one, $f\in C_0^\infty(\Sigma)$ is real, and $\omega$ is Hadamard, then

$$\int_\Sigma d\operatorname{vol}_\Sigma\,f^2(x) \left[D\otimes D\,(\Lambda_\omega-H_2)\right](x,x) \ge-\mathcal B[f,D,H_k]>-\infty,$$

where $k=\max\{n+3,5\}$ and $\mathcal B$ is built from a finite Hadamard-series truncation and the chosen chart. This is the point-split theorem itself Fewster and Smith 2008, §3.1, Theorem 3.1; it should not be replaced by a simpler stress formula without performing the specialization.

For a proper-time-parametrized timelike worldline $\gamma$ whose track lies in such a neighborhood, the tetrad decomposition of the point-split stress tensor gives a bound of the schematic form

$$\int d\tau\,f(\tau)^2 \langle T_{ab}u^au^b\rangle^{\mathrm{ren}}_\omega(\gamma(\tau)) \ge-\mathcal Q[f,\gamma,g,m].$$

Here $\mathcal Q$ includes the Fourier transform of the point-split Hadamard truncation and the finite local stress-renormalization terms. It is independent of $\omega$ within the Hadamard class but depends on the sampling data, local geometry, field parameters, coordinate prescription, and fixed renormalization convention Fewster and Smith 2008, §§3.2.2–3.3.

In the exact benchmark of a massless scalar on an inertial line in four-dimensional Minkowski spacetime, these terms reduce to

$$\int_{-\infty}^{\infty}d\tau\,f^2 \langle T_{00}\rangle_\omega \ge-\frac{1}{16\pi^2} \int_{-\infty}^{\infty}d\tau\,\lvert f''\rvert^2.$$

This check fixes the normalization for the Fourier convention used in the source. It also proves why no pointwise lower bound follows: narrowing the sampling function makes the allowed negative lower bound diverge.

Here the Minkowski vacuum stress is normalized to zero and $f$ is real, smooth, and compactly supported. A change of the finite stress-renormalization prescription must shift the bound by the same sampled local term as the left-hand side. The numerical coefficient is therefore not permission to ignore the subtraction convention.

QEI and ANEC are different claims

Result	Geometry and field	Average	State scope	Limitation
Curved-spacetime absolute QEI	Four-dimensional globally hyperbolic geometry; free minimally coupled scalar	Compactly sampled timelike energy or specified small-domain point split quantity	Hadamard states	Lower bound depends on sampling, geometry, renormalization, and chart prescription
Deformed-half-space ANEC	Minkowski QFT around the vacuum half-space	Complete null-generator integral	States for which the modular argument is defined	First-order shape calculation; not a generic curved result
Modular energy inequality	Fixed algebra and reference state	Difference of reference modular-energy expectations	Normal state pair with defined relative entropy	Becomes local stress energy only when the modular generator is known locally

A QEI permits negative energy but limits its magnitude and duration after timelike sampling. The flat ANEC instead asserts nonnegativity of a complete null integral in a special modular setting. Neither statement can be obtained from the other by replacing a timelike tangent with a null vector or by shrinking $f$ to a delta function.

Scaling check and solved diagnostic

Let $f_{\tau_0}(\tau)=\tau_0^{-1/2}f(\tau/\tau_0)$ in the Minkowski benchmark. Determine the scaling of the lower bound as $\tau_0\to0$. Does the result prove pointwise positivity or a curved-spacetime ANEC?

Solution

Two derivatives give $f_{\tau_0}''=\tau_0^{-5/2}f''(\tau/\tau_0)$. Changing variables in the integral therefore yields $\int d\tau\,\lvert f_{\tau_0}''\rvert^2=\tau_0^{-4}\int ds\,\lvert f''(s)\rvert^2$. This matches the length dimension $L^{-4}$ of a four-dimensional energy density and checks the bound. Because the lower limit tends to $-\infty$ as the sample narrows, no pointwise lower bound follows. The calculation is timelike and flat; it neither derives null ANEC nor extends the deformed-half-space theorem to curved spacetime.

References

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11, no. 3 (1976): 809–833. DOI.
• Casini, Horacio. “Relative Entropy and the Bekenstein Bound.” Classical and Quantum Gravity 25, no. 20 (2008): 205021. DOI; Open PDF.
• Faulkner, Thomas, Robert G. Leigh, Onkar Parrikar, and Huajia Wang. “Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition.” Journal of High Energy Physics 09 (2016): 038. DOI; Open PDF.
• Fewster, Christopher J., and Calvin J. Smith. “Absolute Quantum Energy Inequalities in Curved Spacetime.” Annales Henri Poincaré 9 (2008): 425–455. arXiv.