Generalized Entropy and Quantum Extremal Surfaces

Entropy near a horizon is not one quantity. It may mean a regulated von Neumann entropy of matter, a finite relative entropy of states on a local algebra, a thermodynamic coarse graining, a Noether-charge term, or their jointly renormalized sum. This chapter identifies the data that make each statement meaningful and stops before semiclassical stationarity is promoted to a microscopic or holographic entropy formula.

Helpful background. Replica branched geometries supplies the integer-replica construction; entropy counterterms supplies the regulator-to-continuum logic; relative entropy in QFT supplies the algebraic finite quantity; and the renormalized stress tensor supplies the local backreaction source.

From QFT entropy to a gravitational surface functional

For a codimension-two surface $X$, a state $\rho$, and a declared accessible algebra $\mathcal A_{\rm out}(X)$, the semiclassical object is

$$S_{\rm gen}^{\rm ren}[X;\rho,\mathcal A] =S_{\rm grav}^{\rm ren}[X]+S_{\rm out}^{\rm ren}[X;\rho,\mathcal A].$$

Neither term is generally regulator independent by itself. Matter divergences localized on $X$ renormalize Newton’s constant and higher-curvature couplings; gauge constraints additionally require an algebra and edge prescription. A meaningful comparison holds the state, surface class, algebra, regulator, and renormalized gravitational couplings in one consistent scheme.

The geometric term is derived from the complete diffeomorphism-invariant action, with the stationary Noether construction stated by Iyer and Wald 1994, §VI, pp. 858–861. Extremizing the sum is a semiclassical definition; its use as a holographic entropy prescription is a further proposal Engelhardt and Wall 2015, §§2–3.

For a Euclidean replica family with $Z_n=e^{-I_n}$, the entropy convention used throughout is

$$S=(1-n\partial_n)\ln Z_n\big|_{n=1} =(n\partial_n-1)I_n\big|_{n=1}.$$

The sign and the familiar $+A/(4G_{\rm ren})$ must be derived from the full continued action, including its boundary and defect terms, rather than imported from a different Lorentzian action convention. Integer values of $n$ do not by themselves determine a unique derivative at $n=1$.

The structure map should be read from left to right: specify the entropy object and algebra, renormalize matter and geometry together, then ask whether a replica, horizon-law, or surface-stationarity statement is licensed.

The chapter’s logical structure: entropy data precede joint matter–gravity renormalization, and only then support a typed conclusion. Schematic; not to scale.

Routes through the chapter

1. Black-hole thermodynamics at the QFT interface separates Hawking temperature, state-dependent flux, and geometric entropy.
2. Fine-grained, coarse-grained, and algebraic entropy fixes the entropy object before calculations begin.
3. Horizon entanglement entropy derives the local ultraviolet structure across a smooth horizon.
4. Relative entropy and modular horizon laws develops finite algebraic comparisons and known modular generators.
5. Generalized entropy and UV renormalization combines matter and gravitational counterterms.
6. Species, gauge edges, and contact terms makes the algebra and edge prescription explicit.
7. Noether-charge entropy and higher-curvature terms derives stationary geometric entropy from the action.
8. Replica constructions on fixed and semiclassical backgrounds states the boundary, saddle, symmetry, and continuation assumptions.
9. Conical entropy and effective-action variations tracks bulk, surface, and contact contributions.
10. Semiclassical first laws and physical-process variations distinguishes equilibrium variation from flux-driven change.
11. The generalized second law states precisely which causal-horizon results are proved.
12. Quantum extremal surfaces defines renormalized stationarity and perturbative displacement.
13. Semiclassical black-hole information identifies what evaporation arguments require beyond this chapter.

Domain and failure conditions

This table is the canonical chapter comparison. Each leaf links back here and then gives its own local assumptions.

Object or statement	State and algebra	Regulator and geometry	Licensed conclusion	Status	Decisive failure or handoff
Hawking thermodynamics	Declared black-hole state; exterior observables	Stationary nonextremal Killing horizon; renormalized stress tensor	$T_H=\kappa/(2\pi)$, flux after greybody scattering, stationary first-law data	QFT calculation plus classical geometry	A rapidly evolving trapping horizon lacks the required symmetry
Fine/coarse entropy	State on an algebra; coarse constraints specified	UV regulator only for subregion entropy	Distinguish von Neumann, relative, and thermodynamic entropy	Definition/algebraic theorem	Type-III local algebras defeat naive tensor-factor entropy arithmetic
Horizon entanglement	Hadamard-like short-distance state; chosen algebra	Smooth cut; explicit UV regulator	Local area and curvature divergences; finite differences when controlled	QFT calculation	Regulator-dependent constants are not observables
Relative/modular law	Normal states on one algebra; modular domains valid	Often regulator free	Positivity, monotonicity, $\delta S=\delta\langle K\rangle$ near a reference state	Operator-algebra theorem	A geometric $K$ is unavailable for a generic region/state
Renormalized $S_{\rm gen}$	Matter state and algebra fixed	Matter and gravitational counterterms matched	Finite scheme-consistent generalized-entropy differences/variations	Semiclassical EFT construction	Shifting only one sector produces spurious scheme dependence
Gauge/edge entropy	Center or edge prescription declared	Gauge-compatible regulator and contact terms	Comparable entropy within that prescription	Algebra-dependent construction	Changing the algebra while holding counterterms fixed changes the question
Noether-charge entropy	Classical stationary solution	Diffeomorphism-invariant renormalized action; bifurcate Killing surface	Wald entropy and stationary first law	Classical theorem/semiclassical coupling input	Dynamical cuts carry ambiguity and extra flux terms
Replica entropy	State preparation and replica boundary data	Integer covers; action, defects, and saddles specified	$S=(n\partial_n-1)I_n\rvert_{n=1}$ if continuation exists	Path-integral calculation	Integer data need not select the derivative at $n=1$
Conical effective action	Matter boundary conditions and zero modes fixed	Smoothed cone or distributional prescription	Bulk/surface/contact contribution to entropy	One-loop calculation	Cone and UV limits may not commute
First law	Perturbation of a stationary reference state	Linear order; suitable horizon and asymptotics	Equality among charge, flux, area, and modular variations	Perturbative theorem	Shear, expansion, and quadratic entropy production matter beyond linear order
Generalized second law	Horizon algebra and regular state satisfy stated axioms	Causal horizon; joint renormalization; semiclassical regime	Monotonic $S_{\rm gen}$ for the theorem’s slices	Theorem in specified field classes	Apparent horizons and generic interacting theories are not automatic corollaries
Quantum extremal surface	$\rho$ and $\mathcal A_{\rm out}(X)$ declared	Admissible anchored/homologous surface class; renormalized functional	$\delta_XS_{\rm gen}=0$ and controlled displacement	Semiclassical definition/calculation	Stationarity alone neither minimizes nor supplies an island formula
Information claim	Exterior/radiation algebra and operational task fixed	Evaporation approximation and gravity input stated	A scoped entropy or recovery statement	Model-dependent/open	Mean Hawking flux alone cannot establish a Page curve or microscopic unitarity

The failure map emphasizes the three most common category errors: changing an algebra without changing the entropy question, renormalizing only the matter term, and promoting a stationary surface to a fine-grained entropy prescription.

Within its explicit causal-horizon and horizon-algebra hypotheses, relative-entropy monotonicity does prove generalized-entropy increase Wall 2012, §§2–5; the table does not extend that theorem to apparent horizons or arbitrary gravity theories.

The principal failure modes and their correct downgrades: redefine the question, restore joint renormalization, or hand the claim to a theory with the required gravitational input. Schematic; not to scale.

References

• Engelhardt, N., and A. C. Wall, “Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime,” Journal of High Energy Physics 2015, 073 (2015), doi:10.1007/JHEP01(2015)073.
• Iyer, V., and R. M. Wald, “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy,” Physical Review D 50, 846–864 (1994), doi:10.1103/PhysRevD.50.846.
• Wall, A. C., “A Proof of the Generalized Second Law for Rapidly Changing Fields and Arbitrary Horizon Slices,” Physical Review D 85, 104049 (2012), doi:10.1103/PhysRevD.85.104049.