Generalized Entropy for Candidate Surfaces

A candidate surface acquires a generalized entropy only together with a choice of side. For a codimension-two surface $X$, that choice specifies the bulk region whose quantum algebra is entropically paired with the geometric term. The central object is therefore not “area plus some entropy,” but a renormalized functional of $(X,a_X)$ in a specified state and semiclassical theory.

Required background. Quantum Information supplies reduced states, von Neumann entropy, and the distinction between a tensor factor and an observable algebra.

Helpful background. Semiclassical Gravity supplies renormalization and backreaction; Quantum Fields in Curved Spacetime supplies regulated local field algebras and Hadamard states; Black-Hole Geometry and Mechanics supplies codimension-two surfaces, horizons, and causal regions.

Candidate surfaces and their bulk side

In four spacetime dimensions, “codimension two” means a two-dimensional surface: for example, a sphere on a spatial slice. Cutting the slice along that sphere creates two possible sides. Entropy counts correlations lost when one side is ignored, which is why specifying the surface alone is insufficient. The surface need not coincide with an event horizon.

Fix a semiclassical state $\rho$ of gravity plus quantum fields, a nongravitating boundary or bath region $A$, and an admissible codimension-two surface $X$. Choose an achronal bulk region $a_X$ whose boundary is completed by $X$ and $A$ according to the anchoring and homology rule of the problem. The quantum term is the von Neumann entropy of the algebra assigned to $a_X$,

$$S_{\mathrm{bulk}}(a_X) =-\operatorname{Tr}\rho_{a_X}\log\rho_{a_X},$$

when a regulator makes the tensor-factor notation meaningful. In continuum QFT the local algebra is generally not a finite tensor factor, so one must instead specify a cutoff, split inclusion, or subtraction that makes the entropy difference finite. Natural logarithms are used.

For Einstein gravity, the leading renormalized functional is

$$S_{\mathrm{gen}}[X;A] =\frac{\operatorname{Area}(X)}{4G_{\mathrm{ren}}} +S_{\mathrm{bulk,reg}}(a_X) +S_{\mathrm{ct}}[X].$$

The area is evaluated in the semiclassical geometry associated with $\rho$. Here $G_{\mathrm{ren}}$ denotes the chosen finite renormalized coupling, while $S_{\mathrm{ct}}$ includes the subtractions that cancel the divergences still present in $S_{\mathrm{bulk,reg}}$. One may equivalently combine bare gravitational couplings with regulated matter entropy, but must not include the same coupling renormalization twice. In a theory with higher derivatives, the first term must be replaced by the corresponding Wald-like entropy together with the appropriate extrinsic-curvature corrections. The first quantum correction includes bulk entropy, the area change from backreaction, and local gravitational terms Faulkner, Lewkowycz, and Maldacena 2013, §2.2; quantum extremal surfaces are defined in Engelhardt and Wall 2015, §§2–3.

The notation $S_{\mathrm{gen}}[X;A]$ keeps the anchoring data visible. The same geometric set $X$ can bound different regions in a spacetime with nontrivial topology, and those choices need not have equal bulk entropy.

Why the sum is renormalized

Entanglement across $X$ has local ultraviolet divergences. With a proper-distance cutoff $\epsilon$ in four dimensions, its leading structure is schematically

$$S_{\mathrm{bulk,reg}} =c_2\frac{\operatorname{Area}(X)}{\epsilon^2} +c_{\log}\log(\mu\epsilon) +S_{\mathrm{finite}}+\cdots .$$

The coefficients depend on field content and regulator. The bare gravitational entropy contains compensating coupling dependence. Changing the cutoff moves contributions between the geometric and matter terms while leaving the renormalized sum invariant to the calculated order. Thus neither the regulated area coefficient nor $S_{\mathrm{bulk,reg}}$ alone is the physical candidate entropy.

This cancellation gives an important scheme check. If a finite local counterterm shifts $1/G_{\mathrm{ren}}$ so that the geometric contribution changes by $\Delta S_X$, the definition of the renormalized matter-plus-counterterm contribution must change by $-\Delta S_X$. A calculation that changes only one term has compared two schemes rather than two surfaces.

From a candidate to an entropy prescription

Three operations must not be conflated. First, enumerate surfaces satisfying the boundary, anchoring, and homology conditions. Second, extremize $S_{\mathrm{gen}}$ under local normal deformations. Third, minimize among the resulting quantum extremal surfaces. The first page supplies candidates; the quantum extremality condition and homology rule supply the later filters.

The state and algebra also matter. In a bath model, $A$ has an ordinary nongravitating algebra and $a_X$ may include a gravitating island. In a closed gravitating universe, a naïve spatial tensor factor is obstructed by gauge constraints and gravitational dressing. This page does not derive a microscopic factorization for quantum gravity; it defines the semiclassical observable once a valid algebraic setup has been supplied. The entropy formula and its radiation application are explained in Almheiri et al. 2021, §§6 and 8.

Independent checks and failure modes

Dimensions. With $c=\hbar=1$, $G$ has dimensions of area in four spacetime dimensions, so $\operatorname{Area}(X)/(4G)$ is dimensionless like an entropy.

Classical limit. If the shape derivative of $S_{\mathrm{bulk}}$ is negligible compared with the area variation, extremizing $S_{\mathrm{gen}}$ recovers a classical extremal-area condition. This is a limit of the stationarity equation, not permission to discard the quantum entropy after saddles become nearly degenerate.

Purity check. If a regulated global bulk state is pure and $a_X$ has complement $\bar a_X$, then $S_{\mathrm{bulk}}(a_X)=S_{\mathrm{bulk}}(\bar a_X)$. The complete generalized entropies agree only when the geometric and counterterm assignments correspond to the same surface and admissible complementary problem.

The construction fails as stated if no finite renormalized entropy exists, if $X$ is not admissible, if the semiclassical expansion breaks down at the surface, or if the chosen “region” has no declared gauge-invariant algebra. It also does not determine which candidate dominates.

Exercises

Suppose two regulators give $S_{\mathrm{bulk}}^{(2)}=S_{\mathrm{bulk}}^{(1)}+\lambda\operatorname{Area}(X)$ while describing the same physics. What change is required in the geometric-plus-counterterm contribution, and what would it mean if the total generalized entropy changed?

Solution

The gravitational contribution must shift by $-\lambda\operatorname{Area}(X)$, through the corresponding finite change of the renormalized coupling or surface counterterm. Then $S_{\mathrm{gen}}^{(2)}=S_{\mathrm{gen}}^{(1)}$ to the retained order. If the sum changed, the two calculations would define different renormalized theories or would have omitted a compensating term; the change could not be interpreted as a physical entropy difference between identical candidates.

References

• Almheiri et al. 2021, Ahmed Almheiri, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini, “The Entropy of Hawking Radiation,” Reviews of Modern Physics 93 (2021) 035002.
• Engelhardt and Wall 2015, Netta Engelhardt and Aron C. Wall, “Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime,” Journal of High Energy Physics 01 (2015) 073.
• Faulkner, Lewkowycz, and Maldacena 2013, Thomas Faulkner, Aitor Lewkowycz, and Juan Maldacena, “Quantum Corrections to Holographic Entanglement Entropy,” Journal of High Energy Physics 11 (2013) 074.