Restricted semiclassical entropy dictionary

Fix an effective bulk model on a smooth Lorentzian spacetime $(M,g)$ of dimension $d+1$, with $d\ge 2$ and signature $(-,+,\dots ,+)$, together with a time orientation, meaning a continuous choice of one component of the timelike cone at each point. Here AdS means anti-de Sitter. In this item, asymptotically locally AdS means that $M$ is the interior of a smooth manifold with boundary $\bar{M}$; write $\mathcal{I}:=\mathrm{\partial }\bar{M}$. There is a dimensionless smooth defining function $z$ with $z>0$ on $M$, $z=0$ and $dz\mathrm{\ne }0$ on $\mathcal{I}$, such that $\bar{g}=z^{2}g$ extends smoothly to a nondegenerate Lorentzian metric of the same signature on $\bar{M}$ whose induced boundary metric $\gamma :=\bar{g}{\mathrm{\mid }}_{T\mathcal{I}}$ is Lorentzian. The chosen time orientation extends continuously to $(\bar{M},\bar{g})$ and induces one on $(\mathcal{I},\gamma )$. The model fixes its AdS length $L>0$ in metres, the asymptotic AdS boundary condition, bulk fields, semiclassical approximation, and conformal-boundary data; semiclassical here means that the bulk geometry is treated classically while the bulk entropy is computed from regulated quantum-state data on that geometry. No bulk field equation is inferred here.

Choose a smooth embedded hypersurface ${\mathcal{C}}_{\mathrm{\partial }}\subset \mathcal{I}$ whose induced metric is positive definite and which meets every inextendible future-directed boundary causal curve at exactly one parameter value. A causal curve in $(\bar{M},\bar{g})$ or $(\mathcal{I},\gamma )$ is a locally Lipschitz map from a real interval with a nonzero future-directed tangent almost everywhere and nonpositive squared tangent norm. Inextendibility is tested after an order-preserving locally bi-Lipschitz reparameterization $h:J\to J_0$: a curve $\eta :J\to M_0$ is inextendible exactly when there are no such $h$, larger interval $\tilde{J}⊋J_0$, and causal curve $\tilde{\eta }:\tilde{J}\to M_0$ with $\tilde{\eta }\circ h=\eta$, where $M_0$ is the bulk or boundary spacetime in question. Each bulk candidate slice ${\mathrm{\Sigma }}_X$ is a smooth embedded $d$-dimensional hypersurface in $(\bar{M},\bar{g})$ with positive-definite induced metric and $\mathrm{\partial }{\mathrm{\Sigma }}_X={\mathcal{C}}_{\mathrm{\partial }}$, and every inextendible causal curve in $\bar{M}$ meets it at exactly one parameter value. This is a boundary-inclusive convention: curves may meet or run along $\mathcal{I}$. The existence of these slices is supplied model data; no interior-only Cauchy existence is inferred. Choose a relatively compact open region $A\subset {\mathcal{C}}_{\mathrm{\partial }}$ (its closure is compact) with smooth boundary $\mathrm{\partial }A$. The regulated boundary Hilbert space and its finite-dimensional subspace ${\mathcal{C}}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$ are model data; regulated code subspace means this specified finite-dimensional subspace, and every boundary density matrix in the chosen state family is supported on it. A boundary density matrix ${\rho }_{\mathrm{\partial }}$ is a positive trace-class endomorphism of trace one on the regulated boundary Hilbert space, with range in ${\mathcal{C}}_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$. For such a matrix ${\rho }_{\mathrm{\partial }}$, define its exact finite-cutoff entropy by $S_{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}({\rho }_{\mathrm{\partial }}):=-k_B\mathrm{tr}({\rho }_{\mathrm{\partial }}\log {\rho }_{\mathrm{\partial }})$, using natural logarithms and $0\log 0:=0$. An admissible bulk candidate $X$ is a smooth embedded spacelike $(d-1)$-dimensional submanifold with boundary of $\bar{M}$, with $X\cap \mathcal{I}=\mathrm{\partial }X=\mathrm{\partial }A$, $X\setminus \mathrm{\partial }X\subset M$, and $X\subseteq {\mathrm{\Sigma }}_X$; its induced metric is positive definite and its supplied region $R_X\subseteq {\mathrm{\Sigma }}_X$ and chains satisfy $[x]=[a_X]$ in the ambient locally finite mod-two relative homology group defined by the candidate-surface model. A quantum-stationary candidate (also called a quantum extremal surface) is an admissible $X$ at which the first variation of the scheme-fixed generalized entropy exists and vanishes for every family $F$ specified in the preceding candidate-surface definition: $F$ is jointly smooth, each $F_{ϵ}$ is an admissible embedding with the same anchor and memberwise relative-chain homology condition, and its initial velocity $V$ satisfies $g(V,w)=0$ for every interior tangent vector $w\in TX$ and extends smoothly by zero at the anchor. Let ${\mathcal{Q}}_A$ be the prescribed class of such candidates for the chosen boundary state. Use SI units: $k_B$ is in $\mathrm{J}{\mathrm{K}}^{-1}$, $c$ is in $\mathrm{m}{\mathrm{s}}^{-1}$, and $\mathrm{\hslash }$ is in $\mathrm{J}\mathrm{s}$. Write $G_{{\mathrm{\Lambda }}_n}>0$ for the model-supplied Newton coupling at regulator label ${\mathrm{\Lambda }}_n$, in ${\mathrm{m}}^{\mathrm{d}}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ for this $(d+1)$-dimensional bulk. When the chosen Einstein–Hilbert branch uses the area term, use the same label-dependent physical-metric area ${\mathrm{Area}}_{{\mathrm{\Lambda }}_n}[X]={\int }_{X^{\circ }}{\chi }_{{\mathrm{\Lambda }}_n,X}d{\mathrm{vol}}_{g{\mathrm{\mid }}_{T{X}^{\circ }}}$ defined by the candidate-surface model, with its explicit Countable Choice integration premise and its fixed finite cutoff data. This term is $k_B{c}^3{\mathrm{Area}}_{{\mathrm{\Lambda }}_n}[X]\mathrm{/}(4G_{{\mathrm{\Lambda }}_n}\mathrm{\hslash })$, and all additional finite terms remain in the prescribed $S_{\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r},n}$. The cutoff area is in ${\mathrm{m}}^{\mathrm{d}-1}$, so $[c^3{\mathrm{Area}}_{{\mathrm{\Lambda }}_n}[X]]={\mathrm{m}}^{\mathrm{d}+2}{\mathrm{s}}^{-3}=[G_{{\mathrm{\Lambda }}_n}\mathrm{\hslash }]$ and the factor multiplying $k_B$ is dimensionless. The metric, Borel cutoff, label dependence and subtraction convention are thus fixed before the combined renormalized limit is tested; an unregulated area at the conformal boundary is not asserted finite. For $d=3$ these are the usual four-dimensional SI units. Adopt

$S_{\mathrm{\partial }}^{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}(A):={\min }_{X\in {\mathcal{Q}}_A}{S}_{\mathrm{g}\mathrm{e}\mathrm{n}}^{\mathrm{r}\mathrm{e}\mathrm{n}}[X]$

as the model's semiclassical boundary-entropy prediction only when ${\mathcal{Q}}_A$ is nonempty, the values are finite, and the minimum is attained. It does not identify this prediction with $S_{\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}}({\rho }_{\mathrm{\partial }})$. A path-integral saddle here is a stationary configuration of the chosen gravitational action; saddle dominance means that its contribution controls the specified path-integral approximation over competing configurations. Candidate existence, finite convergence, and dominance of the selected saddle are separate premises. The minimum rule is an adopted physical premise, not a consequence of the finite code entropy identity or the stationary Noether first law. It does not assert that the model describes nature, a universal RT/QES identity, or a continuum factorization of quantum fields.

Sector entropy and relative entropy identities

Regulated generalized entropy for AdS candidate surfaces

Specified regulated generalized entropy

Stationary first law from the covariant boundary identity

Daniel Harlow, Jerusalem Lectures on Black Holes and Quantum Information

Ahmed Almheiri et al., The entropy of Hawking radiation

The finite-sector entropy proposition supplies an exact finite matrix identity in nats; multiplying by $k_B$ converts that entropy convention to physical units, but it does not identify a sector term with geometric area. The Noether first-law supplier gives an Einstein–Hilbert area coefficient only under its stationary hypotheses. Neither result derives this adopted dictionary for a general asymptotically AdS model.