Renormalization of Generalized Entropy

Exterior entanglement entropy is ultraviolet divergent because correlations straddle the entangling surface at arbitrarily short distance. That fact does not make the generalized entropy ill-defined. With one regulator and one effective action used throughout, minimally coupled matter allows the divergent local terms in exterior entropy to be absorbed into Newton’s constant and the higher-curvature couplings. Nonminimal matter requires additional horizon terms, described below. For a fixed physical effective theory and a specified exterior algebra, their sum is finite and independent of the subtraction convention through the calculated order. Neither regulated summand has those properties by itself. Finite physical gravitational couplings still have to be specified.

Required background. Quantum Information supplies density matrices, von Neumann entropy, and the dependence of entropy on a chosen subsystem or operator algebra.

Helpful background. Quantum Fields in Curved Spacetime supplies short-distance correlations and curved-space regulators; Einstein Equations supplies the effective gravitational couplings; Black-Hole Geometry and Mechanics supplies horizon cuts; and Semiclassical Gravity supplies the loop expansion in which matter entropy and geometric counterterms are combined.

Local ultraviolet structure at a horizon cut

Let $\Sigma$ be a smooth spacelike codimension-two cut and regulate proper distances shorter than $\epsilon$. The finite quantity to be constructed is

$$S_{\mathrm{gen}}[\Sigma] =S_{\mathrm{grav}}^{\mathrm{ren}}[\Sigma] +S_{\mathrm{out}}^{\mathrm{ren}}[\Sigma].$$

The displayed geometric-plus-von-Neumann split applies directly to minimally coupled matter. For a scalar with action term $-\frac12\xi\int\sqrt{-g}\,R\phi^2$, the total horizon entropy also includes the matter-dependent Wald contribution

$$S_{\xi}=-2\pi\xi\int_\Sigma d^2y\,\sqrt h\, \langle\phi^2\rangle.$$

The composite operator and its couplings must be renormalized consistently. Ordinary von Neumann entropy alone does not reproduce the nonminimal contribution to Newton’s-constant renormalization, even on a Ricci-flat background. The extra term follows by differentiating the curvature coupling in the Wald entropy formula; it is not an additional statistical entropy of the exterior. See Solodukhin 2011, §§3.16–3.17, Eqs. (141)–(144), (157), PDF. Gauge theories likewise require a specified exterior algebra and compatible edge contributions. The simple cancellation below is developed for minimally coupled matter with these further complications absent.

Before subtraction, the local ultraviolet terms take the schematic form

$$\begin{aligned} S_{\mathrm{out}}(\epsilon) &=c_0N\,\frac{A(\Sigma)}{\epsilon^2} +\log(\mu\epsilon)\int_\Sigma d^2y\,\sqrt h\,\mathcal I_2 +S_{\mathrm{out}}^{\mathrm{finite}}+\cdots,\\ \mathcal I_2 &=c_1R+c_2R_\perp+c_3R_{\perp\perp} +c_4K^i_{ab}K_i^{ab}+c_5K^iK_i . \end{aligned}$$

This is a representative four-dimensional massless-field expansion. Here $N$ counts species schematically; the coefficients depend on spin, coupling, and regulator. Let $n_0$ be a unit timelike normal and $n_1$ a unit spacelike normal, with normal metric $\eta_{ij}=\operatorname{diag}(-1,1)$. Define

$$\begin{aligned} P_\perp^{\mu\nu}&=\eta^{ij}n_i^\mu n_j^\nu,\\ R_\perp&=R_{\mu\nu}P_\perp^{\mu\nu},\qquad R_{\perp\perp}=R_{\mu\nu\rho\sigma} P_\perp^{\mu\rho}P_\perp^{\nu\sigma}. \end{aligned}$$

Use $K^i_{ab}=h_a{}^\mu h_b{}^\nu\nabla_\mu n^i_\nu$ and $K^i=h^{ab}K^i_{ab}$, contracting normal indices with $\eta_{ij}$. This includes the independent normal projection of the Riemann tensor and keeps the timelike-normal sign explicit. Intrinsic surface curvature may replace one combination by the Gauss relation; massive fields can also give $m^2A\log(\mu\epsilon)$ terms. The Euclidean formula uses two positive-norm normals. Its invariant continuation is the expression above, rather than an unsigned sum over Lorentzian normals. Solodukhin 2011, §6.1, Eqs. (276)–(279), PDF gives the curvature and extrinsic-curvature structure.

The expansion separates two ideas that are often conflated. The ultraviolet divergence is local and state independent at leading order, whereas the finite part can carry long-range, state-dependent information. Changing the quantum state while holding its Hadamard short-distance structure fixed therefore changes finite entropy differences without changing the required local counterterms.

Gravitational couplings cancel the divergences

Use the same cutoff to define the one-loop matter effective action,

$$I_{\mathrm{eff}}(\epsilon) =\int d^4x\,\sqrt{-g}\left[ \frac{R}{16\pi G_{\mathrm{bare}}(\epsilon)} +a_{\mathrm{bare}}(\epsilon)R^2 +b_{\mathrm{bare}}(\epsilon)R_{\mu\nu}R^{\mu\nu} +\cdots\right]+I_{\mathrm{nonlocal}} .$$

The area term in gravitational entropy is $A/(4G_{\mathrm{bare}})$. The same heat-kernel coefficients that produce the area divergence of exterior entropy renormalize Newton’s constant, while logarithmic entropy divergences accompany renormalization of $a$, $b$, and the other local curvature couplings. The entropy associated with those higher-derivative terms is the corresponding Noether-charge entropy on a stationary horizon, not merely another multiple of area. The covariant construction is given in Iyer and Wald 1994, §§3–6, PDF.

Thus the regulated sum may be reorganized as

$$\frac{A}{4G_{\mathrm{bare}}(\epsilon)} +S_{\mathrm{out}}(\epsilon) +S_{\mathrm{higher}}^{\mathrm{bare}}(\epsilon) =S_{\mathrm{grav}}^{\mathrm{ren}}(\mu) +S_{\mathrm{out}}^{\mathrm{ren}}(\mu).$$

Each term on the right can depend on renormalization scale and scheme, but their sum cannot. A change of $\mu$ shifts a local contribution from the matter entropy into the gravitational entropy, or conversely. It does not alter $S_{\mathrm{gen}}$ evaluated in the same physical state and geometry.

Finite counterterms and scheme independence

Renormalization removes divergences but does not select a unique split. Moving a finite local term $\Delta I$ between the gravitational and matter parts while keeping the full effective action fixed changes the geometric entropy by $\Delta S_{\mathrm{grav}}$. The definition of the renormalized exterior entropy then changes by $-\Delta S_{\mathrm{grav}}$, so

$$\Delta S_{\mathrm{gen}} =\Delta S_{\mathrm{grav}}+\Delta S_{\mathrm{out}}=0.$$

This cancellation is why a claim about the bare area term alone is not a renormalized physical statement. Actually changing a measured higher-curvature coupling changes the theory and can change its entropy; that is not a subtraction-scheme transformation. Bousso, Fisher, Leichenauer, and Wall make the analogous counterterm cancellation explicit when formulating a cutoff-independent quantum focusing condition Bousso et al. 2016, Appendix A, PDF.

Entropy differences and relative entropy provide an independent check. For sufficiently regular Hadamard states on the same region, the shared local ultraviolet terms cancel in exterior entropy differences. One must also control infrared volume divergences and require the relevant entropy or relative-entropy difference to exist; the Hadamard condition alone does not ensure that. The purely geometric counterterms are also state independent at fixed geometry and cancel in the difference. This reproduces the finite state-dependent content without ever assigning an invariant meaning to either regulated summand separately. The matter-dependent $S_\xi$ in a nonminimally coupled theory must instead be included when taking that difference.

Regime and limitations

The argument is an effective-field-theory statement through the loop order at which all local counterterms have been included. Curvatures and extrinsic curvatures must remain small compared with the cutoff scale, the cut must be smooth on that scale, and the state must have the standard Hadamard short-distance form. The first omitted terms include higher loops and higher-dimension local operators suppressed by the ultraviolet completion scale. Away from a stationary bifurcation surface, the geometric entropy generally needs the extrinsic-curvature terms of the replica prescription; the stationary Wald formula alone is insufficient.

A hard cutoff that violates covariance can obscure the matching, and gauge fields require care with edge modes and contact terms. Singular entangling surfaces introduce corner contributions not displayed above. Most importantly, renormalized generalized entropy is an effective semiclassical observable; this cancellation does not derive a microscopic Hilbert-space factorization in quantum gravity.

Exercise

Suppose $N$ minimally coupled fields give $S_{\mathrm{out}}(\epsilon)=N\gamma A/\epsilon^2+S_{\mathrm{fin}}$ and no curvature-squared term contributes for the chosen stationary cut. Determine the relation between bare and renormalized Newton constants required for $S_{\mathrm{gen}}$ to be finite, and verify that the result is unchanged if a finite amount $\alpha A$ is moved between the two summands.

Solution

Write the regulated generalized entropy as

$$S_{\mathrm{gen}}(\epsilon) =A\left[\frac{1}{4G_{\mathrm{bare}}(\epsilon)} +\frac{N\gamma}{\epsilon^2}\right]+S_{\mathrm{fin}}.$$

Finiteness requires

$$\frac{1}{4G_{\mathrm{ren}}} =\frac{1}{4G_{\mathrm{bare}}(\epsilon)} +\frac{N\gamma}{\epsilon^2}.$$

Then $S_{\mathrm{gen}}=A/(4G_{\mathrm{ren}})+S_{\mathrm{fin}}$. A finite scheme change may set $S_{\mathrm{grav}}'=S_{\mathrm{grav}}+\alpha A$ and $S_{\mathrm{out}}'=S_{\mathrm{out}}-\alpha A$. Their sum is unchanged, showing explicitly that the division into geometric and exterior entropy is scheme dependent while generalized entropy is not.

References

• Bousso, R., Z. Fisher, S. Leichenauer, and A. C. Wall (2016), “Quantum Focusing Conjecture,” Physical Review D 93, 064044. Open PDF.
• Iyer, V., and R. M. Wald (1994), “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy,” Physical Review D 50, 846–864. Open PDF.
• Solodukhin, S. N. (2011), “Entanglement Entropy of Black Holes,” Living Reviews in Relativity 14, 8. Open PDF.