REVIEW
Black Hole Entropy and Entropy of Entanglement
High Energy Physics - Theory · Submitted 3 Mar 1995 · 32 references
Pith's verdict: The gauge-field heat-kernel computation is solid and the horizon surface term is real; the claim that it is absent from the entanglement entropy is inferred rather than computed, and that is the one soft spot.
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Referee report
The paper
The paper claims
For scalar and spinor fields the one-loop correction to a black hole's entropy is exactly the field's entropy of entanglement, but for an Abelian gauge field the two differ by a contact term localized on the horizon that makes the gauge贡献负。
The reading
What carries the argument
The engine is the conical partition function. Entropy is extracted as $(β∂_β - 1)$ applied to the free energy at $β = 2π$, evaluated on a space that is a cone of deficit angle $2π-β$ times a flat transverse space of area $A_⊥$; the response to the deficit angle is captured by the heat kernel at coincident points, which the author writes as a single-particle path integral and evaluates explicitly using Bessel eigenfunctions regular at the tip. The bridge to entanglement entropy is the Unruh statement that the Minkowski vacuum restricted to one side of a boundary is a thermal state in Rindler coordinates at inverse temperature $2π$, so entanglement entropy can be computed as Rindler thermal entropy, which in turn is the determinant of $-∂_θ^2 + H_R^2$ with period $β$ in $θ$ — the same cone. For the gauge field the vector heat kernel is shown to equal $2K_{scalar}$ plus a total derivative in $r$, $2r∂_r s K_{scalar}$, and it is this total derivative, integrated over the tip, that becomes the surface term; the ghosts cancel the bulk of two scalars, leaving $d-2$ physical polarizations plus the boundary piece.
Extended reading notes
Core claim
The central claim is a three-way comparison at one loop. For spin zero and spin one-half, the entropy a field contributes to a black hole equals its entropy of entanglement, so for those fields the distinction between the two notions carries no content. For spin one the free energy on a cone of deficit angle $2π-β$ splits into a bulk piece equal to $d-2$ scalars plus a horizon surface term, $$βF_{gauge} = (d-2)βF_{scalar} + A_⊥(2π-β)∫_{$ε^{2}$}^{∞} ds/(4πs)^{d/2} $e^{{-sm^2}}$,$$ and only the bulk piece matches the entanglement entropy. The surface term is a path integral over particle trajectories whose endpoints lie on the horizon; it cancels against a term at infinity precisely at the on-shell temperature $β=2π$, as it must, since no contact interaction with the horizon should survive when the curvature there vanishes. The author identifies this term as the field theory limit of the Susskind-Uglum coupling of a closed string to an open string stranded on the horizon, and notes that it has no thermal or state-counting interpretation within quantum field theory.
Load-bearing premise
The claim that the horizon surface term is absent from the entanglement entropy of a gauge field rests on showing that the usual equivalence between entanglement entropy and Rindler thermal entropy breaks down for gauge fields, not on computing that entanglement entropy directly in more than two dimensions.
Editorial extensions
If this is right
- The one-loop correction to black hole entropy from a gauge field is negative in $d<8$, so any account that identifies black hole entropy with entanglement entropy alone cannot be complete for gauge fields.
- The surface term vanishes at $β=2π$ but contributes to the entropy through its $β$-derivative, so it is invisible to a naive on-shell free-energy check and only shows up in the thermodynamic variation.
- The non-renormalization of Newton's constant in certain supersymmetric theories is traced to this same horizon contact term rather than to a cancellation among bulk entanglement contributions.
- Because the term is a path integral over paths pinned to the horizon, it is the natural field theory shadow of the stringy closed-string/open-string interaction, giving a concrete low-energy target for any string-derived derivation of black hole entropy.
- The formal equivalence between entanglement entropy and Rindler thermal entropy fails for gauge fields because the Rindler Hamiltonian in covariant gauge has no normalizable eigenstates at the origin, so the equivalence theorem cannot be applied blindly across spin.
Reading between the lines
- The paper's case that the surface term is not entanglement entropy is an argument by exclusion — the equivalence theorem does not apply, therefore the term is not entanglement — rather than a direct computation; a lattice or replica calculation of gauge-field entanglement entropy across a planar cut in four dimensions would settle it either way.
- If the contact term is genuinely non-thermal, it behaves like a contribution to the horizon's own degrees of freedom rather than to the matter, which suggests treating the horizon area term and the gauge contact term as two entries in the same bookkeeping rather than as corrections to one another.
- The identification with the Susskind-Uglum diagram implies a scaling prediction: the contact term scales with horizon area exactly as the classical term does, so a stringy resolution would have to reproduce not just the sign but the precise coefficient $A_⊥(2π-β)∫ ds/(4πs)^{d/2}e^{-sm^2}$.
- Because the effect is tied to the behaviour of the fields in the two dimensions containing the conical singularity, non-Abelian gauge fields and fields with singular boundary behaviour at the tip are the natural next cases to check.
Original abstract
We compare the one-loop corrections to the entropy of a black hole, from quantum fields of spin zero, one-half, and one, to the entropy of entanglement of the fields. For fields of spin zero and one-half the black hole entropy is identical to the entropy of entanglement. For spin one the two entropies differ by a contact interaction with the horizon which appears in the black hole entropy but not in the entropy of entanglement. The contact interaction can be expressed as a path integral over particle paths which begin and end on the horizon; it is the field theory limit of the interaction proposed by Susskind and Uglum which couples a closed string to an open string stranded on the horizon.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We compare the one-loop corrections to the entropy of a black hole, from quantum fields of spin zero, one-half, and one, to the entropy of entanglement of the fields. For fields of spin zero and one-half the black hole entropy is identical to the entropy of entanglement. For spin one the two entropies differ by a contact interaction with the horizon which appears in the black hole entropy but not in the entropy of entanglement. The contact interaction can be expressed as a path integral over particle paths which begin and end on the horizon; it is the field theory limit of the interaction proposed by Susskind and Uglum which couples a closed string to an open string stranded on the horizon.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We compare the one-loop corrections to the entropy of a black hole, from quantum fields of spin zero, one-half, and one, to the entropy of entanglement of the fields. | contribution 0.99 | 0.86 | 0.08 | asserted 0.70 |
| s1 | For fields of spin zero and one-half the black hole entropy is identical to the entropy of entanglement. | result 1.00 | 0.84 | 0.36 | asserted 1.00 |
| s2 | For spin one the two entropies differ by a contact interaction with the horizon which appears in the black hole entropy but not in the entropy of entanglement. | result 1.00 | 0.82 | 0.45 | asserted 1.00 |
| s3 | The contact interaction can be expressed as a path integral over particle paths which begin and end on the horizon; it is the field theory limit of the interaction proposed by Susskind and Uglum which couples a closed string to an open string stranded on the horizon. | interpretation 0.87 | 0.56 | 0.09 | asserted 1.00 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:hep-th/9503016 by the authors.
References
How the literature uses it
93 arXiv papers cite this in 154 sentences
Papers citing it · 102 sentences without a confident label
Builds on it 3 papers
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Building on this framework it was shown in [ref] that in a suitable brick wall-like regularization scheme and for some choice of measure $E_⊥$, the edge correction of [ref] arises as a classical contribution $∫E_⊥ e^-S_E[E_⊥]$ to the thermal statistical partition function.
Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions · QFT considerations · 0.88
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This generalizes Kabat's interpretation [ref] for the photon contact term [ref]: [equation] For the graviton contact term, one might thus be lead to a picture of massless vector particle loops that begin and end on the horizon, and contribute non-trivially to the graviton partition function [equation] With this being said, the physical value of such an interpretation is currently not clear to us.
Gravitons on the edge · Interpretation · 0.99
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However, our computations for the renormalization of Newton's constant, $d_1$, generalize previous flat space discussions in, for example, [ref] .
Non-topological logarithmic corrections in minimal gauged supergravity · Renormalization of the couplings · 0.97
Uses its method or tool 5 papers
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To deal with this we will follow Kabat's method [ref] , where we instead study the graviton path integral on the Rindler cone with the tip deleted.
Gravitons on the edge · Going on-shell via tip-less cones · 0.85
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The same quantity can be computed exactly, ⟨A_μA^μ⟩= ∫_0^∞ds g_μν K^μν_vector(s,x,x) where the vector heat kernel is [ref]
Cosmic string interactions induced by gauge and scalar fields · Gauge field energy · 0.81
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Here we preform the standard replica trick calculation for a free massive scalar [ref] .
Entanglement and RG in the $O(N)$ vector model · Replica trick: Free energy · 0.82
Uses its data or values 1 paper
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For a Maxwell field, the entropy divergence coefficient was found to be [ref] [equation] In addition to the $D - 2$ bosonic degrees of freedom which contribute just like $D - 2$ scalars, there is an additional contact term that makes the entropy negative for $D < 8$.
Do gauge fields really contribute negatively to black hole entropy? · Introduction · 0.95
Compares with it 9 papers
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This agrees with the divergence found by Kabat [ref] , as we will show in the next section.
Entanglement entropy of electromagnetic edge modes · Edge modes · 0.90
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Its leading divergence agrees with the result of Ref. [ref] , but we also establish the existence of subleading and finite terms, some of which are universal, i.e. independent of the regulator scheme.
Geometric entropy and edge modes of the electromagnetic field · Introduction · 0.97
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In all, after the Wick rotation the effective action [ref] transforms into: [equation] Taking the integral over $ω$ one obtains that the free energy is equal to: [equation] This is the result that was found earlier in [ref] .
Free energy and entropy in Rindler and de Sitter space-times · Free energy in the Rindler chart · 0.97
Cites it as support 2 papers
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Heuristic arguments indicate that field-theoretic divergences in the quantum entanglement entropy $S_q$ can be absorbed into renormalization of Newton's constant [ref] suggesting that the total entropy must be finite.
Finite Entanglement Entropy in String Theory · Introduction · 0.96
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Furthermore, as noted in [ref] , the contribution of the photon is gauge independent.
Quantum Gravity at the LHC · A large hidden sector · 0.97
Cites it as background 26 papers
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In section [ref], we review Kabat's derivation of the contact term [ref] .
Do gauge fields really contribute negatively to black hole entropy? · Introduction · 0.91
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The generalized entropy is $S_gen = S_H + S_out$, the second term being the entropy of any matter fields outside of $H$ (see [ref] for a review).[One must renormalize in order to absorb ultraviolet divergences of $S_out$ into counterterms in $S_H$ [ref] .
Generalized second law at linear order for actions that are functions of Lovelock densities · The Second Law in General Relativity · 0.95
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An interesting proposal in the case of the exterior region of a black hole, originally due to Susskind–Uglum [ref] , and developed in an important body of work (see for example [ref] ), is that these two issues are actually related.
Large $N$ von Neumann algebras and the renormalization of Newton's constant · Introduction · 0.83
Discussion
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