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Local bulk operators in AdS/CFT: a boundary view of horizons and locality

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Alex Hamilton, Daniel Kabat, Gilad Lifschytz, David A. Lowe

High Energy Physics - Theory · Submitted 14 Jun 2005 · 30 references

Pith's verdict: A clean, honest AdS2 bulk-reconstruction paper: the global spacelike-support kernel is the real result, and the behind-horizon locality claim is a bit more inherited than the abstract lets on.

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arXiv IDhep-th/0506118 (version 1)
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CitationAlex Hamilton, Daniel Kabat, David A. Lowe, Gilad Lifschytz. "Local bulk operators in AdS/CFT: a boundary view of horizons and locality." arXiv:hep-th/0506118 (2005). https://pith.science/paper/KNAHTVNR
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The paper

The paper claims

A local field in the bulk of anti-de Sitter space can be rebuilt from a smear of boundary data taken only outside its light cone.

The reading

The paper asks a concrete question inside the AdS/CFT correspondence: if every bulk field is supposed to be encoded in a boundary conformal field theory, what exactly is the boundary operator that corresponds to a *local* field at a chosen point in the bulk? The authors answer by writing the bulk field as an integral of the boundary operator against a kernel they call a smearing function, $K$, and then computing $K$ explicitly in three coordinate systems: global AdS$_2$, the Poincaré patch, and the Rindler wedge that models a two-dimensional black hole. The distinguishing feature of their kernels is support: $K$ is nonzero only on boundary points that are spacelike separated from the bulk point, so the reconstruction never draws on boundary data that could have influenced, or been influenced by, the bulk insertion. That is what lets them claim the construction respects bulk locality rather than merely reproducing bulk correlation functions. The result matters because it turns a slogan (the bulk is encoded in the boundary) into a formula, and because it gives a boundary-side picture of what happens when the bulk point is hidden behind a horizon.

What carries the argument

The load-bearing object is the smearing function $K$, defined by the requirement that it solve the bulk wave equation with a spacelike-only step function and reduce, in the boundary limit, to a finite conformal-weight-carrying kernel. Two derivations are given and must agree: a Green's function $G(x|x') = tfrac{1}{2}P_{Delta-1}(sigma),theta(rho-rho'),theta(rho-rho'-|tau-tau'|)$ inserted into Green's identity, and a mode sum in which the polynomial pieces of the kernel are discarded because the boundary field has no Fourier components in the range $-Delta+1, dots, Delta-1$. The AdS-invariant distance $sigma$ carries the covariance, and the antipodal map $sigma(x|Ax') = -sigma(x|x')$ together with $varphi(Ax) = (-1)^{Delta}varphi(x)$ is the device that lets the authors move the part of the kernel that has escaped the Poincaré or Rindler patch back onto the boundary of that patch. The payoff is equation (33): bulk Wightman functions equal boundary correlators of the smeared operators, with the $tau to tau - iepsilon$ prescription of the bulk matching the $s to s - iepsilon$ prescription of the boundary.

Extended reading notes

Core claim

The central claim is that in the semiclassical limit a normalizable free scalar field of conformal dimension $Delta$ at a bulk point $(tau',rho')$ can be written exactly as $varphi(tau',rho') = int dtau, K(tau|tau',rho'), varphi_0^{text{global},R}(tau)$, where $varphi_0$ is the boundary value of the field and the kernel is $K propto left(frac{cos(tau-tau') - sinrho'}{cosrho'}right)^{Delta-1} theta(pi/2 - rho' - |tau-tau'|)$ in global coordinates, with parallel expressions in Poincaré and Rindler coordinates. The step function is the point: the boundary operator has support only at points spacelike separated from the bulk point, and the authors obtain this both from a Green's function that vanishes outside the spacelike region and, independently, from a mode sum where the freedom to drop Fourier components the boundary field does not carry lets them reshape the kernel into the same covariant form. They then push the construction to bulk points outside the Poincaré patch and inside the Rindler horizon of the two-dimensional black hole, where the kernel must be reassembled from pieces that have left the patch, translated back by $2pi$ in global time (or, for non-integer $Delta$, by using the phase picked up by positive- and negative-frequency parts). Behind the horizon the reconstruction necessarily involves both the left and right boundaries, i.e. both copies of the thermofield double, and the authors read this as the statement that an infalling observer's local operator is not an operator in a single copy of the CFT.

Load-bearing premise

The reconstruction is built so that bulk correlation functions come out right by construction, so the real content rests on the claim that the smearing kernel stays supported only at spacelike separation and stays finite, derived for a free scalar field in the strict semiclassical limit with no backreaction and assumed to be independent of the state.

Editorial extensions

If this is right

  • Two smeared boundary operators commute whenever their bulk points are spacelike separated, so bulk microcausality is inherited from boundary data rather than postulated.
  • Light-cone singularities of the bulk theory are generated by the ultraviolet behavior of the boundary theory, so any ultraviolet cutoff on the CFT distorts locality everywhere in the bulk, not just near the boundary.
  • A bulk operator outside the horizon is an operator in one copy of the CFT; a bulk operator behind the horizon requires an operator acting on both copies of the thermofield double.
  • From an outside observer's view, insertions behind the horizon appear as non-thermal modifications of the black hole density matrix obtained by tracing over the second copy.
  • The switch of the black hole time coordinate from timelike to spacelike at the horizon shows up on the boundary as left and right smearing functions being translated in opposite directions under a Rindler time translation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the kernel's spacelike support is what does the work, one could test the construction sharply by asking whether an appropriately smeared boundary operator fails to commute for a pair of bulk points that are timelike separated but not connected by a geodesic, where $sigma < -1$ and the bulk Wightman function is unambiguous.
  • The paper's own remark that the correspondence holds 'by construction' suggests the real content is a statement about the kernel's support and finiteness; a reader could reasonably treat the spacelike-support property, not the correlator identity, as the thing to check first.
  • The two-copy structure behind the horizon suggests a natural comparison with the thermofield-double literature on eternal black holes: the same entanglement that makes the state thermal for one copy is what makes a local operator behind the horizon exist at all.
  • If the kernel is state-independent only at leading order in $1/N$, then the locality statement is a semiclassical approximation rather than an exact property, and the interesting question becomes how the kernel deforms when backreaction is restored.
Original abstract

We develop the representation of local bulk fields in AdS by non-local operators on the boundary, working in the semiclassical limit and using AdS_2 as our main example. In global coordinates we show that the boundary operator has support only at points which are spacelike separated from the bulk point. We construct boundary operators that represent local bulk operators inserted behind the horizon of the Poincare patch and inside the Rindler horizon of a two dimensional black hole. We show that these operators respect bulk locality and comment on the generalization of our construction to higher dimensional AdS black holes.

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,597 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.58 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.08 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We develop the representation of local bulk fields in AdS by non-local operators on the boundary, working in the semiclassical limit and using AdS_2 as our main example. In global coordinates we show that the boundary operator has support only at points which are spacelike separated from the bulk point. We construct boundary operators that represent local bulk operators inserted behind the horizon of the Poincare patch and inside the Rindler horizon of a two dimensional black hole. We show that these operators respect bulk locality and comment on the generalization of our construction to higher dimensional AdS black holes.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We develop the representation of local bulk fields in AdS by non-local operators on the boundary, working in the semiclassical limit and using AdS_2 as our main example. contribution 0.98 0.95 0.22 asserted 0.99
s1 In global coordinates we show that the boundary operator has support only at points which are spacelike separated from the bulk point. result 0.99 0.93 0.21 asserted 1.00
s2 We construct boundary operators that represent local bulk operators inserted behind the horizon of the Poincare patch and inside the Rindler horizon of a two dimensional black hole. contribution 0.91 0.95 0.20 asserted 1.00
s3 We show that these operators respect bulk locality and comment on the generalization of our construction to higher dimensional AdS black holes. result 0.73 0.91 0.31 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.91 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Global smearing function for a ∆ = 3 bulk operator lo
Figure 2. Smearing function for a bulk operator with integer
Figure 3. Smearing function for an integer-dimension bulk o
Figure 4. The Kruskal patch, indicated on the AdS Penrose dia

Figures reproduced from arXiv:hep-th/0506118 by the authors.

References
How later papers used it

192 later papers cite it

Evidence: some. 172 of the citing papers share no author with it, and 34 of those carry a confident label from Pith's citation classifier.

Pith read 8 of the 338 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 1 of 29 papers with this label
How the literature uses it

172 arXiv papers cite this in 289 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 179 sentences without a confident label

Contests it 1 paper
  • This means the boundary-to-bulk map constructed in [ref] , if applied to an interacting CFT, gives rise to a set of bulk observables which are non-local in the sense that they do not commute at spacelike separation.

    Constructing local bulk observables in interacting AdS/CFT · Breakdown of locality · contest 0.79

Builds on it 5 papers
Uses its method or tool 25 papers
  • The conformally coupled scalar in Poincaré AdS can be reconstructed from the dual CFT defined over $ℝ^1,d-1$ using the HKLL smearing function [ref] , [equation] where [equation] where $C(Δ, d)$ is a constant, which depends on the boundary spacetime dimensions and the conformal dimension of the primary[See App. [ref] for a short derivation] after setting $Δ= d+1/2$. .

    CFT reconstruction of local bulk operators in half-Minkowski space · The CFT to half-flat Smearing function · 0.92

  • Following the prescription of [ref] we extract a straightforward geometric picture that determines when a commutator of quasi-local bulk observables is non-vanishing.

    Black hole complementarity from AdS/CFT · introduction · 0.98

  • The holographic encoding map $V:_a⊗_b→_A$ for the ASSR geometry is quite simple: on the two AdS regions we use the HKLL encoding isometry [ref] , while the closed universe is hit by a rank one projection taken to be a $d$-dimensional orthogonal transformation $O$ on the closed universe together with some fixed state $|ψ_0_f$ followed by a projection onto a simple state $0|$ [ref] .

    Observer complementarity for black holes and holography · Antonini-Sasieta-Swingle-Rath geometry · 0.86

Compares with it 8 papers
  • We can use this freedom to eliminate the second line of ([ref]), leaving[Note that the integral is over spacelike-separated points on the right boundary.] ϕ_interior(τ,ρ) = 3 2 ∫_τ- (π2 - ρ)^τ+ (π2 - ρ) dτ' cos(τ- τ') - sinρcosρ ϕ_0^global,R(τ') in agreement with the global smearing function obtained in [ref] .

    Asymmetric interiors for small black holes · Fields inside an eternal black hole · 0.96

  • The relations [ref] resemble the HKLL reconstruction [ref] ,[The similar observation that the Wilson line operator acting on the Ishibashi state can be represented in terms of HKLL construction was made in [ref] .] where a scalar field in with the mass $m^2 = (-1)$ can be obtained by integrating a primary operator of conformal weight $$ with the smearing function which is essentially the bulk-to-boundary propagator over the part of the conformal boundary,

    Holographic reconstruction for AdS Wilson line networks and scalar Witten diagrams · Wilson matrix elements in integral form · 0.99

  • This is the same factor that appeared in [ref] for the massive case.

    Bulk reconstruction and Bogoliubov transformations in AdS$_2$ · Equivalence of boundary representations in AdS$_2$ for massless fields · 0.98

Cites it as background 53 papers

Discussion

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