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Cited by 509Black Holes and Theoretical PhysicsAlgebraic structures and combinatorial modelsNoncommutative and Quantum Gravity Theories

Holographic representation of local bulk operators

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Hamilton, Alex, Kabat, Daniel, Lifschytz, Gilad, Lowe, David A.·2006·Physical Review D
DOI 10.1103/physrevd.74.066009
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Abstract

The Lorentzian anti-de Sitter/conformal field theory correspondence implies a map between local operators in supergravity and nonlocal operators in the CFT. By explicit computation we construct CFT operators which are dual to local bulk fields in the semiclassical limit. The computation is done for general dimension in global, Poincar\'e and Rindler coordinates. We find that the CFT operators can be taken to have compact support in a region of the complexified boundary whose size is set by the bulk radial position. We show that at finite $N$ the number of independent commuting operators localized within a bulk volume saturates the holographic bound.

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Crossref · 10.1103/physrevd.74.066009

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  • At leading order in 1/N, for any bulk field φ(x) with x in the causal wedge W_C[A], φ(x) can be represented via expression (2.2) with the Y integral taken only over D[A].

    “At leading order in 1 /N , the claim is that for any φ ( x ) with x ∈ W C [ A ], we can again represent φ ( x ) via the expression (2.2), but with the Y integral now taken only over D [ A ].”

    From Bulk locality and quantum error correction in AdS/CFT (2015) · cites this paper as [7]

    The paper supports representing free bulk fields by smeared boundary operators at large N, but not the claimed restriction of the integral to the real boundary domain D[A]. In §4.1 (p. 20), it says the Rindler construction requires a complexified boundary and explicitly rules out a well-defined smearing function supported at real Rindler boundary coordinates. (LLM, conf=0.87)

    Checked by a reasoning model — may be wrong.

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  • AdS-Rindler reconstruction provides that any causal wedge W_C[A] containing the locations of the φ_i(x) used in defining H_C allows a set of operators φ_A,i(x) with support only on A whose action on H_C is the same as that of φ_i(x).

    “In fact, this is what the AdS-Rindler reconstruction we reviewed in section 2.2 provides us; any causal wedge W C [ A ] which contains the locations of the φ i ( x )'s used in defining H C will allow a set of operators φ A,i ( x ) with support only on A and whose action on H C is the same as that of φ i ( x ).”

    From Bulk locality and quantum error correction in AdS/CFT (2015) · cites this paper as [7]

    Section 4 (p. 15) represents a bulk field in terms of the Rindler boundary field, supporting the reconstruction aspect of the claim. But §4.1 (p. 20) says the smearing construction requires a complexified boundary and that no well-defined smearing function exists on the real Rindler boundary. The paper also does not establish the stated equivalence of operators acting on H_C for any causal wedge containing the selected bulk points. (LLM, conf=0.86)

    Experimental — accuracy not yet measured

  • HKLL showed that reconstruction of the bulk operator in a Rindler wedge can be made manifestly consistent with the extrapolate dictionary if one works in Rindler coordinates instead of the global coordinates of AdS.

    “HKLL showed that the reconstruction of the bulk operator in a Rindler wedge can be made manifestly consistent with the extrapolate dictionary if we work in Rindler coordinates (which covers W R ) instead of the global coordinates of AdS [104,105].”

    From Holographic spacetime, black holes and quantum error correcting codes: a review (2022) · cites this paper as [104]

  • The AdS-Rindler construction of CFT representations of bulk fields can be implemented purely within the causal wedge.

    “The point then is that the construction of CFT representations of bulk fields in the previous subsection can also be implemented purely within the causal wedge [7].”

    From Bulk locality and quantum error correction in AdS/CFT (2015) · cites this paper as [7]

  • The expression for K is not well-defined.

    “It was immediately realized, however, that in fact this expression for K is not welldefined [7] (see also [51, 52])”

    From Bulk locality and quantum error correction in AdS/CFT (2015) · cites this paper as [7]

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Cites

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Research works printed but not linked: 18

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  7. [8][8] I. R. Klebanov and E. Witten, 'AdS/CFT correspondence and symmetry breaking,' Nucl. Phys. B556 (1999) 89-114, hep-th/9905104. - · arXiv
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  13. [14][14] L. Susskind and E. Witten, 'The holographic bound in anti-de Sitter space,' hep-th/9805114. - · arXiv
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  15. [16][16] A. Erd´ elyi, Higher transcendental functions , vol. I. McGraw-Hill, 1953. -
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Authors
Hamilton, A., Kabat, D., Lifschytz, G., & Lowe, D. A.
Affiliations
  • Columbia University (US)
  • Oranim Academic College of Education (IL)
  • Brown University (US)
ORCID
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
2006-09-29
Volume
74
Issue
6
ISSN
  • 0556-2821 (ISSN-L)
  • 1550-7998
  • 1550-2368
DOI
10.1103/physrevd.74.066009
OpenAlex
W2008982490
Cited by
509
References
2
Citations by year
2026: 31 · 2025: 38 · 2024: 40 · 2023: 55 · 2022: 33 · 2021: 40 · 2020: 47 · 2019: 46 · 2018: 33 · 2017: 37 · 2016: 35 · 2015: 27 · 2014: 17 · 2013: 5 · 2012: 9
FWCI
2.19
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
  • Physical Sciences › Mathematics › Geometry and Topology › Algebraic structures and combinatorial models
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
Keywords
holographic representation, supergravity, semiclassical limit, holographic bound, Rindler coordinates, Poincaré coordinates

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