Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography

Open Access


In order to better understand how AdS holography works for sub-regions, we formulate a holographic version of the Reeh-Schlieder theorem for the simple case of an AdS Klein-Gordon field. This theorem asserts that the set of states constructed by acting on a suitable vacuum state with boundary observables contained within any subset of the boundary is dense in the Hilbert space of the bulk theory. To prove this theorem we need two ingredients which are themselves of interest. First, we prove a purely bulk version of Reeh-Schlieder theorem for an AdS Klein-Gordon field. This theorem relies on the analyticity properties of certain vacuum states. Our second ingredient is a boundary-to-bulk map for local observables on an AdS causal wedge. This mapping is achieved by simple integral kernels which construct bulk observables from convolutions with boundary operators. Our analysis improves on previous constructions of AdS boundary-to-bulk maps in that it is formulated entirely in Lorentz signature without the need for large analytic continuation of spatial coordinates. Both our Reeh-Schlieder theorem and boundary-to-bulk maps may be applied to globally well-defined states constructed from the usual AdS vacuum as well more singular states such as the local vacuum of an AdS causal wedge which is singular on the horizon.


Gauge-gravity correspondence AdS-CFT Correspondence 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    S. Schlieder, Some Remarks about the Localization of States in a Quantum Field Theory, Comm. Math. Phys. 1 (1965) 265.ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    R. Haag, Local quantum physics: fields, particles, algebras. Texts and monographs in physics, Springer-Verlag, Germany (1992).CrossRefMATHGoogle Scholar
  3. [3]
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    E. D’Hoker and D.Z. Freedman, Supersymmetric gauge theories and the AdS /CFT correspondence, hep-th/0201253 [INSPIRE].
  5. [5]
    G.T. Horowitz and J. Polchinski, Gauge/gravity duality, gr-qc/0602037 [INSPIRE].
  6. [6]
    B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The Gravity Dual of a Density Matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    R. Bousso, S. Leichenauer and V. Rosenhaus, Light-sheets and AdS/CFT, Phys. Rev. D 86 (2012) 046009 [arXiv:1203.6619] [INSPIRE].ADSGoogle Scholar
  8. [8]
    R. Emparan, AdS/CFT duals of topological black holes and the entropy of zero energy states, JHEP 06 (1999) 036 [hep-th/9906040] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, Rindler Quantum Gravity, Class. Quant. Grav. 29 (2012) 235025 [arXiv:1206.1323] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    M. Parikh and P. Samantray, Rindler-AdS/CFT, arXiv:1211.7370 [INSPIRE].
  11. [11]
    R. Bousso, B. Freivogel, S. Leichenauer, V. Rosenhaus and C. Zukowski, Null Geodesics, Local CFT Operators and AdS/CFT for Subregions, Phys. Rev. D 88 (2013) 064057 [arXiv:1209.4641] [INSPIRE].ADSGoogle Scholar
  12. [12]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    V.E. Hubeny and M. Rangamani, Causal Holographic Information, JHEP 06 (2012) 114 [arXiv:1204.1698] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    H. Casini and M. Huerta, Remarks on the entanglement entropy for disconnected regions, JHEP 03 (2009) 048 [arXiv:0812.1773] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    P. Hayden, M. Headrick and A. Maloney, Holographic Mutual Information is Monogamous, Phys. Rev. D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].ADSGoogle Scholar
  19. [19]
    I.A. Morrison and M.M. Roberts, Mutual information between thermo-field doubles and disconnected holographic boundaries, arXiv:1211.2887 [INSPIRE].
  20. [20]
    A.C. Wall, Maximin Surfaces and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, arXiv:1211.3494 [INSPIRE].
  21. [21]
    T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative Entropy and Holography, JHEP 08 (2013) 060 [arXiv:1305.3182] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    W.R. Kelly and A.C. Wall, Coarse-grained entropy and causal holographic information in AdS/CFT, JHEP 03 (2014) 118 [arXiv:1309.3610] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    P. Kraus, H. Ooguri and S. Shenker, Inside the horizon with AdS/CFT, Phys. Rev. D 67 (2003) 124022 [hep-th/0212277] [INSPIRE].ADSMathSciNetGoogle Scholar
  25. [25]
    L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The Black hole singularity in AdS/CFT, JHEP 02 (2004) 014 [hep-th/0306170] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    V.E. Hubeny, H. Liu and M. Rangamani, Bulk-cone singularities & signatures of horizon formation in AdS/CFT, JHEP 01 (2007) 009 [hep-th/0610041] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    V.E. Hubeny, Extremal surfaces as bulk probes in AdS/CFT, JHEP 07 (2012) 093 [arXiv:1203.1044] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    I. Bena, On the construction of local fields in the bulk of AdS 5 and other spaces, Phys. Rev. D 62 (2000) 066007 [hep-th/9905186] [INSPIRE].ADSMathSciNetGoogle Scholar
  29. [29]
    A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: A Boundary view of horizons and locality, Phys. Rev. D 73 (2006) 086003 [hep-th/0506118] [INSPIRE].ADSGoogle Scholar
  30. [30]
    A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].ADSMathSciNetGoogle Scholar
  31. [31]
    A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: A Holographic description of the black hole interior, Phys. Rev. D 75 (2007) 106001 [Erratum ibid. D 75 (2007) 129902] [hep-th/0612053] [INSPIRE].
  32. [32]
    D.A. Lowe and S. Roy, Holographic description of asymptotically AdS 2 collapse geometries, Phys. Rev. D 78 (2008) 124017 [arXiv:0810.1750] [INSPIRE].ADSGoogle Scholar
  33. [33]
    D. Kabat, G. Lifschytz and D.A. Lowe, Constructing local bulk observables in interacting AdS/CFT, Phys. Rev. D 83 (2011) 106009 [arXiv:1102.2910] [INSPIRE].ADSGoogle Scholar
  34. [34]
    I. Heemskerk, Construction of Bulk Fields with Gauge Redundancy, JHEP 09 (2012) 106 [arXiv:1201.3666] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    D. Kabat, G. Lifschytz, S. Roy and D. Sarkar, Holographic representation of bulk fields with spin in AdS/CFT, Phys. Rev. D 86 (2012) 026004 [arXiv:1204.0126] [INSPIRE].ADSGoogle Scholar
  36. [36]
    D. Kabat and G. Lifschytz, CFT representation of interacting bulk gauge fields in AdS, Phys. Rev. D 87 (2013) 086004 [arXiv:1212.3788] [INSPIRE].ADSGoogle Scholar
  37. [37]
    S. Leichenauer and V. Rosenhaus, AdS black holes, the bulk-boundary dictionary and smearing functions, Phys. Rev. D 88 (2013) 026003 [arXiv:1304.6821] [INSPIRE].ADSGoogle Scholar
  38. [38]
    P. Breitenlohner and D.Z. Freedman, Positive Energy in anti-de Sitter Backgrounds and Gauged Extended Supergravity, Phys. Lett. B 115 (1982) 197 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    A. Ishibashi and R.M. Wald, Dynamics in nonglobally hyperbolic static space-times. 3. Anti-de Sitter space-time, Class. Quant. Grav. 21 (2004) 2981 [hep-th/0402184] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  41. [41]
    F.G. Friedlander, The wave equation on a curved space-time, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (1975).Google Scholar
  42. [42]
    S. Hollands and R.M. Wald, Local Wick polynomials and time ordered products of quantum fields in curved space-time, Commun. Math. Phys. 223 (2001) 289 [gr-qc/0103074] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  43. [43]
    S. Hollands and R.M. Wald, On the renormalization group in curved space-time, Commun. Math. Phys. 237 (2003) 123 [gr-qc/0209029] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  44. [44]
    S. Hollands and R.M. Wald, Conservation of the stress tensor in interacting quantum field theory in curved spacetimes, Rev. Math. Phys. 17 (2005) 227 [gr-qc/0404074] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    R. Brunetti, K. Fredenhagen and M. Kohler, The Microlocal spectrum condition and Wick polynomials of free fields on curved space-times, Commun. Math. Phys. 180 (1996) 633 [gr-qc/9510056] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  46. [46]
    C.P. Burgess and C.A. Lütken, Propagators and Effective Potentials in Anti-de Sitter Space, Phys. Lett. B 153 (1985) 137 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    R.F. Streater and A.S. Wightman, PCT, spin and statistics, and all that, Advanced book classics. Addison-Wesley, Redwood City, U.S.A. (1989).MATHGoogle Scholar
  48. [48]
    A. Strohmaier, R. Verch and M. Wollenberg, Microlocal analysis of quantum fields on curved space-times: Analytic wavefront sets and Reeh-Schlieder theorems, J. Math. Phys. 43 (2002) 5514 [math-ph/0202003] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  49. [49]
    I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, Bulk and Transhorizon Measurements in AdS/CFT, JHEP 10 (2012) 165 [arXiv:1201.3664] [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    W.G. Unruh and R.M. Wald, What happens when an accelerating observer detects a Rindler particle, Phys. Rev. D 29 (1984) 1047 [INSPIRE].ADSGoogle Scholar
  52. [52]
    N.D. Birrell and P.C.W. Davies, Quantum fields in curved space. Cambridge University Press, Cambridge U.K. (1982).CrossRefMATHGoogle Scholar
  53. [53]
    S. Åminneborg, I. Bengtsson, D. Brill, S. Holst and P. Peldán, Black holes and wormholes in 2 + 1 dimensions, Class. Quant. Grav. 15 (1998) 627.ADSMathSciNetCrossRefMATHGoogle Scholar
  54. [54]
    K. Krasnov, Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 (2000) 929 [hep-th/0005106] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    K. Krasnov, Black hole thermodynamics and Riemann surfaces, Class. Quant. Grav. 20 (2003) 2235 [gr-qc/0302073] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  56. [56]
    K. Skenderis and B.C. van Rees, Holography and wormholes in 2 + 1 dimensions, Commun. Math. Phys. 301 (2011) 583 [arXiv:0912.2090] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  57. [57]
    S.H. Shenker and D. Stanford, Multiple Shocks, arXiv:1312.3296 [INSPIRE].
  58. [58]
    K. Papadodimas and S. Raju, An Infalling Observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    E. Verlinde and H. Verlinde, Behind the Horizon in AdS/CFT, arXiv:1311.1137 [INSPIRE].
  60. [60]
    S.G. Avery and B.D. Chowdhury, No Holography for Eternal AdS Black Holes, arXiv:1312.3346 [INSPIRE].
  61. [61]
    V. Balasubramanian, B. Czech, K. Larjo and J. Simon, Integrability versus information loss: A Simple example, JHEP 11 (2006) 001 [hep-th/0602263] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    V. Balasubramanian, D. Marolf and M. Rozali, Information Recovery From Black Holes, Gen. Rel. Grav. 38 (2006) 1529 [hep-th/0604045] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  63. [63]
    D. Marolf, Unitarity and Holography in Gravitational Physics, Phys. Rev. D 79 (2009) 044010 [arXiv:0808.2842] [INSPIRE].ADSGoogle Scholar
  64. [64]
    D. Marolf, Holographic Thought Experiments, Phys. Rev. D 79 (2009) 024029 [arXiv:0808.2845] [INSPIRE].ADSGoogle Scholar
  65. [65]
    D. Marolf, Holography without strings?, Class. Quant. Grav. 31 (2014) 015008 [arXiv:1308.1977] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  66. [66]
    L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin/Heidelberg, 2nd ed. (1990).MATHGoogle Scholar
  67. [67]
    M.J. Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys. 179 (1996) 529 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  68. [68]
    R. Verch, Wavefront sets in algebraic quantum field theory, Commun. Math. Phys. 205 (1999) 337 [math-ph/9807022] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Department of PhysicsMcGill UniversityMontrealCanada

Personalised recommendations