Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Schlieder, Some Remarks about the Localization of States in a Quantum Field Theory, Comm. Math. Phys. 1 (1965) 265.
R. Haag, Local quantum physics: fields, particles, algebras. Texts and monographs in physics, Springer-Verlag, Germany (1992).
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].
E. D’Hoker and D.Z. Freedman, Supersymmetric gauge theories and the AdS /CFT correspondence, hep-th/0201253 [INSPIRE].
G.T. Horowitz and J. Polchinski, Gauge/gravity duality, gr-qc/0602037 [INSPIRE].
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].
R. Bousso, S. Leichenauer and V. Rosenhaus, Light-sheets and AdS/CFT, Phys. Rev. D 86 (2012) 046009 [arXiv:1203.6619] [INSPIRE].
R. Emparan, AdS/CFT duals of topological black holes and the entropy of zero energy states, JHEP 06 (1999) 036 [hep-th/9906040] [INSPIRE].
B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, Rindler Quantum Gravity, Class. Quant. Grav. 29 (2012) 235025 [arXiv:1206.1323] [INSPIRE].
M. Parikh and P. Samantray, Rindler-AdS/CFT, arXiv:1211.7370 [INSPIRE].
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].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
V.E. Hubeny and M. Rangamani, Causal Holographic Information, JHEP 06 (2012) 114 [arXiv:1204.1698] [INSPIRE].
H. Casini and M. Huerta, Remarks on the entanglement entropy for disconnected regions, JHEP 03 (2009) 048 [arXiv:0812.1773] [INSPIRE].
P. Hayden, M. Headrick and A. Maloney, Holographic Mutual Information is Monogamous, Phys. Rev. D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].
I.A. Morrison and M.M. Roberts, Mutual information between thermo-field doubles and disconnected holographic boundaries, arXiv:1211.2887 [INSPIRE].
A.C. Wall, Maximin Surfaces and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, arXiv:1211.3494 [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative Entropy and Holography, JHEP 08 (2013) 060 [arXiv:1305.3182] [INSPIRE].
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].
P. Kraus, H. Ooguri and S. Shenker, Inside the horizon with AdS/CFT, Phys. Rev. D 67 (2003) 124022 [hep-th/0212277] [INSPIRE].
L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The Black hole singularity in AdS/CFT, JHEP 02 (2004) 014 [hep-th/0306170] [INSPIRE].
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].
V.E. Hubeny, Extremal surfaces as bulk probes in AdS/CFT, JHEP 07 (2012) 093 [arXiv:1203.1044] [INSPIRE].
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].
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].
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].
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].
D.A. Lowe and S. Roy, Holographic description of asymptotically AdS 2 collapse geometries, Phys. Rev. D 78 (2008) 124017 [arXiv:0810.1750] [INSPIRE].
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].
I. Heemskerk, Construction of Bulk Fields with Gauge Redundancy, JHEP 09 (2012) 106 [arXiv:1201.3666] [INSPIRE].
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].
D. Kabat and G. Lifschytz, CFT representation of interacting bulk gauge fields in AdS, Phys. Rev. D 87 (2013) 086004 [arXiv:1212.3788] [INSPIRE].
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].
P. Breitenlohner and D.Z. Freedman, Positive Energy in anti-de Sitter Backgrounds and Gauged Extended Supergravity, Phys. Lett. B 115 (1982) 197 [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].
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].
F.G. Friedlander, The wave equation on a curved space-time, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge U.K. (1975).
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].
S. Hollands and R.M. Wald, On the renormalization group in curved space-time, Commun. Math. Phys. 237 (2003) 123 [gr-qc/0209029] [INSPIRE].
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].
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].
C.P. Burgess and C.A. Lütken, Propagators and Effective Potentials in Anti-de Sitter Space, Phys. Lett. B 153 (1985) 137 [INSPIRE].
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).
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].
I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, Bulk and Transhorizon Measurements in AdS/CFT, JHEP 10 (2012) 165 [arXiv:1201.3664] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
W.G. Unruh and R.M. Wald, What happens when an accelerating observer detects a Rindler particle, Phys. Rev. D 29 (1984) 1047 [INSPIRE].
N.D. Birrell and P.C.W. Davies, Quantum fields in curved space. Cambridge University Press, Cambridge U.K. (1982).
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.
K. Krasnov, Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 (2000) 929 [hep-th/0005106] [INSPIRE].
K. Krasnov, Black hole thermodynamics and Riemann surfaces, Class. Quant. Grav. 20 (2003) 2235 [gr-qc/0302073] [INSPIRE].
K. Skenderis and B.C. van Rees, Holography and wormholes in 2 + 1 dimensions, Commun. Math. Phys. 301 (2011) 583 [arXiv:0912.2090] [INSPIRE].
S.H. Shenker and D. Stanford, Multiple Shocks, arXiv:1312.3296 [INSPIRE].
K. Papadodimas and S. Raju, An Infalling Observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
E. Verlinde and H. Verlinde, Behind the Horizon in AdS/CFT, arXiv:1311.1137 [INSPIRE].
S.G. Avery and B.D. Chowdhury, No Holography for Eternal AdS Black Holes, arXiv:1312.3346 [INSPIRE].
V. Balasubramanian, B. Czech, K. Larjo and J. Simon, Integrability versus information loss: A Simple example, JHEP 11 (2006) 001 [hep-th/0602263] [INSPIRE].
V. Balasubramanian, D. Marolf and M. Rozali, Information Recovery From Black Holes, Gen. Rel. Grav. 38 (2006) 1529 [hep-th/0604045] [INSPIRE].
D. Marolf, Unitarity and Holography in Gravitational Physics, Phys. Rev. D 79 (2009) 044010 [arXiv:0808.2842] [INSPIRE].
D. Marolf, Holographic Thought Experiments, Phys. Rev. D 79 (2009) 024029 [arXiv:0808.2845] [INSPIRE].
D. Marolf, Holography without strings?, Class. Quant. Grav. 31 (2014) 015008 [arXiv:1308.1977] [INSPIRE].
L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin/Heidelberg, 2nd ed. (1990).
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].
R. Verch, Wavefront sets in algebraic quantum field theory, Commun. Math. Phys. 205 (1999) 337 [math-ph/9807022] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1403.3426
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Morrison, I.A. Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography. J. High Energ. Phys. 2014, 53 (2014). https://doi.org/10.1007/JHEP05(2014)053
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2014)053