A note on the two point function on the boundary of AdS spacetime

Document Type : Regular article

Author

Departamento de Fisica, Facultad de Ciencias, UNAM, Mexico

Abstract

We calculate by a new way the two point function on the boundary of AdS spacetime in 1+2 dimensions for the massless conformal real scalar field. The result agrees with the answer provided by the Boundary-limit Holography and Witten recipe. This is done in Poincar'{e} coordinates. The basic ingredients of this new method are conformal techniques, quantum fields defined on a half of Minkowski spacetime and a limit inspired by the Boundary-limit Holography. We also show that a state in AdS, the global vacuum, in three dimensions induces a state on the conformal boundary of AdS spacetime, which in turn induces a state on the BTZ black hole. On the other hand the same state in AdS induces a state on the BTZ black hole which in turn induces a state on its conformal boundary. The two ways of getting the state on the conformal boundary of the BTZ black hole coincide for the massless conformal real scalar field. We point out that the normalizable modes in the AdS/CFT correspondence for the BTZ black hole give a similar contribution as the non-normalizable modes used in the Witten prescription. We also give some clues on why the Witten and the Boundary-limit Holography prescription coincide.

Keywords

[1] J. M. Maldacena, ”The large-N limit of superconformal field theories and supergravity”, Adv. Theor. Math. Phys. 2, 231 (1998).
[2] E. Witten, ”Anti de Sitter Space and Holography”, Adv. Theor. Math. Phys. 2, 253 (1998).
[3] S. S. Gubser, I. R. Klebanov, A. M. Polyakov, ”Gauge Theory Correlators from Non-Critical String Theory”, Phys. Lett. B 428, 105 (1998).
[4] G. A. Kerimov, ”Scalar field theory in the AdS/CFT correspondence: An operator formulation”, Mod. Phys. Lett. A 22, 2287 (2007).
[5] K. H. Rehren, ”Algebraic Holography”, Ann. H. Poinc. 1, 607 (2000).
[6] M. Bertola, J. Bros, U. Moschella, R. Schaeffer, ”A general construction of conformal field theories from scalar anti-de Sitter quantum field theories”, Nucl. Phys. B 587, 619 (2000).
[7] S. J. Avis, C. J. Isham, D. Storey, ”Quantum field theory in anti-de Sitter space-time”, Phys. Rev. D 18, 3565 (1978).
[8] P. Breitenlohner and D. Z. Freedman, ”Positive energy in Anti-de Sitter backgrounds and gauged extended supergravity”, Phys. Lett. B 115, 197 (1982)
[9] P. Breitenlohner and D. Z. Freedman, ”Stability in gauged extended supergravity”, Ann. Phys. 144 249 (1982).
[10] I. R. Klebanov, E. Witten, ”AdS/CFT correspondence and symmetry breaking”, Nucl. Phys. B 556, 89 (1999).
[11] M. Duetsch, K. H. Rehren, ”A comment on the dual field in the AdS-CFT correspondence”, Lett. Math. Phys. 62, 171 (2002).
[12] V. Balasubramanian, P. Kraus, A. Lawrence, ”Bulk versus boundary dynamics in anti-de Sitter spacetime”, Phys. Rev. D 59, 046003 (1999).
[13] U. H. Danielsson, E. Keski-Vakkuri, M. Kruczenski, ”Vacua, Propagators, and Holographic Probes in AdS/CFT”, J. High Energy Phys. 9901, 002 (1999).
[14] E. Keski-Vakkuri, ”Bulk and boundary dynamics in BTZ black holes”, Phys. Rev. D59, 104001 (1999).
[15] L. Ort´ız, ”Hawking temperature in the eternal BTZ black hole: an example of holography in AdS spacetime” Gen. Relativ. Gravit. 45, 427 (2013)
Volume 1, Issue 1
November 2021
Pages 37-46
  • Receive Date: 10 September 2021
  • Revise Date: 10 October 2021
  • Accept Date: 10 October 2021