Abstract
We propose an S matrix approach to the quantum black hole in which causality, unitarity and their interrelation play a prominent role. Assuming the ’t Hooft S matrix ansatz for a gravitating region surrounded by an asymptotically flat space-time we find a non-local transformation which changes the standard causality requirement but is a symmetry of the unitarity condition of the S matrix. This new S matrix then implies correlations between the in and out states of the theory with the involvement of a third entity which in the case of a quantum black hole, we argue is the horizon S matrix. Effects of spacetime curvature and horizon are in fact introduced by this procedure which is seen to be a generalization of the Bogoliubov transformation. The analysis is performed within the Bogoliubov S matrix framework by considering a spacetime consisting of causal complements with a boundary in between. No particular metric or lagrangian dynamics need be invoked even to obtain an evolution equation for the full S matrix. Hawking’s results are reproduced by restricting to low energy incoming modes at the horizon and the generalized hamiltonian of the horizon S matrix in this case is shown to be the generator of the Bogoliubov transformation. The modification of Bogoliubov causality at intermediate stages of black hole evaporation allows for a temporary violation of quantum mechanical no cloning theorems. In this way we find that the tension between information preservation and complementarity may be resolved provided the full quantum gravity theory either through symmetries or fine tuning forbids the occurrence of closed time like curves of information flow. Then, even if causality is violated near the horizon at any intermediate stage, a standard causal ordering may be preserved for the observer outside the black hole. The usefulness of our formulation is that it appears well suited to understand unitarity at any intermediate stage of black hole evaporation. Moreover, it is applicable generally to all theories with long range correlations including the final state projection models. As a nontrivial check, we use it in the perturbative context to analyze infrared divergences in QED and thereby reproduce the Faddeev-Kulish theory of asymptotic dynamics.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
J.B. Hartle and S.W. Hawking, Path integral derivation of black hole radiance, Phys. Rev. D 13 (1976) 2188 [INSPIRE].
G. ’t Hooft, Ambiguity of the equivalence principle and Hawking’s temperature, J. Geom. Phys. 1 (1984) 45 [INSPIRE].
C.R. Stephens, G. ’t Hooft and B.F. Whiting, Black hole evaporation without information loss, Class. Quant. Grav. 11 (1994) 621 [gr-qc/9310006] [INSPIRE].
G. ’t Hooft, The scattering matrix approach for the quantum black hole: an overview, Int. J. Mod. Phys. A 11 (1996) 4623 [gr-qc/9607022] [INSPIRE].
G. ’t Hooft, On the quantum structure of a black hole, Nucl. Phys. B 256 (1985) 727 [INSPIRE].
G. ’t Hooft, The black hole interpretation of string theory, Nucl. Phys. B 335 (1990) 138 [INSPIRE].
G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [INSPIRE].
L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
G. ’t Hooft, The black hole horizon as a quantum surface, Phys. Scripta T 36 (1991) 247 [INSPIRE].
L. Susskind, L. Thorlacius and J. Uglum, The stretched horizon and black hole complementarity, Phys. Rev. D 48 (1993) 3743 [hep-th/9306069] [INSPIRE].
Y. Kiem, H.L. Verlinde and E.P. Verlinde, Black hole horizons and complementarity, Phys. Rev. D 52 (1995) 7053 [hep-th/9502074] [INSPIRE].
S.D. Mathur, What happens at the horizon?, Int. J. Mod. Phys. D 22 (2013) 1341016 [arXiv:1308.2785] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An apologia for firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
S.L. Braunstein, S. Pirandola and K. Życzkowski, Better late than never: information retrieval from black holes, Phys. Rev. Lett. 110 (2013) 101301 [arXiv:0907.1190] [INSPIRE].
S.D. Mathur, The information paradox: a pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
S.B. Giddings and Y. Shi, Quantum information transfer and models for black hole mechanics, Phys. Rev. D 87 (2013) 064031 [arXiv:1205.4732] [INSPIRE].
Y. Takahashi and H. Umezawa, Thermo field dynamics, Int. J. Mod. Phys. B 10 (1996) 1755 [INSPIRE].
W. Israel, Thermo field dynamics of black holes, Phys. Lett. A 57 (1976) 107 [INSPIRE].
G.T. Horowitz and J.M. Maldacena, The black hole final state, JHEP 02 (2004) 008 [hep-th/0310281] [INSPIRE].
D. Gottesman and J. Preskill, Comment on ‘the black hole final state’, JHEP 03 (2004) 026 [hep-th/0311269] [INSPIRE].
S. Lloyd and J. Preskill, Unitarity of black hole evaporation in final-state projection models, arXiv:1308.4209 [INSPIRE].
F. Buscemi, All entangled quantum states are nonlocal, Phys. Rev. Lett. 108 (2012) 200401 [arXiv:1106.6095].
M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge U.K. (2013).
S. Kamefuchi, L. O’Raifeartaigh and A. Salam, Change of variables and equivalence theorems in quantum field theories, Nucl. Phys. 28 (1961) 529 [INSPIRE].
N.N. Bogoliubov and D.V. Shirkov, Introduction to the theory of quantized fields, Interscience publishers Inc., New York U.S.A. (1959).
D. Deutsch, Quantum mechanics near closed timelike lines, Phys. Rev. D 44 (1991) 3197 [INSPIRE].
S. Lloyd, L. Maccone, R. Garcia-Patron, V. Giovannetti and Y. Shikano, Quantum mechanics of time travel through post-selected teleportation, Phys. Rev. D 84 (2011) 025007 [arXiv:1007.2615] [INSPIRE].
P.P. Kulish and L.D. Faddeev, Asymptotic conditions and infrared divergences in quantum electrodynamics, Theor. Math. Phys. 4 (1970) 745 [Teor. Mat. Fiz. 4 (1970) 153] [INSPIRE].
S.B. Giddings, Nonviolent information transfer from black holes: a field theory parametrization, Phys. Rev. D 88 (2013) 024018 [arXiv:1302.2613] [INSPIRE].
W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7 (1954) 649 [INSPIRE].
I. Ojima, Gauge fields at finite temperatures: thermo field dynamics, KMS condition and their extension to gauge theories, Annals Phys. 137 (1981) 1 [INSPIRE].
N.P. Landsman and C.G. van Weert, Real and imaginary time field theory at finite temperature and density, Phys. Rept. 145 (1987) 141 [INSPIRE].
B. Swingle, Entanglement renormalization and holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [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].
M.M. Caldarelli, D. Klemm and P.J. Silva, Chronology protection in anti-de Sitter, Class. Quant. Grav. 22 (2005) 3461 [hep-th/0411203] [INSPIRE].
G. Milanesi and M. O’Loughlin, Singularities and closed time-like curves in type IIB 1/2 BPS geometries, JHEP 09 (2005) 008 [hep-th/0507056] [INSPIRE].
B.S. DeWitt, Quantum field theory in curved space-time, Phys. Rept. 19 (1975) 295 [INSPIRE].
T. Dray and G. Hooft, The gravitational shock wave of a massless particle, Nucl. Phys. B 253 (1985) 173 [INSPIRE].
D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [INSPIRE].
K. Papadodimas and S. Raju, An infalling observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
J. Polchinski, String theory, volumes I and II, Cambridge University Press, Cambridge U.K. (1998).
P.J. Cameron, Aspects of infinite permutation groups, Cambridge University Press, Cambridge U.K. (2005), pg. 1.
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
J. Ware, R. Saotome and R. Akhoury, Construction of an asymptotic S matrix for perturbative quantum gravity, JHEP 10 (2013) 159 [arXiv:1308.6285] [INSPIRE].
K. Papadodimas and S. Raju, State-dependent bulk-boundary maps and black hole complementarity, Phys. Rev. D 89 (2014) 086010 [arXiv:1310.6335] [INSPIRE].
K. Papadodimas and S. Raju, The black hole interior in AdS/CFT and the information paradox, Phys. Rev. Lett. 112 (2014) 051301 [arXiv:1310.6334] [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: 1311.5613v2
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
Akhoury, R. Unitary S matrices with long-range correlations and the quantum black hole. J. High Energ. Phys. 2014, 169 (2014). https://doi.org/10.1007/JHEP08(2014)169
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2014)169