Abstract
We revisit the old black hole S-Matrix construction and its new partial wave expansion of ’t Hooft. Inspired by old ideas from non-critical string theory & c = 1 Matrix Quantum Mechanics, we reformulate the scattering in terms of a quantum mechanical model — of waves scattering off inverted harmonic oscillator potentials — that exactly reproduces the unitary black hole S-Matrix for all spherical harmonics; each partial wave corresponds to an inverted harmonic oscillator with ground state energy that is shifted relative to the s-wave oscillator. Identifying a connection to 2d string theory allows us to show that there is an exponential degeneracy in how a given total initial energy may be distributed among many partial waves of the 4d black hole.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [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].
L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.D. Mathur, Confusions and questions about the information paradox, http://www.physics.ohio-state.edu/~mathur/confusions2.pdf.
S.D. Mathur, The fuzzball paradigm for black holes: FAQ, http://www.physics.ohio-state.edu/~mathur/faq2.pdf.
S.D. Mathur, The Information paradox: A Pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
S.D. Mathur, The Fuzzball proposal for black holes: An Elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
I. Bena and N.P. Warner, Black holes, black rings and their microstates, Lect. Notes Phys. 755 (2008) 1 [hep-th/0701216] [INSPIRE].
V. Balasubramanian, J. de Boer, S. El-Showk and I. Messamah, Black Holes as Effective Geometries, Class. Quant. Grav. 25 (2008) 214004 [arXiv:0811.0263] [INSPIRE].
K. Skenderis and M. Taylor, The fuzzball proposal for black holes, Phys. Rept. 467 (2008) 117 [arXiv:0804.0552] [INSPIRE].
S.D. Mathur, Fuzzballs and the information paradox: A Summary and conjectures, arXiv:0810.4525 [INSPIRE].
G. ’t Hooft, Scattering matrix for a quantized black hole, in proceedings of the International School of Cosmology and Gravitation: Black Hole Physics, Erice, Italy, May 12–22 1991 [INSPIRE].
G. ’t Hooft, Unitarity of the black hole scattering matrix, in proceedings of the International Conference on Fundamental Aspects of Quantum Theory to Celebrate the 60th Birthday of Yakir Aharonov, Columbia, South Carolina, December 10–12 1992 [INSPIRE].
G. ’t Hooft, Diagonalizing the Black Hole Information Retrieval Process, arXiv:1509.01695 [INSPIRE].
G. ’t Hooft, Black hole unitarity and antipodal entanglement, Found. Phys. 46 (2016) 1185 [arXiv:1601.03447] [INSPIRE].
G. ’t Hooft, The Quantum Black Hole as a Hydrogen Atom: Microstates Without Strings Attached, arXiv:1605.05119 [INSPIRE].
P.C. Aichelburg and R.U. Sexl, On the Gravitational field of a massless particle, Gen. Rel. Grav. 2 (1971) 303 [INSPIRE].
T. Dray and G. ’t Hooft, The Gravitational Shock Wave of a Massless Particle, Nucl. Phys. B 253 (1985) 173 [INSPIRE].
I.R. Klebanov, String theory in two-dimensions, in proceedings of the Spring School on String Theory and Quantum Gravity (to be followed by Workshop), Trieste, Italy, April 15–23 1991 [hep-th/9108019] [INSPIRE].
G.W. Moore, M.R. Plesser and S. Ramgoolam, Exact S-matrix for two-dimensional string theory, Nucl. Phys. B 377 (1992) 143 [hep-th/9111035] [INSPIRE].
K. Schoutens, H.L. Verlinde and E.P. Verlinde, Quantum black hole evaporation, Phys. Rev. D 48 (1993) 2670 [hep-th/9304128] [INSPIRE].
E.P. Verlinde and H.L. Verlinde, A Unitary S-matrix and two-dimensional black hole formation and evaporation, Nucl. Phys. B 406 (1993) 43 [hep-th/9302022] [INSPIRE].
S.Yu. Alexandrov, V.A. Kazakov and I.K. Kostov, Time dependent backgrounds of 2D string theory, Nucl. Phys. B 640 (2002) 119 [hep-th/0205079] [INSPIRE].
J.L. Karczmarek, J.M. Maldacena and A. Strominger, Black hole non-formation in the matrix model, JHEP 01 (2006) 039 [hep-th/0411174] [INSPIRE].
J.J. Friess and H.L. Verlinde, Hawking effect in 2D string theory, hep-th/0411100 [INSPIRE].
J.M. Maldacena and N. Seiberg, Flux-vacua in two dimensional string theory, JHEP 09 (2005) 077 [hep-th/0506141] [INSPIRE].
C.G. Callan Jr., S.B. Giddings, J.A. Harvey and A. Strominger, Evanescent black holes, Phys. Rev. D 45 (1992) R1005(R) [hep-th/9111056] [INSPIRE].
G.W. Gibbons and P.K. Townsend, Black holes and Calogero models, Phys. Lett. B 454 (1999) 187 [hep-th/9812034] [INSPIRE].
V. Kazakov, I.K. Kostov and D. Kutasov, A Matrix model for the two-dimensional black hole, Nucl. Phys. B 622 (2002) 141 [hep-th/0101011] [INSPIRE].
J.M. Magan, Black holes as random particles: entanglement dynamics in infinite range and matrix models, JHEP 08 (2016) 081 [arXiv:1601.04663] [INSPIRE].
A. Jansen and J.M. Magan, Black hole collapse and democratic models, Phys. Rev. D 94 (2016) 104007 [arXiv:1604.03772] [INSPIRE].
S. Banerjee, J.-W. Bryan, K. Papadodimas and S. Raju, A toy model of black hole complementarity, JHEP 05 (2016) 004 [arXiv:1603.02812] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
K. Papadodimas and S. Raju, An Infalling Observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, arXiv:1601.05000 [INSPIRE].
S. Weinberg, Six-dimensional Methods for Four-dimensional Conformal Field Theories, Phys. Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].
S.W. Hawking, The Information Paradox for Black Holes, arXiv:1509.01147 [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Soft Hair on Black Holes, Phys. Rev. Lett. 116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].
J.M. Maldacena, Long strings in two dimensional string theory and non-singlets in the matrix model, JHEP 09 (2005) 078 [Int. J. Geom. Meth. Mod. Phys. 3 (2006) 1] [hep-th/0503112] [INSPIRE].
E.J. Martinec, Matrix models and 2D string theory, in proceedings of the 9th Frontiers of Mathematical Physics Summer School on Strings, Gravity and Cosmology, Vancouver, Canada, August 2–13 2004, pp. 403–457 [in proceedings of the NATO Advanced Study Institute: Marie Curie Training Course: Applications of Random Matrices in Physics, Les Houches, France, June 6–25 2004, pp. 403–457] [hep-th/0410136] [INSPIRE].
G.W. Moore and N. Seiberg, From loops to fields in 2D quantum gravity, Int. J. Mod. Phys. A 7 (1992) 2601 [INSPIRE].
J. McGreevy and H.L. Verlinde, Strings from tachyons: The c = 1 matrix reloaded, JHEP 12 (2003) 054 [hep-th/0304224] [INSPIRE].
P.H. Ginsparg and G.W. Moore, Lectures on 2D gravity and 2D string theory, hep-th/9304011 [INSPIRE].
Y. Nakayama, Liouville field theory: A Decade after the revolution, Int. J. Mod. Phys. A 19 (2004) 2771 [hep-th/0402009] [INSPIRE].
M.S. de Bianchi, Time-delay of classical and quantum scattering processes: a conceptual overview and a general definition, Cent. Eur. J. Phys. 10 (2012) 282 [arXiv:1010.5329].
C. de Carvalho and H. Nussenzveig, Time delay, Phys. Rept. 364 (2002) 83.
T. Banks, Holographic Space-time Models in 1 + 1 Dimensions, arXiv:1506.05777 [INSPIRE].
V. Kazakov, Bosonic strings and string field theories in one-dimensional target space, in proceedings of the Cargese Study Institute: Random Surfaces, Quantum Gravity and Strings, Cargese, France, May 27–June 2 1990 [INSPIRE].
C. Asplund and D. Berenstein, Non-adiabaticity and improved back-reaction, arXiv:1009.4667 [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), talk given at KITP Program: Entanglement in Strongly-Correlated Quantum Matter, April 7–May 27 2015 and online at http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
A. Kitaev, A simple model of quantum holography (part 2), talk given at KITP Program: Entanglement in Strongly-Correlated Quantum Matter, April 7–May 27 2015 and online at http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
D. Anninos, S.A. Hartnoll, L. Huijse and V.L. Martin, Large-N matrices from a nonlocal spin system, Class. Quant. Grav. 32 (2015) 195009 [arXiv:1412.1092] [INSPIRE].
D. Anninos, F. Denef and R. Monten, Grassmann Matrix Quantum Mechanics, JHEP 04 (2016) 138 [arXiv:1512.03803] [INSPIRE].
J.M. Maldacena and D. Stanford, Comments on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
G.W. Gibbons, The Elliptic Interpretation of Black Holes and Quantum Mechanics, Nucl. Phys. B 271 (1986) 497 [INSPIRE].
N.G. Sanchez and B.F. Whiting, Quantum Field Theory and the Antipodal Identification of Black Holes, Nucl. Phys. B 283 (1987) 605 [INSPIRE].
M.K. Parikh, I. Savonije and E.P. Verlinde, Elliptic de Sitter space: dS/Z 2, Phys. Rev. D 67 (2003) 064005 [hep-th/0209120] [INSPIRE].
J.M. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
D. Boulatov and V. Kazakov, One-dimensional string theory with vortices as the upside down matrix oscillator, Int. J. Mod. Phys. A 8 (1993) 809 [hep-th/0012228] [INSPIRE].
B. de Wit, J. Hoppe and H. Nicolai, On the Quantum Mechanics of Supermembranes, Nucl. Phys. B 305 (1988) 545 [INSPIRE].
W. Taylor, M(atrix) theory: Matrix quantum mechanics as a fundamental theory, Rev. Mod. Phys. 73 (2001) 419 [hep-th/0101126] [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: 1607.07885
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
Betzios, P., Gaddam, N. & Papadoulaki, O. The black hole S-Matrix from quantum mechanics. J. High Energ. Phys. 2016, 131 (2016). https://doi.org/10.1007/JHEP11(2016)131
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2016)131