Abstract
In [1] we gave a precise holographic calculation of chaos at the scrambling time scale. We studied the influence of a small perturbation, long in the past, on a two-sided correlation function in the thermofield double state. A similar analysis applies to squared commutators and other out-of-time-order one-sided correlators [2-6]. The essential bulk physics is a high energy scattering problem near the horizon of an AdS black hole. The above papers used Einstein gravity to study this problem; in the present paper we consider stringy and Planckian corrections. Elastic stringy corrections play an important role, effectively weakening and smearing out the development of chaos. We discuss their signature in the boundary field theory, commenting on the extension to weak coupling. Inelastic effects, although important for the evolution of the state, leave a parametrically small imprint on the correlators that we study. We briefly discuss ways to diagnose these small corrections, and we propose another correlator where inelastic effects are order one.
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S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
S.H. Shenker and D. Stanford, Multiple Shocks, JHEP 12 (2014) 046 [arXiv:1312.3296] [INSPIRE].
D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
A. Kitaev, Hidden Correlations in the Hawking Radiation and Thermal Noise, talk given at Fundamental Physics Prize Symposium, Nov. 10, 2014.
A. Kitaev, talk at Stanford SITP seminars, Nov. 11, 2014.
A. Kitaev, talk at Stanford SITP seminars, Dec. 18, 2014.
L. Susskind, Strings, black holes and Lorentz contraction, Phys. Rev. D 49 (1994) 6606 [hep-th/9308139] [INSPIRE].
A. Mezhlumian, A.W. Peet and L. Thorlacius, String thermalization at a black hole horizon, Phys. Rev. D 50 (1994) 2725 [hep-th/9402125] [INSPIRE].
K. Schoutens, H.L. Verlinde and E.P. Verlinde, Quantum black hole evaporation, Phys. Rev. D 48 (1993) 2670 [hep-th/9304128] [INSPIRE].
P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].
C. Dankert, R. Cleve, J. Emerson, and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Phys. Rev. A 80 (2009) 012304 [quant-ph/0606161].
J. Emerson, E. Livine, and S. Lloyd, Convergence conditions for random quantum circuits, Phys. Rev. A 72 (2005) 060302 [quant-ph/0503210].
A.W. Harrow and R.A. Low, Random quantum circuits are approximate 2-designs, Comm. Math. Phys. 291 (2009) 257 [arXiv:0802.1919].
L. Arnaud and D. Braun, Efficiency of producing random unitary matrices with quantum circuits, Phys. Rev. A 78 (2008) 062329 [arXiv:0807.0775].
W.G. Brown and L. Viola, Convergence rates for arbitrary statistical moments of random quantum circuits, Phys. Rev. Lett. 104 (2010) 250501 [arXiv:0910.0913].
I.T. Diniz and D. Jonathan, Comment on the paper ‘Random quantum circuits are approximate 2-designs, Comm. Math. Phys. 304 (2011) 281 [arXiv:1006.4202].
W. Brown and O. Fawzi, Scrambling speed of random quantum circuits, arXiv:1210.6644 [INSPIRE].
N. Lashkari, D. Stanford, M. Hastings, T. Osborne and P. Hayden, Towards the Fast Scrambling Conjecture, JHEP 04 (2013) 022 [arXiv:1111.6580] [INSPIRE].
Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [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].
D. Marolf and J. Polchinski, Gauge/Gravity Duality and the Black Hole Interior, Phys. Rev. Lett. 111 (2013) 171301 [arXiv:1307.4706] [INSPIRE].
L. Susskind, The Transfer of Entanglement: The Case for Firewalls, arXiv:1210.2098 [INSPIRE].
B. Freivogel and L. Susskind, A Framework for the landscape, Phys. Rev. D 70 (2004) 126007 [hep-th/0408133] [INSPIRE].
R. Bousso and L. Susskind, The Multiverse Interpretation of Quantum Mechanics, Phys. Rev. D 85 (2012) 045007 [arXiv:1105.3796] [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].
A. Lawrence and E. Silverstein, in progress and private communication.
E. Silverstein, unpublished notes, February 2013.
E. Silverstein, Backdraft: String Creation in an Old Schwarzschild Black Hole, arXiv:1402.1486 [INSPIRE].
M. Dodelson and E. Silverstein, work in progress.
B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, Rindler Quantum Gravity, Class. Quant. Grav. 29 (2012) 235025 [arXiv:1206.1323] [INSPIRE].
W. Israel, Thermo field dynamics of black holes, Phys. Lett. A 57 (1976) 107 [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
D.A. Roberts and D. Stanford, Two-dimensional conformal field theory and the butterfly effect, arXiv:1412.5123 [INSPIRE].
A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Sov. Phys. JETP 28 (1969) 1200.
S. Jackson, L. McGough and H. Verlinde, Conformal Bootstrap, Universality and Gravitational Scattering, arXiv:1412.5205 [INSPIRE].
L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions, JHEP 08 (2007) 019 [hep-th/0611122] [INSPIRE].
L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal Approximation in AdS/CFT: Conformal Partial Waves and Finite N Four-Point Functions, Nucl. Phys. B 767 (2007) 327 [hep-th/0611123] [INSPIRE].
L. Cornalba, M.S. Costa and J. Penedones, Eikonal approximation in AdS/CFT: Resumming the gravitational loop expansion, JHEP 09 (2007) 037 [arXiv:0707.0120] [INSPIRE].
G. ’t Hooft, Graviton Dominance in Ultrahigh-Energy Scattering, Phys. Lett. B 198 (1987) 61 [INSPIRE].
D. Amati, M. Ciafaloni and G. Veneziano, Classical and Quantum Gravity Effects from Planckian Energy Superstring Collisions, Int. J. Mod. Phys. A 3 (1988) 1615 [INSPIRE].
H.L. Verlinde and E.P. Verlinde, Scattering at Planckian energies, Nucl. Phys. B 371 (1992) 246 [hep-th/9110017] [INSPIRE].
D.N. Kabat and M. Ortiz, Eikonal quantum gravity and Planckian scattering, Nucl. Phys. B 388 (1992) 570 [hep-th/9203082] [INSPIRE].
R.C. Brower, M.J. Strassler and C.-I. Tan, On the eikonal approximation in AdS space, JHEP 03 (2009) 050 [arXiv:0707.2408] [INSPIRE].
R.C. Brower, J. Polchinski, M.J. Strassler and C.-I. Tan, The Pomeron and gauge/string duality, JHEP 12 (2007) 005 [hep-th/0603115] [INSPIRE].
G. ’t Hooft, The black hole interpretation of string theory, Nucl. Phys. B 335 (1990) 138 [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].
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].
K. Sfetsos, On gravitational shock waves in curved space-times, Nucl. Phys. B 436 (1995) 721 [hep-th/9408169] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
A. Giveon, D. Kutasov and N. Seiberg, Comments on string theory on AdS 3, Adv. Theor. Math. Phys. 2 (1998) 733 [hep-th/9806194] [INSPIRE].
J. de Boer, H. Ooguri, H. Robins and J. Tannenhauser, String theory on AdS 3, JHEP 12 (1998) 026 [hep-th/9812046] [INSPIRE].
D. Kutasov and N. Seiberg, More comments on string theory on AdS 3, JHEP 04 (1999) 008 [hep-th/9903219] [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS 3 and \( \mathrm{S}\mathrm{L}\left(2,\mathbb{R}\right) \) WZW model 1.: The Spectrum, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS 3 and the \( \mathrm{S}\mathrm{L}\left(2,\mathbb{R}\right) \) WZW model. Part 3. Correlation functions, Phys. Rev. D 65 (2002) 106006 [hep-th/0111180] [INSPIRE].
D. Amati and C. Klimčík, Strings in a Shock Wave Background and Generation of Curved Geometry from Flat Space String Theory, Phys. Lett. B 210 (1988) 92 [INSPIRE].
S.B. Giddings, Locality in quantum gravity and string theory, Phys. Rev. D 74 (2006) 106006 [hep-th/0604072] [INSPIRE].
S.B. Giddings, D.J. Gross and A. Maharana, Gravitational effects in ultrahigh-energy string scattering, Phys. Rev. D 77 (2008) 046001 [arXiv:0705.1816] [INSPIRE].
L.N. Lipatov, Reggeization of the Vector Meson and the Vacuum Singularity in Nonabelian Gauge Theories, Sov. J. Nucl. Phys. 23 (1976) 338 [INSPIRE].
E.A. Kuraev, L.N. Lipatov and V.S. Fadin, The Pomeranchuk Singularity in Nonabelian Gauge Theories, Sov. Phys. JETP 45 (1977) 199 [Zh. Eksp. Teor. Fiz. 72 (1977) 377].
I.I. Balitsky and L.N. Lipatov, The Pomeranchuk Singularity in Quantum Chromodynamics, Sov. J. Nucl. Phys. 28 (1978) 822 [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
B. Basso, S. Caron-Huot and A. Sever, Adjoint BFKL at finite coupling: a short-cut from the collinear limit, JHEP 01 (2015) 027 [arXiv:1407.3766] [INSPIRE].
M. Alfimov, N. Gromov and V. Kazakov, QCD Pomeron from AdS/CFT Quantum Spectral Curve, arXiv:1408.2530 [INSPIRE].
T. Banks and G. Festuccia, The Regge Limit for Green Functions in Conformal Field Theory, JHEP 06 (2010) 105 [arXiv:0910.2746] [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].
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Shenker, S.H., Stanford, D. Stringy effects in scrambling. J. High Energ. Phys. 2015, 132 (2015). https://doi.org/10.1007/JHEP05(2015)132
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DOI: https://doi.org/10.1007/JHEP05(2015)132