Abstract
Schwarzian quantum mechanics describes the collective IR mode of the SYK model and captures key features of 2D black hole dynamics. Exact results for its correlation functions were obtained in [1]. We compare these results with bulk gravity expectations. We find that the semi-classical limit of the OTO four-point function exactly matches with the scattering amplitude obtained from the Dray-’t Hooft shockwave \( \mathcal{S} \)-matrix. We show that the two point function of heavy operators reduces to the semi-classical saddle-point of the Schwarzian action. We also explain a previously noted match between the OTO four point functions and 2D conformal blocks. Generalizations to higher-point functions are discussed.
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References
T.G. Mertens, G.J. Turiaci and H.L. Verlinde, Solving the Schwarzian via the conformal bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [INSPIRE].
T. Dray and G. ’t Hooft, The gravitational shock wave of a massless particle, Nucl. Phys. B 253 (1985) 173 [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].
K. Schoutens, H.L. Verlinde and E.P. Verlinde, Quantum black hole evaporation, Phys. Rev. D 48 (1993) 2670 [hep-th/9304128] [INSPIRE].
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].
S. Jackson, L. McGough and H. Verlinde, Conformal bootstrap, universality and gravitational scattering, Nucl. Phys. B 901 (2015) 382 [arXiv:1412.5205] [INSPIRE].
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
A. Kitaev, Hidden correlations in the Hawking radiation and themral noise, talk given at the Fundamental Physics Prize Symposium, November 10, Stanford University, Stanford U.S.A. (2014).
A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, talk given at KITP seminar, February 12, Caltech, Pasadena, U.S.A. (2015).
A. Kitaev, A simple model of quantum holography, talks given at KITP, April 7 and May 27 (2015).
A. Kitaev and S.J. Suh, The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual, JHEP 05 (2018) 183 [arXiv:1711.08467] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
A. Jevicki, K. Suzuki and J. Yoon, Bi-local holography in the SYK model, JHEP 07 (2016) 007 [arXiv:1603.06246] [INSPIRE].
A. Jevicki and K. Suzuki, Bi-local holography in the SYK model: perturbations, JHEP 11 (2016) 046 [arXiv:1608.07567] [INSPIRE].
J.S. Cotler et al., Black holes and random matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS 2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
R. Jackiw, Lower dimensional gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
C. Teitelboim, Gravitation and Hamiltonian structure in two space-time dimensions, Phys. Lett. B 126 (1983) 41.
K. Jensen, Chaos in AdS 2 holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional nearly Anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS 2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
M. Cvetič and I. Papadimitriou, AdS 2 holographic dictionary, JHEP 12 (2016) 008 [Erratum ibid. 01 (2017) 120] [arXiv:1608.07018] [INSPIRE].
P. Nayak et al., On the dynamics of near-extremal black holes, JHEP 09 (2018) 048 [arXiv:1802.09547] [INSPIRE].
G. Turiaci and H. Verlinde, On CFT and quantum chaos, JHEP 12 (2016) 110 [arXiv:1603.03020] [INSPIRE].
G. Turiaci and H. Verlinde, Towards a 2d QFT analog of the SYK model, JHEP 10 (2017) 167 [arXiv:1701.00528] [INSPIRE].
D. Bagrets, A. Altland and A. Kamenev, Sachdev-Ye-Kitaev model as Liouville quantum mechanics, Nucl. Phys. B 911 (2016) 191 [arXiv:1607.00694] [INSPIRE].
D. Bagrets, A. Altland and A. Kamenev, Power-law out of time order correlation functions in the SYK model, Nucl. Phys. B 921 (2017) 727 [arXiv:1702.08902] [INSPIRE].
D. Stanford and E. Witten, Fermionic localization of the Schwarzian theory, JHEP 10 (2017) 008 [arXiv:1703.04612] [INSPIRE].
Z. Yang, The quantum gravity dynamics of near extremal black holes, arXiv:1809.08647.
P. Gao, D.L. Jafferis and A. Wall, Traversable wormholes via a double trace deformation, JHEP 12 (2017) 151 [arXiv:1608.05687] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Diving into traversable wormholes, Fortsch. Phys. 65 (2017) 1700034 [arXiv:1704.05333] [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].
W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [INSPIRE].
T.G. Mertens, The Schwarzian theory — Origins, JHEP 05 (2018) 036 [arXiv:1801.09605] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152 [INSPIRE].
J.-L. Gervais and A. Neveu, The dual string spectrum in Polyakov’s quantization. 1, Nucl. Phys. B 199 (1982) 59 [INSPIRE].
J.-L. Gervais and A. Neveu, Dual string spectrum in Polyakov’s quantization. 2. Mode separation, Nucl. Phys. B 209 (1982) 125 [INSPIRE].
J.L. Gervais and A. Neveu, New quantum solution of Liouville field theory, Phys. Lett. B 123 (1983) 86.
J.-L. Gervais and A. Neveu, New quantum treatment of Liouville field theory, Nucl. Phys. B 224 (1983) 329 [INSPIRE].
H. Dorn and G. Jorjadze, Boundary Liouville theory: Hamiltonian description and quantization, SIGMA 3 (2007) 012 [hep-th/0610197] [INSPIRE].
H. Dorn and G. Jorjadze, Operator approach to boundary Liouville theory, Annals Phys. 323 (2008) 2799 [arXiv:0801.3206] [INSPIRE].
B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [INSPIRE].
W. Groenevelt, The Wilson function transform, math.CA/0306424.
W. Groenevelt, Wilson function transforms related to Racah coefficients, math.CA/0501511.
B. Le Floch and G.J. Turiaci, AGT/ℤ2, JHEP 12 (2017) 099 [arXiv:1708.04631] [INSPIRE].
H. Chen et al., Degenerate operators and the 1/c expansion: Lorentzian resummations, high order computations and super-Virasoro blocks, JHEP 03 (2017) 167 [arXiv:1606.02659] [INSPIRE].
Y. Gu, A. Lucas and X.-L. Qi, Spread of entanglement in a Sachdev-Ye-Kitaev chain, JHEP 09 (2017) 120 [arXiv:1708.00871] [INSPIRE].
I. Kourkoulou and J. Maldacena, Pure states in the SYK model and nearly-AdS 2 gravity, arXiv:1707.02325 [INSPIRE].
A. Eberlein, V. Kasper, S. Sachdev and J. Steinberg, Quantum quench of the Sachdev-Ye-Kitaev Model, Phys. Rev. B 96 (2017) 205123 [arXiv:1706.07803] [INSPIRE].
J. Sonner and M. Vielma, Eigenstate thermalization in the Sachdev-Ye-Kitaev model, JHEP 11 (2017) 149 [arXiv:1707.08013] [INSPIRE].
N. Seiberg, Notes on quantum Liouville theory and quantum gravity, Prog. Theor. Phys. Suppl. 102 (1990) 319 [INSPIRE].
J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv:1804.00491 [INSPIRE].
F.M. Haehl and M. Rozali, Fine grained chaos in AdS 2 gravity, Phys. Rev. Lett. 120 (2018) 121601 [arXiv:1712.04963] [INSPIRE].
Y.-H. Qi, Y. Seo, S.-J. Sin and G. Song, Schwarzian correction to quantum correlation in SYK model, arXiv:1804.06164 [INSPIRE].
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Lam, H.T., Mertens, T.G., Turiaci, G.J. et al. Shockwave S-matrix from Schwarzian quantum mechanics. J. High Energ. Phys. 2018, 182 (2018). https://doi.org/10.1007/JHEP11(2018)182
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DOI: https://doi.org/10.1007/JHEP11(2018)182