Abstract
The emergence of the bulk Hilbert space is a mysterious concept in holography. In [1], the SYK model was solved in the double scaling limit by summing chord diagrams. Here, we explicitly construct the bulk Hilbert space of double scaled SYK by slicing open these chord diagrams; this Hilbert space resembles that of a lattice field theory where the length of the lattice is dynamical and determined by the chord number. Under a calculable bulk-to-boundary map, states of fixed chord number map to particular entangled 2-sided states with a corresponding size. This bulk reconstruction is well-defined even when quantum gravity effects are important. Acting on the double scaled Hilbert space is a Type II1 algebra of observables, which includes the Hamiltonian and matter operators. In the appropriate quantum Schwarzian limit, we also identify the JT gravitational algebra including the physical SL(2, ℝ) symmetry generators, and obtain explicit representations of the algebra using chord diagram techniques.
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M. Berkooz, M. Isachenkov, V. Narovlansky and G. Torrents, Towards a full solution of the large N double-scaled SYK model, JHEP 03 (2019) 079 [arXiv:1811.02584] [INSPIRE].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, http://online.kitp.ucsb.edu/online/entangled15/kitaev/, University of California, Santa Barbara, CA, U.S.A., 7 April 2015.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/, University of California, Santa Barbara, CA, U.S.A., 27 May 2015.
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [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.S. Cotler et al., Black holes and random matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
M. Berkooz, P. Narayan and J. Simon, Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, JHEP 08 (2018) 192 [arXiv:1806.04380] [INSPIRE].
S. Leutheusser and H. Liu, Causal connectability between quantum systems and the black hole interior in holographic duality, arXiv:2110.05497 [INSPIRE].
S. Leutheusser and H. Liu, Emergent times in holographic duality, arXiv:2112.12156 [INSPIRE].
E. Witten, Gravity and the crossed product, JHEP 10 (2022) 008 [arXiv:2112.12828] [INSPIRE].
V. Chandrasekaran, R. Longo, G. Penington and E. Witten, An algebra of observables for de Sitter space, arXiv:2206.10780 [INSPIRE].
H.W. Lin, J. Maldacena and Y. Zhao, Symmetries near the horizon, JHEP 08 (2019) 049 [arXiv:1904.12820] [INSPIRE].
D. Harlow and J.-Q. Wu, Algebra of diffeomorphism-invariant observables in Jackiw-Teitelboim gravity, JHEP 05 (2022) 097 [arXiv:2108.04841] [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. Harlow and D. Jafferis, The factorization problem in Jackiw-Teitelboim gravity, JHEP 02 (2020) 177 [arXiv:1804.01081] [INSPIRE].
M. Berkooz, N. Brukner, V. Narovlansky and A. Raz, The double scaled limit of super-symmetric SYK models, JHEP 12 (2020) 110 [arXiv:2003.04405] [INSPIRE].
H.W. Lin, J. Maldacena, L. Rozenberg and J. Shan, Looking at supersymmetric black holes for a very long time, arXiv:2207.00408 [INSPIRE].
A. Blommaert, T.G. Mertens and H. Verschelde, The Schwarzian theory — a Wilson line perspective, JHEP 12 (2018) 022 [arXiv:1806.07765] [INSPIRE].
A. Blommaert, T.G. Mertens and H. Verschelde, Fine structure of Jackiw-Teitelboim quantum gravity, JHEP 09 (2019) 066 [arXiv:1812.00918] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
M. Pluma and R. Speicher, A dynamical version of the SYK model and the q-Brownian motion, Random Matrices: Theory and Applications 11 (2022) 2250031 [arXiv:1905.12999] [INSPIRE].
H.W. Lin, D. Stanford and Z. Yang, in preparation (2022).
J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv:1804.00491 [INSPIRE].
I. Kourkoulou and J. Maldacena, Pure states in the SYK model and nearly-AdS2 gravity, arXiv:1707.02325 [INSPIRE].
A. Goel, H.T. Lam, G.J. Turiaci and H. Verlinde, Expanding the black hole interior: partially entangled thermal states in SYK, JHEP 02 (2019) 156 [arXiv:1807.03916] [INSPIRE].
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].
P. Saad, Late time correlation functions, baby universes, and ETH in JT gravity, arXiv:1910.10311 [INSPIRE].
G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, JHEP 03 (2022) 205 [arXiv:1911.11977] [INSPIRE].
H.W. Lin, J. Maldacena, L. Rozenberg and J. Shan, Holography for people with no time, arXiv:2207.00407 [INSPIRE].
D.E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi and E. Altman, A universal operator growth hypothesis, Phys. Rev. X 9 (2019) 041017 [arXiv:1812.08657] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, Operator complexity: a journey to the edge of Krylov space, JHEP 06 (2021) 062 [arXiv:2009.01862] [INSPIRE].
V. Balasubramanian, P. Caputa, J.M. Magan and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D 106 (2022) 046007 [arXiv:2202.06957] [INSPIRE].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Holographic representation of local bulk operators, Phys. Rev. D 74 (2006) 066009 [hep-th/0606141] [INSPIRE].
D. Stanford and L. Susskind, Complexity and shock wave geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
L.V. Iliesiu, M. Mezei and G. Sárosi, The volume of the black hole interior at late times, JHEP 07 (2022) 073 [arXiv:2107.06286] [INSPIRE].
D. Stanford and Z. Yang, Firewalls from wormholes, arXiv:2208.01625 [INSPIRE].
X.-L. Qi and A. Streicher, Quantum epidemiology: operator growth, thermal effects, and SYK, JHEP 08 (2019) 012 [arXiv:1810.11958] [INSPIRE].
M. Bozejko, B. Kummerer and R. Speicher, Q Gaussian processes: noncommutative and classical aspects, Commun. Math. Phys. 185 (1997) 129 [funct-an/9604010] [INSPIRE].
E. Ricard, Factoriality of q-Gaussian von Neumann algebras, math.FA/0311413.
P. Sniady, Factoriality of Boejko-Speicher von Neumann algebras, Commun. Math. Phys. 246 (2004) 561.
M.A. Olshanetsky and V.B.K. Rogov, Liouville quantum mechanics on a lattice from geometry of quantum Lorentz group, J. Phys. A 27 (1994) 4669 [hep-th/9310084] [INSPIRE].
A. Almheiri, X. Dong and D. Harlow, Bulk locality and quantum error correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
V. Chandrasekaran and A. Levine, Quantum error correction in SYK and bulk emergence, JHEP 06 (2022) 039 [arXiv:2203.05058] [INSPIRE].
D.L. Jafferis, D.K. Kolchmeyer, B. Mukhametzhanov and J. Sonner, JT gravity with matter, generalized ETH and random matrices, arXiv:2209.02131 [INSPIRE].
L. Susskind, Entanglement and chaos in de Sitter space holography: an SYK example, JHAP 1 (2021) 1 [arXiv:2109.14104] [INSPIRE].
H. Verlinde, Duality between SYK and 2 + 1D de Sitter gravity, unpublished work (2019).
H. Lin and L. Susskind, Infinite temperature’s not so hot, arXiv:2206.01083 [INSPIRE].
Y.D. Lensky and X.-L. Qi, Rescuing a black hole in the large-q coupled SYK model, JHEP 04 (2021) 116 [arXiv:2012.15798] [INSPIRE].
Z. Yang, The quantum gravity dynamics of near extremal black holes, JHEP 05 (2019) 205 [arXiv:1809.08647] [INSPIRE].
A. Kitaev and S.J. Suh, Statistical mechanics of a two-dimensional black hole, JHEP 05 (2019) 198 [arXiv:1808.07032] [INSPIRE].
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Lin, H.W. The bulk Hilbert space of double scaled SYK. J. High Energ. Phys. 2022, 60 (2022). https://doi.org/10.1007/JHEP11(2022)060
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DOI: https://doi.org/10.1007/JHEP11(2022)060