Abstract
We study canonical quantization of Jackiw-Teibelboim (JT) gravity coupled to a massless scalar field. We provide concrete expressions of matter SL(2, R) charges and the boundary matter operators in terms of the creation and annihilation operators in the scalar field. The matter charges are represented in the form of an oscillator (Jordon-Schwinger) realization of the SL(2, R) algebra. We also show how the gauge constraints are implemented classically, by matching explicitly classical solutions of Schwarzian dynamics with bulk solutions. We introduce n-point transition functions defined by insertions of boundary matter operators along the two-sided Lorentzian evolution, which may fully spell out the quantum dynamics in the presence of matter. For the Euclidean case, we proceed with a two-sided picture of the disk geometry and consider the two-sided 2-point correlation function where initial and final states are arranged by inserting matter operators in a specific way. For some simple initial states, we evaluate the correlation function perturbatively. We also discuss some basic features of the two-sided correlation functions with additional insertions of boundary matter operators along the two-sided evolution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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 [INSPIRE].
T.G. Mertens and G.J. Turiaci, Solvable models of quantum black holes: a review on Jackiw-Teitelboim gravity, arXiv:2210.10846 [https://doi.org/10.48550/arXiv.2210.10846].
G. Sárosi, AdS2 holography and the SYK model, PoS Modave2017 (2018) 001 [arXiv:1711.08482] [INSPIRE].
D.A. Trunin, Pedagogical introduction to the Sachdev-Ye-Kitaev model and two-dimensional dilaton gravity, Usp. Fiz. Nauk 191 (2021) 225 [arXiv:2002.12187] [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].
K. Jensen, Chaos in AdS2 holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
D. Stanford and E. Witten, Fermionic localization of the Schwarzian theory, JHEP 10 (2017) 008 [arXiv:1703.04612] [INSPIRE].
D. Harlow and D. Jafferis, The factorization problem in Jackiw-Teitelboim gravity, JHEP 02 (2020) 177 [arXiv:1804.01081] [INSPIRE].
G. Penington and E. Witten, Algebras and states in JT gravity, arXiv:2301.07257 [https://doi.org/10.48550/arXiv.2301.07257].
D.L. Jafferis and D.K. Kolchmeyer, Entanglement entropy in Jackiw-Teitelboim gravity, arXiv:1911.10663 [https://doi.org/10.48550/arXiv.1911.10663].
D. Marolf and A.C. Wall, Eternal black holes and superselection in AdS/CFT, Class. Quant. Grav. 30 (2013) 025001 [arXiv:1210.3590] [INSPIRE].
S. Leutheusser and H. Liu, Causal connectability between quantum systems and the black hole interior in holographic duality, arXiv:2110.05497 [https://doi.org/10.48550/arXiv.2110.05497].
S. Leutheusser and H. Liu, Emergent times in holographic duality, arXiv:2112.12156 [https://doi.org/10.48550/arXiv.2112.12156].
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, JHEP 02 (2023) 082 [arXiv:2206.10780] [INSPIRE].
V. Chandrasekaran, G. Penington and E. Witten, Large N algebras and generalized entropy, arXiv:2209.10454 [https://doi.org/10.1007/JHEP04(2023)009] [INSPIRE].
D.K. Kolchmeyer, Von Neumann algebras in JT gravity, arXiv:2303.04701 [https://doi.org/10.48550/arXiv.2303.04701].
D. Bak, C. Kim, S.-H. Yi and J. Yoon, Python’s lunches in Jackiw-Teitelboim gravity with matter, JHEP 2022 (2022) 175 [arXiv:2112.04224].
H.W. Lin, J. Maldacena, L. Rozenberg and J. Shan, Looking at supersymmetric black holes for a very long time, arXiv:2207.00408 [https://doi.org/10.48550/arXiv.2207.00408].
A. Almheiri and J. Polchinski, Models of AdS2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [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].
H.W. Lin, J. Maldacena and Y. Zhao, Symmetries near the horizon, JHEP 08 (2019) 049 [arXiv:1904.12820] [INSPIRE].
D. Marolf and I.A. Morrison, Group averaging for de Sitter free fields, Class. Quant. Grav. 26 (2009) 235003 [arXiv:0810.5163] [INSPIRE].
D. Bak, C. Kim and S.-H. Yi, Structure of deformations in Jackiw-Teitelboim black holes with matter, arXiv:2209.01394 [https://doi.org/10.48550/arXiv.2209.01394].
M. Spradlin and A. Strominger, Vacuum states for AdS2 black holes, JHEP 11 (1999) 021 [hep-th/9904143] [INSPIRE].
D. Bak, C. Kim and S.-H. Yi, Bulk view of teleportation and traversable wormholes, JHEP 08 (2018) 140 [arXiv:1805.12349] [INSPIRE].
D. Harlow, Jerusalem lectures on black holes and quantum information, Rev. Mod. Phys. 88 (2016) 015002 [arXiv:1409.1231] [INSPIRE].
C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, Print-86-1309, Princeton University (1986) [INSPIRE].
R.M. Wald, Quantum field theory in curved space-time and black hole thermodynamics, University of Chicago Press, Chicago, IL, U.S.A. (1995) [INSPIRE].
J.S. Cotler et al., Black holes and random matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
G. Mandal, P. Nayak and S.R. Wadia, Coadjoint orbit action of Virasoro group and two-dimensional quantum gravity dual to SYK/tensor models, JHEP 11 (2017) 046 [arXiv:1702.04266] [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].
T.G. Mertens, The Schwarzian theory — origins, JHEP 05 (2018) 036 [arXiv:1801.09605] [INSPIRE].
A. Blommaert, T.G. Mertens and H. Verschelde, The Schwarzian theory — a Wilson line perspective, JHEP 12 (2018) 022 [arXiv:1806.07765] [INSPIRE].
D. Bak, M. Gutperle and A. Karch, Time dependent black holes and thermal equilibration, JHEP 12 (2007) 034 [arXiv:0708.3691] [INSPIRE].
A. Kitaev and S.J. Suh, Statistical mechanics of a two-dimensional black hole, JHEP 05 (2019) 198 [arXiv:1808.07032] [INSPIRE].
Z. Yang, The quantum gravity dynamics of near extremal black holes, JHEP 05 (2019) 205 [arXiv:1809.08647] [INSPIRE].
L.V. Iliesiu, S.S. Pufu, H. Verlinde and Y. Wang, An exact quantization of Jackiw-Teitelboim gravity, JHEP 11 (2019) 091 [arXiv:1905.02726] [INSPIRE].
H.W. Lin, The bulk Hilbert space of double scaled SYK, JHEP 11 (2022) 060 [arXiv:2208.07032] [INSPIRE].
A. Kitaev, Notes on \( \overset{\sim }{\textrm{SL}} \)(2, R) representations, arXiv:1711.08169 [https://doi.org/10.48550/arXiv.1711.08169].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2303.05057
Rights and permissions
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.
About this article
Cite this article
Bak, D., Kim, C. & Yi, SH. Quantization of Jackiw-Teitelboim gravity with a massless scalar. J. High Energ. Phys. 2023, 45 (2023). https://doi.org/10.1007/JHEP05(2023)045
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2023)045