Abstract
We investigate the simple model of Pennington, Shenker, Stanford and Yang for modeling the density matrix of Hawking radiation, but further include dynamics for EOW branes behind the horizon. This allows interactions that scatter one interior state to another, and also allows EOW loops. At strong coupling, we find that EOW states are no longer random; the ensemble has collapsed, and coupling constants encode the microscopic matrix elements of Hawking radiation. This suggests strong interior dynamics are important for understanding evaporating black holes, without any ensemble average. In this concrete model the density matrix of the radiation deviates from the thermal state, small off-diagonal fluctuations encode equivalences between naively orthogonal states, and bound the entropy from above. For almost evaporated black holes the off-diagonal terms become as large as the diagonal ones, eventually giving a pure state. We also find the unique analytic formula for all Renyi entropies.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, The entropy of Hawking radiation, Rev. Mod. Phys. 93 (2021) 035002 [arXiv:2006.06872] [INSPIRE].
G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 (2020) 002 [arXiv:1905.08255] [INSPIRE].
A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063 [arXiv:1905.08762] [INSPIRE].
A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [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].
D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
P. Saad, Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity, arXiv:1910.10311 [INSPIRE].
D. Marolf and H. Maxfield, Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information, JHEP 08 (2020) 044 [arXiv:2002.08950] [INSPIRE].
D. Stanford, More quantum noise from wormholes, arXiv:2008.08570 [INSPIRE].
A. Blommaert, T.G. Mertens and H. Verschelde, Eigenbranes in Jackiw-Teitelboim gravity, JHEP 02 (2021) 168 [arXiv:1911.11603] [INSPIRE].
A. Blommaert, Dissecting the ensemble in JT gravity, arXiv:2006.13971 [INSPIRE].
J. Pollack, M. Rozali, J. Sully and D. Wakeham, Eigenstate Thermalization and Disorder Averaging in Gravity, Phys. Rev. Lett. 125 (2020) 021601 [arXiv:2002.02971] [INSPIRE].
N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality, JHEP 01 (2021) 130 [arXiv:2006.04839] [INSPIRE].
A. Maloney and E. Witten, Averaging over Narain moduli space, JHEP 10 (2020) 187 [arXiv:2006.04855] [INSPIRE].
A. Belin and J. de Boer, Random statistics of OPE coefficients and Euclidean wormholes, Class. Quant. Grav. 38 (2021) 164001 [arXiv:2006.05499] [INSPIRE].
J. Cotler and K. Jensen, AdS3 gravity and random CFT, JHEP 04 (2021) 033 [arXiv:2006.08648] [INSPIRE].
T. Anous, J. Kruthoff and R. Mahajan, Density matrices in quantum gravity, SciPost Phys. 9 (2020) 045 [arXiv:2006.17000] [INSPIRE].
Y. Chen, V. Gorbenko and J. Maldacena, Bra-ket wormholes in gravitationally prepared states, JHEP 02 (2021) 009 [arXiv:2007.16091] [INSPIRE].
H. Liu and S. Vardhan, Entanglement entropies of equilibrated pure states in quantum many-body systems and gravity, PRX Quantum 2 (2021) 010344 [arXiv:2008.01089] [INSPIRE].
D. Marolf and J.E. Santos, AdS Euclidean wormholes, Class. Quant. Grav. 38 (2021) 224002 [arXiv:2101.08875] [INSPIRE].
V. Meruliya, S. Mukhi and P. Singh, Poincaré Series, 3d Gravity and Averages of Rational CFT, JHEP 04 (2021) 267 [arXiv:2102.03136] [INSPIRE].
S.B. Giddings and G.J. Turiaci, Wormhole calculus, replicas, and entropies, JHEP 09 (2020) 194 [arXiv:2004.02900] [INSPIRE].
D. Stanford and E. Witten, JT gravity and the ensembles of random matrix theory, Adv. Theor. Math. Phys. 24 (2020) 1475 [arXiv:1907.03363] [INSPIRE].
K. Okuyama and K. Sakai, JT gravity, KdV equations and macroscopic loop operators, JHEP 01 (2020) 156 [arXiv:1911.01659] [INSPIRE].
A. Belin, J. De Boer, P. Nayak and J. Sonner, Charged eigenstate thermalization, Euclidean wormholes and global symmetries in quantum gravity, SciPost Phys. 12 (2022) 059 [arXiv:2012.07875] [INSPIRE].
H. Verlinde, Deconstructing the Wormhole: Factorization, Entanglement and Decoherence, arXiv:2105.02142 [INSPIRE].
S. Collier and A. Maloney, Wormholes and spectral statistics in the Narain ensemble, JHEP 03 (2022) 004 [arXiv:2106.12760] [INSPIRE].
P. Saad, S. Shenker and S. Yao, Comments on wormholes and factorization, arXiv:2107.13130 [INSPIRE].
R. Bousso and E. Wildenhain, Gravity/ensemble duality, Phys. Rev. D 102 (2020) 066005 [arXiv:2006.16289] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, The Worldsheet Dual of the Symmetric Product CFT, JHEP 04 (2019) 103 [arXiv:1812.01007] [INSPIRE].
L. Eberhardt, Partition functions of the tensionless string, JHEP 03 (2021) 176 [arXiv:2008.07533] [INSPIRE].
L. Eberhardt, Summing over Geometries in String Theory, JHEP 05 (2021) 233 [arXiv:2102.12355] [INSPIRE].
A. Blommaert and J. Kruthoff, Gravity without averaging, SciPost Phys. 12 (2022) 073 [arXiv:2107.02178] [INSPIRE].
P. Saad, S.H. Shenker, D. Stanford and S. Yao, Wormholes without averaging, arXiv:2103.16754 [INSPIRE].
K. Papadodimas and S. Raju, An Infalling Observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
K. Papadodimas and S. Raju, The unreasonable effectiveness of exponentially suppressed corrections in preserving information, Int. J. Mod. Phys. D 22 (2013) 1342030 [INSPIRE].
D. Marolf and H. Maxfield, Observations of Hawking radiation: the Page curve and baby universes, JHEP 04 (2021) 272 [arXiv:2010.06602] [INSPIRE].
D. Marolf and H. Maxfield, The page curve and baby universes, Int. J. Mod. Phys. D 30 (2021) 2142027 [arXiv:2105.12211] [INSPIRE].
A. Goel, L.V. Iliesiu, J. Kruthoff and Z. Yang, Classifying boundary conditions in JT gravity: from energy-branes to α-branes, JHEP 04 (2021) 069 [arXiv:2010.12592] [INSPIRE].
D. Stanford, Z. Yang and S. Yao, Subleading Weingartens, JHEP 02 (2022) 200 [arXiv:2107.10252] [INSPIRE].
I. Kourkoulou and J. Maldacena, Pure states in the SYK model and nearly-AdS2 gravity, arXiv:1707.02325 [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 AdS2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
K. Jensen, Chaos in AdS2 Holography, Phys. Rev. Lett. 117 (2016) 111601 [arXiv:1605.06098] [INSPIRE].
P. Gao, D.L. Jafferis and D.K. Kolchmeyer, An effective matrix model for dynamical end of the world branes in Jackiw-Teitelboim gravity, JHEP 01 (2022) 038 [arXiv:2104.01184] [INSPIRE].
E. Brezin and V. Kazakov, Exactly solvable field theories of closed strings, in The Large N Expansion In Quantum Field Theory And Statistical Physics: From Spin Systems to 2-Dimensional Gravity, World Scientific, (1993), pp. 711–717.
M.R. Douglas and S.H. Shenker, Strings in Less Than One-Dimension, Nucl. Phys. B 335 (1990) 635 [INSPIRE].
D.J. Gross and A.A. Migdal, Nonperturbative Two-Dimensional Quantum Gravity, Phys. Rev. Lett. 64 (1990) 127 [INSPIRE].
T.G. Mertens and G.J. Turiaci, Liouville quantum gravity — holography, JT and matrices, JHEP 01 (2021) 073 [arXiv:2006.07072] [INSPIRE].
G.J. Turiaci, M. Usatyuk and W.W. Weng, 2D dilaton-gravity, deformations of the minimal string, and matrix models, Class. Quant. Grav. 38 (2021) 204001 [arXiv:2011.06038] [INSPIRE].
D. Stanford and N. Seiberg, Unpublished.
T.G. Mertens, Degenerate operators in JT and Liouville (super)gravity, JHEP 04 (2021) 245 [arXiv:2007.00998] [INSPIRE].
K. Hosomichi, Minimal Open Strings, JHEP 06 (2008) 029 [arXiv:0804.4721] [INSPIRE].
I.K. Kostov, Boundary correlators in 2-D quantum gravity: Liouville versus discrete approach, Nucl. Phys. B 658 (2003) 397 [hep-th/0212194] [INSPIRE].
J.M. Maldacena, G.W. Moore, N. Seiberg and D. Shih, Exact vs. semiclassical target space of the minimal string, JHEP 10 (2004) 020 [hep-th/0408039] [INSPIRE].
V. Fateev, A.B. Zamolodchikov and A.B. Zamolodchikov, Boundary Liouville field theory. 1. Boundary state and boundary two point function, hep-th/0001012 [INSPIRE].
B. Ponsot and J. Teschner, Boundary Liouville field theory: Boundary three point function, Nucl. Phys. B 622 (2002) 309 [hep-th/0110244] [INSPIRE].
J. Polchinski, Combinatorics of boundaries in string theory, Phys. Rev. D 50 (1994) R6041 [hep-th/9407031] [INSPIRE].
M.L. Mehta, Random matrices, Elsevier, (2004).
Harish-Chandra, Differential Operators on a Semisimple Lie Algebra, Am. J. Math. 79 (1957) 87 [INSPIRE].
C. Itzykson and J.B. Zuber, The Planar Approximation. 2, J. Math. Phys. 21 (1980) 411 [INSPIRE].
Y. Gu, Moments of random matrices and weingarten functions, Ph.D. Thesis, Queen’s University, Kingston, Ontario, Canada (2013).
D.A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP 04 (2017) 121 [arXiv:1610.04903] [INSPIRE].
S.R. Coleman, Black Holes as Red Herrings: Topological Fluctuations and the Loss of Quantum Coherence, Nucl. Phys. B 307 (1988) 867 [INSPIRE].
S.B. Giddings and A. Strominger, Loss of Incoherence and Determination of Coupling Constants in Quantum Gravity, Nucl. Phys. B 307 (1988) 854 [INSPIRE].
S.B. Giddings and A. Strominger, Baby Universes, Third Quantization and the Cosmological Constant, Nucl. Phys. B 321 (1989) 481 [INSPIRE].
J.M. Maldacena and L. Maoz, Wormholes in AdS, JHEP 02 (2004) 053 [hep-th/0401024] [INSPIRE].
L.V. Iliesiu, M. Kologlu and G.J. Turiaci, Supersymmetric indices factorize, arXiv:2107.09062 [INSPIRE].
R. Speicher, Free probability theory, arXiv:0911.0087.
P. Cvitanović, PLANAR PERTURBATION EXPANSION, Phys. Lett. B 99 (1981) 49 [INSPIRE].
J. Kudler-Flam, Relative Entropy of Random States and Black Holes, Phys. Rev. Lett. 126 (2021) 171603 [arXiv:2102.05053] [INSPIRE].
J. Kudler-Flam, V. Narovlansky and S. Ryu, Distinguishing Random and Black Hole Microstates, PRX Quantum 2 (2021) 040340 [arXiv:2108.00011] [INSPIRE].
N. Engelhardt, S. Fischetti and A. Maloney, Free energy from replica wormholes, Phys. Rev. D 103 (2021) 046021 [arXiv:2007.07444] [INSPIRE].
F. Haake, S. Gnutzmann and M. Kuś, Quantum Signatures of Chaos 4th edition, Springer series in synergetics, Springer, Dordrecht, The Netherlands (2018), [DOI].
I. Bena and N.P. Warner, Black holes, black rings and their microstates, Lect. Notes Phys. 755 (2008) 1 [hep-th/0701216] [INSPIRE].
S.D. Mathur, The information paradox: A pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
S.D. Mathur, Fuzzballs and black hole thermodynamics, arXiv:1401.4097 [INSPIRE].
T.G. Mertens and G.J. Turiaci, Defects in Jackiw-Teitelboim Quantum Gravity, JHEP 08 (2019) 127 [arXiv:1904.05228] [INSPIRE].
E. Witten, Matrix Models and Deformations of JT Gravity, Proc. Roy. Soc. Lond. A 476 (2020) 20200582 [arXiv:2006.13414] [INSPIRE].
H. Maxfield and G.J. Turiaci, The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral, JHEP 01 (2021) 118 [arXiv:2006.11317] [INSPIRE].
V.A. Kazakov, A Simple Solvable Model of Quantum Field Theory of Open Strings, Phys. Lett. B 237 (1990) 212 [INSPIRE].
S.R. Das, A. Dhar, A.M. Sengupta and S.R. Wadia, New Critical Behavior in d = 0 Large N Matrix Models, Mod. Phys. Lett. A 5 (1990) 1041 [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].
A. Blommaert, T.G. Mertens and H. Verschelde, Unruh detectors and quantum chaos in JT gravity, JHEP 03 (2021) 086 [arXiv:2005.13058] [INSPIRE].
Z. Yang, The Quantum Gravity Dynamics of Near Extremal Black Holes, JHEP 05 (2019) 205 [arXiv:1809.08647] [INSPIRE].
A. Blommaert, T.G. Mertens and H. Verschelde, Fine Structure of Jackiw-Teitelboim Quantum Gravity, JHEP 09 (2019) 066 [arXiv:1812.00918] [INSPIRE].
A. Blommaert, T.G. Mertens and H. Verschelde, The Schwarzian Theory — A Wilson Line Perspective, JHEP 12 (2018) 022 [arXiv:1806.07765] [INSPIRE].
G.W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].
M. Mirzakhani, Simple geodesics and weil-petersson volumes of moduli spaces of bordered riemann surfaces, Invent. Math. 167 (2007) 179.
R. Dijkgraaf and E. Witten, Developments in Topological Gravity, Int. J. Mod. Phys. A 33 (2018) 1830029 [arXiv:1804.03275] [INSPIRE].
T.G. Mertens, The Schwarzian theory — origins, JHEP 05 (2018) 036 [arXiv:1801.09605] [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].
E. Witten, On quantum gauge theories in two-dimensions, Commun. Math. Phys. 141 (1991) 153 [INSPIRE].
N.I. Vilenkin, Special functions and the theory of group representations, vol. 22, American Mathematical Society (1978).
A. Blommaert, T.G. Mertens and H. Verschelde, Edge dynamics from the path integral — Maxwell and Yang-Mills, JHEP 11 (2018) 080 [arXiv:1804.07585] [INSPIRE].
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: 2108.02210
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
Blommaert, A., Usatyuk, M. Microstructure in matrix elements. J. High Energ. Phys. 2022, 70 (2022). https://doi.org/10.1007/JHEP09(2022)070
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2022)070