Abstract
We consider operator growth for generic large-N gauge theories at finite temperature. Our analysis is performed in terms of Fourier modes, which do not mix with other operators as time evolves, and whose correlation functions are determined by their two-point functions alone, at leading order in the large-N limit. The algebra of these modes allows for a simple analysis of the operators with whom the initial operator mixes over time, and guarantees the existence of boundary CFT operators closing the bulk Poincaré algebra, describing the experience of infalling observers. We discuss several existing approaches to operator growth, such as number operators, proper energies, the many-body recursion method, quantum circuit complexity, and comment on its relation to classical chaos in black hole dynamics. The analysis evades the bulk vs boundary dichotomy and shows that all such approaches are the same at both sides of the holographic duality, a statement that simply rests on the equality between operator evolution itself. In the way, we show all these approaches have a natural formulation in terms of the Gelfand-Naimark-Segal (GNS) construction, which maps operator evolution to a more conventional quantum state evolution, and provides an extension of the notion of operator growth to QFT.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D.A. Roberts, D. Stanford and A. Streicher, Operator growth in the SYK model, JHEP 06 (2018) 122 [arXiv:1802.02633] [INSPIRE].
X.-L. Qi and A. Streicher, Quantum Epidemiology: Operator Growth, Thermal Effects and SYK, JHEP 08 (2019) 012 [arXiv:1810.11958] [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].
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
A.I. Larkin and Y.N. Ovchinnikov, Quasiclassical Method in the Theory of Superconductivity, JETP 28 (1969) 1200.
S.H. Shenker and D. Stanford, Stringy effects in scrambling, JHEP 05 (2015) 132 [arXiv:1412.6087] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
L. Susskind, Why do Things Fall?, arXiv:1802.01198 [INSPIRE].
J.M. Magán, Black holes, complexity and quantum chaos, JHEP 09 (2018) 043 [arXiv:1805.05839] [INSPIRE].
H.W. Lin, J. Maldacena and Y. Zhao, Symmetries Near the Horizon, JHEP 08 (2019) 049 [arXiv:1904.12820] [INSPIRE].
J.L.F. Barbón, J. Martín-García and M. Sasieta, Momentum/Complexity Duality and the Black Hole Interior, arXiv:1912.05996 [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].
J.L.F. Barbón, E. Rabinovici, R. Shir and R. Sinha, On The Evolution Of Operator Complexity Beyond Scrambling, JHEP 10 (2019) 264 [arXiv:1907.05393] [INSPIRE].
P. Bueno, J.M. Magán and C.S. Shahbazi, Complexity measures in QFT and constrained geometric actions, arXiv:1908.03577 [INSPIRE].
M.A. Nielsen, A geometric approach to quantum circuit lower bounds, quant-ph/0502070.
M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, quant-ph/0701004.
V. Viswanath and G. Müller, The Recursion Method: Application to Many-Body Dynamics, vol. 23, Springer Science & Business Media, (2008).
A. Streicher, SYK Correlators for All Energies, JHEP 02 (2020) 048 [arXiv:1911.10171] [INSPIRE].
A. Lucas, Non-perturbative dynamics of the operator size distribution in the Sachdev-Ye-Kitaev model, arXiv:1910.09539 [INSPIRE].
Y.D. Lensky, X.-L. Qi and P. Zhang, Size of bulk fermions in the SYK model, arXiv:2002.01961 [INSPIRE].
A. Mousatov, Operator Size for Holographic Field Theories, arXiv:1911.05089 [INSPIRE].
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
S. El-Showk and K. Papadodimas, Emergent Spacetime and Holographic CFTs, JHEP 10 (2012) 106 [arXiv:1101.4163] [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, Black Hole Interior in the Holographic Correspondence and the Information Paradox, Phys. Rev. Lett. 112 (2014) 051301 [arXiv:1310.6334] [INSPIRE].
K. Papadodimas and S. Raju, State-Dependent Bulk-Boundary Maps and Black Hole Complementarity, Phys. Rev. D 89 (2014) 086010 [arXiv:1310.6335] [INSPIRE].
J. De Boer, R. Van Breukelen, S.F. Lokhande, K. Papadodimas and E. Verlinde, Probing typical black hole microstates, JHEP 01 (2020) 062 [arXiv:1901.08527] [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888.
J.L.F. Barbón and J.M. Magán, Chaotic Fast Scrambling At Black Holes, Phys. Rev. D 84 (2011) 106012 [arXiv:1105.2581] [INSPIRE].
J.L.F. Barbón and J.M. Magán, Fast Scramblers Of Small Size, JHEP 10 (2011) 035 [arXiv:1106.4786] [INSPIRE].
J.L.F. Barbón and J.M. Magán, Fast Scramblers, Horizons and Expander Graphs, JHEP 08 (2012) 016 [arXiv:1204.6435] [INSPIRE].
J. de Boer, R. Van Breukelen, S.F. Lokhande, K. Papadodimas and E. Verlinde, On the interior geometry of a typical black hole microstate, JHEP 05 (2019) 010 [arXiv:1804.10580] [INSPIRE].
R. Haag, Local quantum physics: Fields, particles, algebras, Springer, (1992).
E. Witten, APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory, Rev. Mod. Phys. 90 (2018) 045003 [arXiv:1803.04993] [INSPIRE].
J. De Boer and L. Lamprou, Holographic Order from Modular Chaos, arXiv:1912.02810 [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
L.C.B. Crispino, A. Higuchi and G.E.A. Matsas, The Unruh effect and its applications, Rev. Mod. Phys. 80 (2008) 787 [arXiv:0710.5373] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [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].
D. Harlow, TASI Lectures on the Emergence of Bulk Physics in AdS/CFT, PoS(TASI2017)002 [arXiv:1802.01040] [INSPIRE].
J.M. Magán, Decoherence and microscopic diffusion at the Sachdev-Ye-Kitaev model, Phys. Rev. D 98 (2018) 026015 [arXiv:1612.06765] [INSPIRE].
A. Romero-Bermúdez, K. Schalm and V. Scopelliti, Regularization dependence of the OTOC. Which Lyapunov spectrum is the physical one?, JHEP 07 (2019) 107 [arXiv:1903.09595] [INSPIRE].
M.A. Nielsen, M.R. Dowling, M. Gu and A.C. Doherty, Quantum Computation as Geometry, Science 311 (2006) 1133 [quant-ph/0603161].
A. Bernamonti, F. Galli, J. Hernandez, R.C. Myers, S.-M. Ruan and J. Simón, First Law of Holographic Complexity, Phys. Rev. Lett. 123 (2019) 081601 [arXiv:1903.04511] [INSPIRE].
A. Ashtekar and T.A. Schilling, Geometrical formulation of quantum mechanics, gr-qc/9706069 [INSPIRE].
A.L. Fitzpatrick and J. Kaplan, Scattering States in AdS/CFT, arXiv:1104.2597 [INSPIRE].
M. Nozaki, T. Numasawa and T. Takayanagi, Holographic Local Quenches and Entanglement Density, JHEP 05 (2013) 080 [arXiv:1302.5703] [INSPIRE].
K. Goto, M. Miyaji and T. Takayanagi, Causal Evolutions of Bulk Local Excitations from CFT, JHEP 09 (2016) 130 [arXiv:1605.02835] [INSPIRE].
K. Goto and T. Takayanagi, CFT descriptions of bulk local states in the AdS black holes, JHEP 10 (2017) 153 [arXiv:1704.00053] [INSPIRE].
S. Terashima, AdS/CFT Correspondence in Operator Formalism, JHEP 02 (2018) 019 [arXiv:1710.07298] [INSPIRE].
D. Berenstein and J. Simón, Localized states in global AdS space, Phys. Rev. D 101 (2020) 046026 [arXiv:1910.10227] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
B. Czech, J. De Boer, D. Ge and L. Lamprou, A modular sewing kit for entanglement wedges, JHEP 11 (2019) 094 [arXiv:1903.04493] [INSPIRE].
I. Gelfand and M. Neumark, On the imbedding of normed rings into the ring of operators in hilbert space, Rec. Math. [Mat. Sbornik] N.S. 12(54) (1943) 197.
I.E. Segal, Irreducible representations of operator algebras, Bull. Am. Math. Soc. 53 (1947) 73.
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
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 2002.03865
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Magán, J.M., Simón, J. On operator growth and emergent Poincaré symmetries. J. High Energ. Phys. 2020, 71 (2020). https://doi.org/10.1007/JHEP05(2020)071
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2020)071