Euclidean and Lorentzian Actions of the Classicalized Holographic Tensor Network

Document Type : Regular article


Graduate School of Human and Environmental Studies, Kyoto University


In three spacetime dimensions, we propose a generally covariant Lorentzian action of the classicalized holographic tensor network (cHTN) as the holographic reduction of the Einstein--Hilbert action of gravity in the presence of a negative cosmological constant.
In this article, first, we investigate the properties of this Lorentzian action in the ground state.
Next, based on the Euclidean action of the cHTN, we derive the gravity perturbation induced by a massive particle at rest in the cHTN as the Unruh effect.
Finally, we view our holographic formulation of spacetime as a non-equilibrium second law subject to general covariance.


Main Subjects


Article PDF

[1] K. V. Kucha˘r, “Time and interpretations of quantum gravity”, Int. J. Mod. Phys. Proc. Suppl. D 20, 3 (2011).
[2] C. J. Isham, : Canonical quantum gravity and the problem of time. In: L. A. Ibort, M. A. Rodr´iguez, (eds.) Integrable Systems, Quantum Groups and Quantum Field Theories, pp. 157-287. Kluwer, Dordrecht (1993).
[3] G. ’t Hooft, arXiv:gr-qc/9310026.
[4] L. Susskind, “The world as a hologram”, J. Math. Phys. 36, 6377 (1995).
[5] R. Bousso, “The holographic principle”, Rev. Mod. Phys. 74, 825 (2002).
[6] E. Baum, “Zero cosmological constant from minimum action”, Phys. Lett. B 133, 185 (1983).
[7] S. W. Hawking, “The cosmological constant is probably zero”, Phys. Lett. B 134, 403 (1984).
[8] A. Vilenkin, “Predictions from quantum cosmology”, Phys. Rev. Lett. 74, 846 (1995).
[9] E. Konishi, “Holographic interpretation of Shannon entropy of coherence of quantum pure states”, EPL 129, 11006 (2020).
[10] E. Konishi, “Addendum: Holographic interpretation of Shannon entropy of coherence of quantum pure states”, EPL 132, 59901 (2020).
[11] J. M. Maldacena, “The large-N limit of superconformal field theories and supergravity”, Adv. Theor. Math. Phys. 2, 231 (1998).
[12] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators from non-critical string theory”, Phys. Lett. B 428, 105 (1998).
[13] E. Witten, “Anti de Sitter space and holography”, Adv. Theor. Math. Phys. 2, 253 (1998).
[14] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, “Large-N field theories, string theory and gravity”, Phys. Rep. 323, 183 (2000).
[15] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from the anti-de Sitter space/conformal field theory correspondence”, Phys. Rev. Lett. 96, 181602 (2006).
[16] S. Ryu and T. Takayanagi, “Aspects of holographic entanglement entropy”, J. High Energy Phys. 08, 045 (2006).
[17] V. E. Hubeny, M. Rangamani and T. Takayanagi, “A covariant holographic entanglement entropy proposal”, J. High Energy Phys. 07, 062 (2007).
[18] M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy. Lect. Notes Phys., Vol. 931 Springer (2017).
[19] B. Swingle, “Entanglement renormalization and holography”, Phys. Rev. D 86, 065007 (2012).
[20] H. Matsueda, M. Ishibashi and Y. Hashizume, “Tensor network and a black hole”, Phys. Rev. D 87, 066002 (2013).
[21] N. Bao, C. Cao, S. M. Carroll, A. Chatwin-Davies and N. Hunter-Jones, “Consistency conditions for an AdS multiscale entanglement renormalization ansatz correspondence”, Phys. Rev. D 91, 125036 (2015).
[22] B. Swingle, “Spacetime from entanglement”, Annu. Rev. Condens. Matter Phys. 9, 345 (2018).
[23] A. Jahn and J. Eisert, “Holographic tensor network models and quantum error correc- tion: a topical review”, Quantum Sci. Technol. 6, 033002 (2021).
[24] B. d’Espagnat, Conceptual Foundations of Quantum Mechanics. 2nd edn. W. A. Ben- jamin, Reading, Massachusetts (1976).
[25] E. Konishi, “Imaginary-time path-integral in bulk space from the holographic principle”, JHAP 1, (1) 47-56 (2021).
[26] E. Konishi, “de Sitter spacetime from holographic flat spacetime with inexact bulk quantum mechanics”, JHAP 2, (3) 71-80 (2022).
[27] E. Konishi, “Time parametrization in long-range interacting Bose–Einstein conden-sates”, J. Phys. Commun. 5, 095012 (2021).
[28] J. von Neumann, Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton, NJ (1955).
[29] J. von Neumann, “On rings of operators. Reduction theory”, Ann. Math. 50, 2 (1949).
[30] G. W. Gibbons and S. W. Hawking, “Action integrals and partition functions in quantum gravity”, Phys. Rev. D 15, 2752 (1977).
[31] D. Harlow and E. Shaghoulian, “Euclidean gravity and holography”, Int. J. Mod. Phys. D 2141005 (2021).
[32] E. Witten, : A note on complex spacetime metrics. In: A. Niemi, K. K. Phua, A. Shapere, (eds.) Frank Wilczek: 50 Years of Theoretical Physics, pp. 245-280. World Scientific, Singapore (2022).
[33] E. Konishi, “Projection hypothesis from the von Neumann-type interaction with a Bose–Einstein condensate”, EPL 136, 10004 (2021).
[34] P. C. W. Davies, “Scalar particle production in Schwarzschild and Rindler metrics”, J. Phys. A 8, 609 (1975).
[35] W. G. Unruh, “Notes on black-hole evaporation”, Phys. Rev. D 14, 870 (1976).
[36] G. L. Sewell, “Quantum fields on manifolds: PCT and gravitationally induced thermal states”, Ann. Phys. 141, 201 (1982).
[37] L. C. B. Crispino, A. Higuchi and G. E. A. Matsas, “The Unruh effect and its applications”, Rev. Mod. Phys. 80, 787 (2008).
[38] D. Harlow, “Jerusalem lectures on black holes and quantum information”, Rev. Mod. Phys. 88, 015002 (2016).
[39] G. Vidal, “Entanglement renormalization”, Phys. Rev. Lett. 99, 220405 (2007).
[40] G. Vidal, “Class of quantum many-body states that can be efficiently simulated”, Phys. Rev. Lett. 101, 110501 (2008).
[41] H. N˘astase, Introduction to the AdS/CFT Correspondence. Cambridge University Press, Cambridge (2015).
[42] E. Konishi, “Random walk of bipartite spins in a classicalized holographic tensor network”, Results in Physics 19, 103410 (2020).
[43] A. Baldazzi, R. Percacci and V. Skrinjar, “Wicked metrics”, Class. Quantum Grav. 36, 105008 (2019).
[44] J. M. R. Parrondo, J. M. Horowitz and T. Sagawa, “Thermodynamics of information”, Nat. Phys. 11, 131 (2015).
[45] S. Deser and O. Levin, “Accelerated detectors and temperature in (anti-) de Sitter spaces”, Class. Quantum Grav. 14, L163 (1997).
[46] T. Jacobson, “Comment on accelerated detectors and temperature in (anti-) de Sitter spaces”, Class. Quantum Grav. 15, 251 (1998).
[47] S. W. Hawking, “Black holes in general relativity”, Commun. Math. Phys. 25, 152 (1972).
[48] J. M. Bardeen, B. Carter and S. W. Hawking, “The four laws of black hole mechanics”, Commun. Math. Phys. 31, 161 (1973).
Volume 2, Issue 4
November 2022
Pages 1-10
  • Receive Date: 10 October 2022
  • Revise Date: 25 November 2022
  • Accept Date: 25 November 2022