On-shell Equation of the Lorentzian Classicalized Holographic Tensor Network

Document Type : Letter

Author

Graduate School of Human and Environmental Studies, Kyoto University

Abstract

In the Lorentzian classicalized holographic tensor network (cHTN), we derive its relativistic on-shell equation from its Lorentzian action in the presence of a relativistic massive particle in the bulk spacetime: $-\sigma \hbar \theta=Mc^2$.

Here, $\sigma$ is the von Neumann entropy of the cHTN per site in nats, $\theta$ is the real-proper-time expansion of the cHTN defined along the world line of the particle, and $M$ is the non-zero mass of the particle.

We explain the physical properties, interpretation, and consequences of this equation.

Specifically, from this equation we derive the properties of the on-shell proper acceleration of another massive particle in the bulk spacetime as those of the gravitational acceleration induced by the original massive particle.

Keywords

Main Subjects

 

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 [1] G. ’t Hooft, arXiv:gr-qc/9310026. DOI: 10.48550/arXiv.gr-qc/9310026.
[2] L. Susskind, "The world as a hologram", J. Math. Phys.
36, 6377 (1995). DOI: 10.1063/1.531249.
[3] R. Bousso, "The holographic principle", Rev. Mod. Phys.
74, 825 (2002). DOI: 10.1103/RevModPhys.74.825.
[4] J. M. Maldacena, "The large-
N limit of superconformal field theories and supergravity", Adv. Theor. Math. Phys. 2, 231 (1998). DOI: 10.1023/A:1026654312961.
[5] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, "Gauge theory correlators from non-critical string theory", Phys. Lett. B
428, 105 (1998). DOI: 10.1016/S0370-2693(98)00377-3.
[6] E. Witten, "Anti de Sitter space and holography", Adv. Theor. Math. Phys.
2, 253 (1998). DOI: 10.4310/ATMP.1998.v2.n2.a2.
 [7] 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). DOI: 10.1016/S0370-1573(99)00083-6.
[8] H. NĖ˜astase, Introduction to the AdS/CFT Correspondence. Cambridge University Press, Cambridge (2015). DOI: 10.1017/CBO9781316090954.
[9] B. Swingle, "Entanglement renormalization and holography", Phys. Rev. D
86, 065007 (2012). DOI: 10.1103/PhysRevD.86.065007.
[10] G. Vidal, "Entanglement renormalization", Phys. Rev. Lett.
99, 220405 (2007). DOI: 10.1103/PhysRevLett.99.220405.
[11] G. Vidal, "Class of quantum many-body states that can be efficiently simulated", Phys. Rev. Lett.
101, 110501 (2008). DOI: 10.1103/PhysRevLett.101.110501.
[12] 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). DOI: 10.1103/PhysRevLett.96.181602.
[13] S. Ryu and T. Takayanagi, "Aspects of holographic entanglement entropy", J. High Energy Phys.
08, 045 (2006). DOI: 10.1088/1126-6708/2006/08/045.
[14] V. E. Hubeny, M. Rangamani and T. Takayanagi, "A covariant holographic entanglement entropy proposal", J. High Energy Phys.
07, 062 (2007). DOI: 10.1088/1126-6708/2007/07/062.
[15] M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy. Lect. Notes Phys., Vol.
931 Springer (2017). DOI: 10.1007/978-3-319-52573-0.
[16] B. Chen, B. Czech and Z. Wang, "Quantum information in holographic duality", Rep. Prog. Phys.
85, 046001 (2022). DOI: 10.1088/1361-6633/ac51b5.
[17] H. Matsueda, M. Ishibashi and Y. Hashizume, "Tensor network and a black hole", Phys. Rev. D
87, 066002 (2013). DOI: 10.1103/PhysRevD.87.066002.
[18] 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). DOI: 10.1103/PhysRevD.91.125036.
[19] B. Swingle, "Spacetime from entanglement", Annu. Rev. Condens. Matter Phys.
9, 345 (2018). DOI: 10.1146/annurev-conmatphys-033117-054219.
[20] A. Jahn and J. Eisert, "Holographic tensor network models and quantum error correction: a topical review", Quantum Sci. Technol.
6, 033002 (2021). DOI: 10.1088/2058-9565/ac0293.
[21] E. Konishi, "Holographic interpretation of Shannon entropy of coherence of quantum pure states", EPL
129, 11006 (2020). DOI: 10.1209/0295-5075/129/11006.
[22] E. Konishi, "Addendum: Holographic interpretation of Shannon entropy of coherence of quantum pure states", EPL
132, 59901 (2020). DOI: 10.1209/0295-5075/132/59901.
[23] E. Konishi, "Imaginary-time path-integral in bulk space from the holographic principle", JHAP
1, (1) 47-56 (2021). DOI: 10.22128/jhap.2021.432.1001.
 [24] W. H. Zurek, "Decoherence and the transition from quantum to classical", Phys. Today 44, (10) 36-44 (1991). DOI: 10.1063/1.881293.
[25] E. Konishi, "Random walk of bipartite spins in a classicalized holographic tensor network", Results in Physics
19, 103410 (2020). DOI: 10.1016/j.rinp.2020.103410.
[26] E. Konishi, "Quantum measuring systems: considerations from the holographic principle", JHAP
3, (1) 31-38 (2023). DOI: 10.22128/jhap.2023.652.1039.
[27] E. Konishi, "Euclidean and Lorentzian actions of the classicalized holographic tensor network", JHAP
2, (4) 1-10 (2022). DOI: 10.22128/jhap.2022.621.1036.
[28] J. von Neumann, "On rings of operators. Reduction theory", Ann. Math.
50, 2 (1949). DOI: 10.2307/1969463.
[29] J. von Neumann, "Proof of the ergodic theorem and the
H-theorem in quantum mechanics", Eur. Phys. J. H 35, 201 (2010). DOI: 10.1140/epjh/e2010-00008-5.
[30] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000). DOI: 10.1017/CBO9780511976667.
[31] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields. ButterworthHeinemann, Oxford (1975).
[32] P. C. W. Davies, "Scalar particle production in Schwarzschild and Rindler metrics", J. Phys. A
8, 609 (1975). DOI: 10.1088/0305-4470/8/4/022.
[33] W. G. Unruh, "Notes on black-hole evaporation", Phys. Rev. D
14, 870 (1976). DOI: 10.1103/PhysRevD.14.870.
[34] G. L. Sewell, "Quantum fields on manifolds: PCT and gravitationally induced thermal states", Ann. Phys.
141, 201 (1982). DOI: 10.1016/0003-4916(82)90285-8.
[35] L. C. B. Crispino, A. Higuchi and G. E. A. Matsas, "The Unruh effect and its applications", Rev. Mod. Phys. 80, 787 (2008). DOI: 10.1103/RevModPhys.80.787.
[36] D. Harlow, "Jerusalem lectures on black holes and quantum information", Rev. Mod. Phys.
88, 015002 (2016). DOI: 10.1103/RevModPhys.88.015002.
[37] A. Bagchi and R. Fareghbal, "BMS/GCA redux: towards flatspace holography from non-relativistic symmetries", J. High Energy Phys.
10, 092 (2012). DOI: 10.1007/JHEP10(2012)092.
[38] E. Konishi, "de Sitter spacetime from holographic flat spacetime with inexact bulk quantum mechanics", JHAP
2, (3) 71-80 (2022). DOI: 10.22128/jhap.2022.563.1026.
Volume 3, Issue 3
September 2023
Pages 37-44
  • Receive Date: 29 April 2023
  • Revise Date: 25 May 2023
  • Accept Date: 11 September 2023