The quantum cat map on the modular discretization of extremal black hole horizons

Minos Axenides,Stam Nicolis,Emmanuel Floratos

doi:10.1140/epjc/s10052-018-5850-9

Abstract

Based on our recent work on the discretization of the radial hbox {AdS}_2 geometry of extremal BH horizons, we present a toy model for the chaotic unitary evolution of infalling single-particle wave packets. We construct explicitly the eigenstates and eigenvalues for the single-particle dynamics for an observer falling into the BH horizon, with as time evolution operator the quantum Arnol’d cat map (QACM). Using these results we investigate the validity of the eigenstate thermalization hypothesis (ETH), as well as that of the fast scrambling time bound (STB). We find that the QACM, while possessing a linear spectrum, has eigenstates, which are random and satisfy the assumptions of the ETH. We also find that the thermalization of infalling wave packets in this particular model is exponentially fast, thereby saturating the STB, under the constraint that the finite dimension of the single-particle Hilbert space takes values in the set of Fibonacci integers.

Highlights

In Appendix A we collect all the necessary material for the detailed construction of the Weyl representation of SL2( p) and PSL2( p) and we present the technical details for the analytic construction of the eigenstates and eigenvalues of the quantum Arnol’d cat map (QACM)
In this work we proposed a toy model for the chaotic scattering of single-particle wave packets in the modular discretization of the radial AdS2 space-time geometry of extremal extremal BHs
We obtained an AdS2/CFT1 holography and we provided the eigenstates and eigenvalues for the quantum chaotic Arnol’d cat map, as well as the single-particle S-matrix

Summary

Introduction

Avery interesting revival of the old relation between the near-horizon shock-wave BH geometries with gravitational memory effects and the information paradox has recently appeared [1,2,3,4,5,6]. This is the so-called modular discretization, AdS2[ p], for every prime integer p [28,29,30,31] In this framework, the entropy of the black hole, is identified with the Kolmogorov–Sinai entropy of the deterministic, chaotic, dynamics of the geometric, microscopic, degrees of freedom, defining the near-horizon geometry. This discrete geometry provides a natural framework for describing the single-particle dynamics, via observers, with time evolution operators that are elements of the isometry group This is a discrete analog of the superconformal quantum mechanics of probes near the horizons of large extremal black holes [32]. The motion of a particle under ACM, along the spatial direction, x2, of AdS2[N ] can be seen to be fully chaotic and mixing; cf. Fig. 3

Chaotic eigenstates of the QACM on AdS2

ETH and the scrambling time bound for the QACM

Summary and conclusions