The Arnol'd cat: Failure of the correspondence principle

Ford, Joseph; Mantica, Giorgio; Ristow, Gerald H.

doi:10.1016/0167-2789(91)90012-x

Cited by 123 publications

(103 citation statements)

References 31 publications

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“…Its dynamics are algorithmically equivalent to classical motion on a regular lattice, whose spacing is inversely proportional to the Planck constant [3,4]. When the spacing diminishes, the lattice becomes denser in continuous, classical phase-space.…”

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confidence: 99%

“…When the spacing diminishes, the lattice becomes denser in continuous, classical phase-space. Yet, it has long been recognized that chaos cannot be naively revived in such a limit procedure [4,5,6]. A way out of this impasse is obtained by randomly perturbing the dynamics [7]: is this addition enough to bring back the full algorithmic content, that is the distinctive signature of chaos [8] ?…”

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confidence: 99%

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Coarse-Grained Probabilistic Automata Mimicking Chaotic Systems

et al. 2003

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Coarse-Grained Probabilistic Automata Mimicking Chaotic Systems

et al. 2003

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“…The uniform hyperbolicity also precludes any dependence on initial conditions. The dynamics derive from the kicked oscillator Hamiltonian [14] …”

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Parameter Scaling in the Decoherent Quantum-Classical Transition for Chaotic Systems

2003

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The quantum to classical transition has been shown to depend on a number of parameters. Key among these are a scale length for the action,h, a measure of the coupling between a system and its environment, D, and, for chaotic systems, the classical Lyapunov exponent, λ. We propose computing a measure, reflecting the proximity of quantum and classical evolutions, as a multivariate function of (h, λ, D) and searching for transformations that collapse this hyper-surface into a function of a composite parameter ζ =h α λ β D γ . We report results for the quantum Cat Map, showing extremely accurate scaling behavior over a wide range of parameters and suggest that, in general, the technique may be effective in constructing universality classes in this transition.

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“…Our present work builds on our earlier work on Finite Quantum Mechanics (FQM) where we introduced the discretized toroidal membrane, Z N × Z N [31] which elucidates the matrix model truncation of the membrane dynamics [32,33] rendering the discrete membrane as a quantum phase space of finite quantum mechanics, with canonical transformation group SL(2, Z N ) [34][35][36][37][38].…”

Section: Jhep02(2014)109mentioning

confidence: 99%

Modular discretization of the AdS2/CFT1 holography

Axenides

Floratos

Nicolis

2014

J. High Energ. Phys.

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We propose a finite discretization for the black hole, near horizon, geometry and dynamics. We realize our proposal, in the case of extremal black holes, for which the radial and temporal near horizon geometry is known to be AdS 2 = SL(2, R)/SO(1, 1, R). We implement its discretization by replacing the set of real numbers R with the set of integers modulo N with AdS 2 going over to the finite geometry AdS 2 [N ] = SL(2, Z N )/SO(1, 1, Z N ). We model the dynamics of the microscopic degrees of freedom by generalized Arnol'd cat maps, A ∈ SL(2, Z N ) which are isometries of the geometry, at both the classical and quantum levels. These are well known to exhibit fast quantum information processing through the well studied properties of strong arithmetic chaos, dynamical entropy, nonlocality and factorization in the cutoff discretization N . We construct, finally, a new kind of unitary and holographic correspondence, for AdS 2 [N ]/CFT 1 [N ], via coherent states of the bulk and boundary geometries.

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The Arnol'd cat: Failure of the correspondence principle

Cited by 123 publications

References 31 publications

Coarse-Grained Probabilistic Automata Mimicking Chaotic Systems

Coarse-Grained Probabilistic Automata Mimicking Chaotic Systems

Parameter Scaling in the Decoherent Quantum-Classical Transition for Chaotic Systems

Modular discretization of the AdS2/CFT1 holography

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