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The classical limit of non-integrable quantum systems, a route to quantum chaos

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  • Castagnino, Mario
  • Lombardi, Olimpia

Abstract

The classical limit of non-integrable quantum systems is studied. We define non-integrable quantum systems as those, which have, as their classical limit, a non-integrable classical system. This quantum systems will be the candidates to be the models of quantum chaos. In order to obtain this limit, the self-induced decoherence approach and the corresponding classical limit are generalized from integrable to non-integrable systems. In this approach, the lost of information, usually conceived as the result of a coarse-graining or the trace of an environment, is produced by a particular choice of the algebra of observables and the systematic use of mean values, that project the unitary evolution onto an effective non-unitary one. By means of our method, we can obtain the classical limit of the quantum state of a non-integrable system, which turns out to be a set of unstable, potentially chaotic classical trajectories contained in the Wigner transformation of the quantum state.

Suggested Citation

  • Castagnino, Mario & Lombardi, Olimpia, 2006. "The classical limit of non-integrable quantum systems, a route to quantum chaos," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 879-898.
    • Handle: RePEc:eee:chsofr:v:28:y:2006:i:4:p:879-898
    DOI: 10.1016/j.chaos.2005.08.149
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    References listed on IDEAS

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    1. Laura, Roberto & Castagnino, Mario & Betan, Rodolfo M.Id, 1999. "Perturbative method for generalized spectral decompositions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 271(3), pages 357-386.
    2. Castagnino, Mario, 2004. "The classical–statistical limit of quantum mechanics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 335(3), pages 511-517.
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    Cited by:

    1. Castagnino, Mario & Lombardi, Olimpia, 2009. "Towards a definition of the quantum ergodic hierarchy: Ergodicity and mixing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(4), pages 247-267.
    2. El Fakkousy, Idriss & Zouhairi, Bouchta & Benmalek, Mohammed & Kharbach, Jaouad & Rezzouk, Abdellah & Ouazzani-Jamil, Mohammed, 2022. "Classical and quantum integrability of the three-dimensional generalized trapped ion Hamiltonian," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    3. Castagnino, M. & Gadella, M. & Lara, L.P., 2006. "On local Hamiltonians and dissipative systems," Chaos, Solitons & Fractals, Elsevier, vol. 30(3), pages 542-551.

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    2. Castagnino, Mario & Lombardi, Olimpia, 2009. "Towards a definition of the quantum ergodic hierarchy: Ergodicity and mixing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(4), pages 247-267.

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