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Poincaré's theorem and unitary transformations for classical and quantum systems

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  • Petrosky, Tomio Y.
  • Prigogine, Ilya

Abstract

Poincaré's celebrated theorem on the nonexistence of analytical invariants of motion is extended to the case of a continuous spectrum to deal with large classical and quantum systems. It is shown that Poincaré's theorem applies to situations where there exist continuous sets of resonances. This condition is equivalent to the nonvanishing of the asymptotic collision operator as defined in modern kinetic theory. Typical examples are systems presenting relaxation processes or exhibiting unstable quantum levels. As the result of Poincaré's theorem, the unitary transformation, leading to a cyclic Hamiltonian in classical mechanics or to the diagonalization of the Hamiltonian operator in quantum mechanics, diverges. We obtain therefore a dynamical classification of large classical or quantum systems. This is of special interest for quantum systems as, historically, quantum mechanics has been formulated following closely the patterns of classical integrable systems. The well known results of Friedrichs concerning the coupling of discrete states with a continuum are recovered. However, the role of the collision operator suggests new ways of eliminating the divergence in the unitary transformation theory.

Suggested Citation

  • Petrosky, Tomio Y. & Prigogine, Ilya, 1988. "Poincaré's theorem and unitary transformations for classical and quantum systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 147(3), pages 439-460.
  • Handle: RePEc:eee:phsmap:v:147:y:1988:i:3:p:439-460
    DOI: 10.1016/0378-4371(88)90164-1
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    References listed on IDEAS

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    1. Prigogine, Ilya & Petrosky, Tomio Y., 1988. "An alternative to quantum theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 147(3), pages 461-486.
    2. I. Prigogine & F. C. Andrews, 1960. "A Boltzmann-Like Approach for Traffic Flow," Operations Research, INFORMS, vol. 8(6), pages 789-797, December.
    3. Robert Herman & Tenny Lam & Ilya Prigogine, 1972. "Kinetic Theory of Vehicular Traffic: Comparison with Data," Transportation Science, INFORMS, vol. 6(4), pages 440-452, November.
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    Cited by:

    1. Ben-Ya'acov, Uri, 1995. "Lorentz symmetry of subdynamics in relativistic systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 222(1), pages 307-329.

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