Quantum signatures of chaos or quantum chaos?
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CLEI Theory
Quantum Signatures of Chaos or Quantum Chaos? V. E. Bunakov* St. Petersburg State University, Universitetskaya naberezhnaya 7-9, St. Petersburg, 199034 Russia Petersburg Nuclear Physics Institute, National Research Center Kurchatov Institute, Gatchina, 188300 Russia Received January 26, 2016
Abstract—A critical analysis of the present-day concept of chaos in quantum systems as nothing but a “quantum signature” of chaos in classical mechanics is given. In contrast to the existing semi-intuitive guesses, a definition of classical and quantum chaos is proposed on the basis of the Liouville–Arnold theorem: a quantum chaotic system featuring N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) specified by the symmetry of the Hamiltonian of the system. Quantitative measures of quantum chaos that, in the classical limit, go over to the Lyapunov exponent and the classical stability parameter are proposed. The proposed criteria of quantum chaos are applied to solving standard problems of modern dynamical chaos theory. DOI: 10.1134/S1063778816060053
1. PROBLEMS OF CHAOS IN CLASSICAL MECHANICS
1.1. Searches for the Most General Physical Reasons behind Chaos For the first time, contemporary physics came across the concept of chaos when L. Boltzmann developed the kinetic theory of gases and derived respective kinetic equations. In doing this, he had to invoke the language of statistical mechanics and to introduce the hypothesis of molecular chaos. According to this hypothesis, the system being considered forgets fast the details of its previous evolution (weakening of correlations, according to the terminology frequently used at the present time). Although the validity of kinetic theory is no longer questioned, some issues concerning the derivation of kinetic equations have been hotly debated to date (for an overview, see, for example, [1, 2]). Since then, the majority of physicists had thought that it was a large number of degrees of freedom in macroscopic systems that dictated the abandonment of the idea of solving dynamical equations and resort to a probabilistic statistical description of the evolution of such systems. The prevalent opinion was that the behavior of simple systems featuring a small number of degrees of freedom is fully deterministic—it was only necessary in that case to go over to canonical variables and to solve respective equations of motion for preset initial conditions. As far back as the end of XIX century, however, mathematicians and mechanicians began to realize *
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that the motion of systems featuring even a very small number of degrees of freedom may be unpredictable if their trajectories are unstable against very small variations in initial conditions. For unstable systems, the distance between initially very close trajectories in the respective phase space increases with time in proportion to exp(Λt). The rate Λ at which the trajectories diverge is referred to as Lyapunov’s exponent. If the phase space of an
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