Quantum Chaos, Irreversible Classical Dynamics, and Random Matrix Theory

Abstract

The Bohigas-Giannoni-Schmit conjecture stating that the statistical spectral properties of systems which are chaotic in their classical limit coincide with random matrix theory is proved. For this purpose a new semiclassical field theory for individual chaotic systems is constructed in the framework of the non-linear σ-model. The low lying modes are shown to be associated with the Perron-Frobenius spectrum of the underlying irreversible classical dynamics. It is shown that the existence of a gap in the Perron-Frobenius spectrum results in a RMT behavior. Moreover, our formalism offers a way of calculating system specific corrections beyond RMT.The theory of random matrices [1] emerged from the need to characterize complex quantum systems in which knowledge of the Hamiltonian is minimal, e.g. complex nuclei. The basic hypothesis is that the Hamiltonian may be treated as one drawn from an ensemble of random matrices with appropriate symmetries. It has been proposed by invoking the complexity of systems which have many degrees of freedom with unknown interaction coupling among them.The study of the statistical quantum properties of systems with small number of degrees of freedom, within the framework of random matrix theory (RMT), has developed along two parallel lines. The first was by considering an ensemble of random systems such as disordered metallic grains [2]. Randomness in this case is introduced on the level of the Hamiltonian itself, e.g. as a consequence of the unknown impurity configuration. In the second approach, RMT was used in order to understand the level statistics of non-stochastic systems which are chaotic in their classical limit such as the Sinai or the stadium billiards [3]. Here "randomness" is generated by the underlying deterministic classical dynamics itself. Nevertheless, it has been conjectured [3] that "spectrum fluctuations of quantal time-reversal invariant systems whose classical analogues are strongly chaotic have the Gaussian Orthogonal Ensembles pattern".Despite being supported by extensive numerical studies, the origin of the success of RMT as well as its domain of validity are still not completely resolved. In this letter we show that, in the semiclassical limit, this conjecture is indeed valid for systems with exponential decay of classical correlation functions in time. Moreover, the formalism which we introduce below offers a way of calculating system specific corrections beyond RMT.So far, the main attempts to establish the relationship between non-stochastic chaotic systems and RMT, have been based on periodic orbit theory [4]. Gutzwiller's trace formula expresses the semiclassical density of states as an infinite sum over the classical periodic orbits of the system. However, the number of periodic orbits is exponentially large and clearly contains information that is redundant from quantum mechanical point of view. This detailed information conceals the way of drawing a connection between the quantum behavior of chaotic systems and RMT. Indeed, the success of the periodic...

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