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 (RMT) is proved. A new semiclassical field theory for individual chaotic systems is constructed in the framework of a nonlinear $\sigma$ model. The low lying modes are shown to be associated with the Perron-Frobenius (PF) spectrum of the underlying irreversible classical dynamics. It is shown that the existence of a gap in the PF spectrum results in RMT behavior. Moreover, our formalism offers a way of calculating system specific corrections beyond RMT.