Quantum Chaos, Irreversible Classical Dynamics and Random Matrix Theory

TL;DR

This paper proves the Bohigas--Giannoni--Schmit conjecture by establishing a link between classical chaos, quantum spectral statistics, and random matrix theory using a new semiclassical field theory approach.

Contribution

It introduces a novel semiclassical field theory based on the non-linear sigma-model to connect classical irreversible dynamics with quantum spectral properties.

Findings

Proves the Bohigas--Giannoni--Schmit conjecture.

Shows a gap in the Perron-Frobenius spectrum leads to RMT behavior.

Provides a method to calculate system-specific corrections beyond RMT.

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 $\sigma$-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.