Action Correlations and Random Matrix Theory

Abstract

The correlations in the spectra of quantum systems are intimately related to correlations which are of genuine classical origin, and which appear in the spectra of actions of the classical periodic orbits of the corresponding classical systems. We review this duality and the semiclassical theory which brings it about. The conjecture that the quantum spectral statistics are described in terms of Random Matrix Theory, leads to the proposition that the classical two-point correlation function is given also in terms of a universal function. We study in detail the spectrum of actions of the Baker map, and use it to illustrate the steps needed to reveal the classical correlations, their origin and their relation to symbolic dynamics.

26 pages, 15 figures

References in corpus (1)

• The Leading Off-Diagonal Correction to the Form Factor of Large Graphs

Cited by in corpus (10)

• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Periodic-Orbit Theory of Universality in Quantum Chaos
• Spectral form factor of hyperbolic systems: leading off-diagonal approximation
• Shot noise from action correlations
• Classical basis for quantum spectral fluctuations in hyperbolic systems
• Semiclassical approach to fidelity amplitude
• Clustering of periodic orbits and ensembles of truncated unitary matrices
• Duality between quantum and classical dynamics for integrable billiards
• Developments in Random Matrix Theory
• Counting of Sieber-Richter pairs of periodic orbits