Emergence of quantum dynamics from chaos: The case of prequantum cat maps

TL;DR

This paper explores how quantum dynamics can emerge from classical chaos using prequantum cat maps, establishing a link between transfer operator resonances and quantum eigenvalues in higher dimensions.

Contribution

It extends Faure and Tsujii's quantization framework to higher-dimensional torus automorphisms, connecting classical resonances with quantum spectra.

Findings

Explicit relation between transfer operator resonances and quantum eigenvalues

Generalization of previous results to higher dimensions

Insight into quantum-classical correspondence in chaotic systems

Abstract

Faure and Tsujii recently proposed a new quantization theory for symplectic Anosov diffeomorphisms. It combines prequantization with the study of the Pollicott--Ruelle resonances of an associated transfer operator. We apply this framework to the hyperbolic symplectic automorphisms of the $2n$-dimensional torus, the so-called cat maps. Our main result gives an explicit relation between the resonances of the prequantum transfer operator and the eigenvalues of the standard quantum cat maps, generalizing the case $n=1$ previously treated by Faure.