The Cat and the Butterfly: quantum properties of chaotic systems

In this article I present an introduction to the problem of quantum chaos, i.e. the characterization and treatment of quantum systems whose classical analogue has chaotic dynamics. Two of the main approaches to this problem are discussed: one is statistical, making use of random matrices, while the other is based on the semiclassical approximation. Results obtained by these approaches are mentioned for closed systems (spectral correlation functions), transport (moments like conductance and shot noise) and resonances (fractal Weyl law and fractal wave functions).本文旨在介绍量子混沌（quantum chaos）问题，即对经典对应系统具有混沌动力学特性的量子系统开展表征与处理的相关研究。本文将讨论该问题的两种主流研究路径：其一为基于随机矩阵（random matrices）的统计方法，其二为基于半经典近似（semiclassical approximation）的理论方法。此外还将介绍上述两种方法在封闭系统（谱关联函数）、输运过程（电导、散粒噪声等输运矩量）以及共振现象（分形外尔定律与分形波函数）相关研究中取得的成果。