Quantum-Classical Correspondence and Quantum Chaos

In this article, a theory which we have recently developed is reviewed. This theory addresses the long standing problem of quantum-classical correspondence. It begins with the axiomatic structure of quantum mechanics, from which the basic concept of dynamical degrees of freedom emerges and the associated physical geometry of an arbitrary quantum system is constructed. Such a geometrical space possesses both complex and symplectic structures, the two basic ingredients to guarantee the rigorous correspondence between quantum and classical kinematics. Also, it is remarkable that the quantum-classical correspondence of dynamics can be mediated by a quenched index obtained from the geometry. The intermediate stage from quantum to classical dynamics is described by the semiquantal dynamics. The main property of the semiquantal dynamics is that it manifests the effect of quantum fluctuation on classical trajectories, and quantum fluctuation depends explicitly on the quenched index. This offers a way to study “chaos in quantum systems”. We also prove a theorem which links dynamical symmetry and quantum integrability, and conclude that “quantum chaos” is related to dynamical symmetry breaking. The generic rule of quantum chaos is explored. Throughout the article, many basic quantum systems are used to illustrate these important points.