Quantum Chaos

In classical mechanics, chaos originates from nonlinearity and is extremely sensitive to initial conditions. For different but exceedingly close initial states could give rise to distinctly different states in the time evolution. It is in this sense that the results are nondeterministic. As linear quantum evolution includes classical physics as a limiting case, how chaos can be manifested in quantum dynamics thus becomes a puzzle. To this end, there are two ways of exploring quantum chaos. One is via the quantum-classical correspondence realization, and the other is the study of the universal level statistics of quantum systems whose classical counterparts are chaotic. In this chapter, we shall focus on the first approach as coherent states naturally connect quantum dynamics with classical mechanics. We shall discuss how a quantum system can be represented by the associated Lie group and spectral generating algebra, from which the concept of classical-like degrees of freedom can be defined, and the corresponding coset space of the coherent states can serve as the quantum counterpart of phase space. Meanwhile, quantum integrability can be defined via the concept of dynamical symmetry. Thus, the universal properties of quantum chaos associated with dynamical symmetry breaking are obtained within the framework of coherent state representation.