The many faces of quantum chaos

Various approaches to quantum chaos are reviewed and compared. It is not difficult to generate quantum evolutions (unitary mappings of Hilbert space) which are chaotic. Some of the algorithms achieving this result are quite formal and apparently devoid of physical interest. In the more promising approaches, the following property stands out: a quantum system whose classical analog is chaotic displays hypersensitivity to small perturbations of its Hamiltonian. The long range evolution of such a system is unpredictable in the presence of small, uncontrollable perturbations. This unpredictability is the hallmark of physical chaos.