Quantum-Classical Correspondence: Dynamical Quantization and the Classical Limit

Joshua Wilkie, Paul W. Brumer

A generalized approach to the quantization of a large class of maps on a torus, i.e., quantization via the von Neumann equation, is described and a number of issues related to the quantization of model systems are discussed. The approach yields well-behaved mixed quantum states for tori for which the corresponding Schr\"odinger equation has no solutions, as well as an extended spectrum for tori where the Schr\"odinger equation can be solved. Quantum-classical correspondence is demonstrated for the class of mappings considered, with the Wigner-Weyl density p(p,q,t) going to the correct classical limit. An application to the cat map yields, in a direct manner, nonchaotic quantum dynamics, plus the exact chaotic classical propagator in the correspondence limit.

V. Ceauşescu, Andrei H Gheorghe

Patrizia Castiglione, Giovanni Jona-Lasinio, Carlo Presilla A

We present numerical evidence that in a system of interacting bosons there exists a correspondence between the spectral properties of the exact quantum Hamiltonian and the dynamical chaos of the associated mean-field evolution. This correspondence, analogous to the usual quantum-classical correspondence, is related to the formal parallel between the second quantization of the mean field, which generates the exact dynamics of the quantum N-body system, and the first quantization of classical canonical coordinates. The limit of infinite density and the thermodynamic limit are then briefly discussed.

Yu. M. Shirokov

Gerd Niestegge

Eric A. Galapon

We raise the problem of constructing quantum observables that have classical counterparts without quantization. Specifically, we seek to define and motivate a solution to the quantum-classical correspondence problem independent from quantization and discuss the general insufficiency of prescriptive quantization, particularly, the Weyl quantization. We demonstrate our points by constructing time of arrival operators without quantization and from these recover their classical counterparts.