Gaussian Dynamics is Classical Dynamics

Abstract

A direct comparison of quantum and classical dynamical systems can be accomplished through the use of distribution functions. This is useful for both fundamental investigations such as the nature of the quantum-classical transition as well as for applications such as quantum feedback control. By affording a clear separation between kinematical and dynamical quantum effects, the Wigner distribution is particularly valuable in this regard. Here we discuss some consequences of the fact that when closed-system classical and quantum dynamics are treated in Gaussian approximation, they are in fact identical. Thus, it follows that several results in the so-called `semiquantum' chaos actually arise from approximating the classical, and not the quantum dynamics. (Similarly, opposing claims of quantum suppression of chaos are also suspect.) As a simple byproduct of the analysis we are able to show how the Lyapunov exponent appears in the language of phase space distributions in a way that clearly underlines the difference between quantum and classical dynamical situations. We also informally discuss some aspects of approximations that go beyond the Gaussian approximation, such as the issue of when quantum nonlinear dynamical corrections become important compared to nonlinear classical corrections.