Linear and Nonlinear Dynamical Chaos

TL;DR

This paper explores the concept of dynamical chaos across classical and quantum mechanics, discussing its properties, contradictions, and implications for fundamental physical principles like measurement and causality.

Contribution

It provides a comprehensive analysis of chaos in classical and quantum systems, highlighting new insights into quantum chaos and its relation to fundamental physics.

Findings

Classical chaos exhibits local instability and robustness.

Quantum chaos is a distinct dynamical phenomenon.

Connections between chaos, measurement, and causality are discussed.

Abstract

Interrelations between dynamical and statistical laws in physics, on the one hand, and between the classical and quantum mechanics, on the other hand, are discussed with emphasis on the new phenomenon of dynamical chaos. The principal results of the studies into chaos in classical mechanics are presented in some detail, including the strong local instability and robustness of the motion, continuity of both the phase space as well as the motion spectrum, and time reversibility but nonrecurrency of statistical evolution, within the general picture of chaos as a specific case of dynamical behavior. Analysis of the apparently very deep and challenging contradictions of this picture with the quantum principles is given. The quantum view of dynamical chaos, as an attempt to resolve these contradictions guided by the correspondence principle and based upon the characteristic time scales of quantum evolution, is explained. The picture of the quantum chaos as a new generic dynamical phenomenon is outlined together with a few other examples of such a chaos including linear (classical) waves and digital computer. I conclude with discussion of the two fundamental physical problems: the quantum measurement (\psi-collapse), and the causality principle which both appear to be related to the phenomenon of dynamical chaos.