Relations between different notions of degrees of freedom of a quantum system and its classical model

TL;DR

This paper explores the relationships between various notions of degrees of freedom in quantum and classical systems, analyzing their differences and similarities through theoretical definitions and illustrative examples.

Contribution

It formulates and compares three types of degrees of freedom in classical and quantum systems, focusing on the dynamical degrees of freedom and their relation to classical properties.

Findings

Relations between quantum and classical dynamical degrees of freedom are characterized.

A conjecture links quantumness generation to classical dynamical properties.

Illustrative systems support the proposed relations and conjecture.

Abstract

There are at least three different notions of degrees of freedom (DF) that are important in comparison of quantum and classical dynamical systems. One is related to the type of dynamical equations and inequivalent initial conditions, the other to the structure of the system and the third to the properties of dynamical orbits. In this paper, definitions and comparison in classical and quantum systems of the tree types of DF are formulated and discussed. In particular, we concentrate on comparison of the number of the so called dynamical DF in a quantum system and its classical model. The comparison involves analyzes of relations between integrability of the classical model, dynamical symmetry and separability of the quantum and the corresponding classical systems and dynamical generation of appropriately defined quantumness. The analyzes is conducted using illustrative typical systems. A conjecture summarizing the observed relation between generation of quantumness by the quantum dynamics and dynamical properties of the classical model is formulated.