On the classical limit of quantum mechanics, fundamental graininess and chaos: compatibility of chaos with the correspondence principle

TL;DR

This paper reviews the classical limit of quantum mechanics, examining how chaos and fundamental graininess challenge the correspondence principle, and introduces a formalism to analyze phase space surfaces and trajectory deformations.

Contribution

It presents a formalism for the classical limit that accounts for chaos and graininess, analyzing phase space surfaces and trajectory deformation in quantum chaos.

Findings

Regular cells become amoeboid in non-integrable systems

Deformations pose a threat to the correspondence principle

Timescales of quantum chaos are crucial for classical limit

Abstract

The aim of this paper is to review the classical limit of Quantum Mechanics and to precise the well known threat of chaos (and fundamental graininess)to the correspondence principle. We will introduce a formalism for this classical limit that allows us to find the surfaces defined by the constants of the motion in phase space. Then in the integrable case we will find the classical trajectories, and in the non-integrable one the fact that regular initial cells become "amoeboid-like". This deformations and their consequences can be considered as a threat to the correspondence principle unless we take into account the characteristic timescales of quantum chaos. Essentially we present an analysis of the problem similar to the one of Omn\`{e}s [10,11], but with a simpler mathematical structure.