On obtaining classical mechanics from quantum mechanics

TL;DR

This paper explores how to systematically derive classical phase spaces and dynamics from quantum systems, including non-linear cases, using a quotienting procedure on the quantum phase space.

Contribution

It introduces a method to extract both linear and non-linear classical phase spaces from quantum structures, extending previous approaches to more general quantum systems.

Findings

Systematic quotienting procedure yields classical phase spaces from quantum Hilbert spaces.

Method applies to finite, separable, and non-separable Hilbert spaces, including spin systems and polymer quantization.

Effective classical dynamics can be constructed to mirror quantum behavior under certain conditions.

Abstract

Constructing a classical mechanical system associated with a given quantum mechanical one, entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum mechanical system naturally has the structure of an infinite dimensional symplectic manifold (`quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straight forwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and recovers the linear classical phase space $\mathbb{R}^{\mathrm{2N}}$. We report on how the procedure also allows extraction of non-linear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin-j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (Polymer quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality.