Quantization of kinematics on configuration manifolds

TL;DR

This paper reviews the topological aspects of quantum kinematics on configuration manifolds, classifying various quantizations and introducing quantum Borel kinematics with topological invariants.

Contribution

It provides a comprehensive classification of quantum kinematics on manifolds, including systems with gauge fields, and introduces quantum Borel kinematics as a global approach.

Findings

Complete classification theorems in special cases

Introduction of quantum Borel kinematics with topological invariants

Demonstration of applications through selected examples

Abstract

The review is devoted to topological global aspects of quantal description. The treatment concentrates on quantizations of kinematical observables --- generalized positions and momenta. A broad class of quantum kinematics is rigorously constructed for systems, the configuration space of which is either a homogeneous space of a Lie group or a connected smooth finite-dimensional manifold without boundary. The class also includes systems in an external gauge field for an Abelian or a compact gauge group. Conditions for equivalence and irreducibility of generalized quantum kinematics are investigated with the aim of classification of possible quantizations. Complete classification theorems are given in two special cases. It is attempted to motivate the global approach based on a generalization of imprimitivity systems called {\em quantum Borel kinematics}. These are classified by means of global invariants --- quantum numbers of topological origin. Selected examples are presented which demonstrate the richness of applications of Borel quantization. The review aims to provide an introductory survey of the subject and to be sufficiently selfcontained as well, so that it can serve as a standard reference concerning Borel quantization for systems admitting localization on differentiable manifolds.