An Introduction to Coordinate-free Quantization and its Application to Constrained Systems

TL;DR

This paper introduces a coordinate-free approach to quantization that incorporates a flat phase space metric and extends to constrained systems, offering a covariant alternative to traditional Cartesian-based methods.

Contribution

It presents a covariant, coordinate-free quantization method with Wiener measure regularization and demonstrates its extension to constrained systems, broadening quantization techniques.

Findings

Coordinate-free quantization with flat phase space metric

Extension to systems with constraints

Illustrative examples of constrained systems

Abstract

Canonical quantization entails using Cartesian coordinates, and Cartesian coordinates exist only in flat spaces. This situation can either be questioned or accepted. In this paper we offer a brief and introductory overview of how a flat phase space metric can be incorporated into a covariant, coordinate-free quantization procedure involving a continuous-time (Wiener measure) regularization of traditional phase space path integrals. Additionally we show how such procedures can be extended to incorporate systems with constraints and illustrate that extension for special systems.