Canonical quantization of classical mechanics in curvilinear coordinates. Invariant quantization procedure

TL;DR

This paper introduces an invariant quantization method for classical mechanics in curvilinear coordinates, deriving explicit quantum operators and discussing extensions to non-flat spaces and associated ambiguities.

Contribution

It presents a novel invariant quantization procedure applicable to classical mechanics in curvilinear coordinates, with explicit operator forms and discussion of non-flat space extensions.

Findings

Explicit position and momentum operators in arbitrary curvilinear coordinates

Quantization formalism extended to non-flat configuration spaces

Discussion of ambiguities in the quantization process

Abstract

In the paper is presented an invariant quantization procedure of classical mechanics on the phase space over flat configuration space. Then, the passage to an operator representation of quantum mechanics in a Hilbert space over configuration space is derived. An explicit form of position and momentum operators as well as their appropriate ordering in arbitrary curvilinear coordinates is demonstrated. Finally, the extension of presented formalism onto non-flat case and related ambiguities of the process of quantization are discussed.