Incompressible Canonical Quantization

TL;DR

This paper introduces incompressible momentum observables and constructs a set of canonically conjugate variables in manifolds, simplifying quantum descriptions and addressing momentum operator ambiguities related to particle statistics.

Contribution

It presents a novel framework for defining incompressible momentum in manifolds, linking momentum operator extensions to particle types, and unifying quantum descriptions with Euclidean space.

Findings

Incompressible momentum observables can be constructed under certain metric conditions.

Quantum motion in manifolds can be described using variables analogous to Cartesian coordinates.

The non-uniqueness of momentum operators relates to particle fermion or boson nature.

Abstract

The notion of incompressible momentum observables is introduced. It is shown that when the metric in a manifold has a certain form, a set of canonically conjugate variables Xk and Pk in which Pk are incompressible, can be constructed. Based on this set of variables, the quantum mechanical description of the motion of a particle in a manifold, is identical to that associated with the familiar canonically conjugate variables xk and pk in an Euclidean space with Cartesian coordinates. The controversy related to non-uniqueness of momentum operators when the range of a coordinate is a finite interval is reduced to two possible extensions. This suggests relating these two types of extensions to the type of particle as being fermion or boson.