When Canonical Quantization Fails, Here is How to Fix It

TL;DR

This paper addresses the limitations of canonical quantization, proposing an alternative approach called Enhanced Quantization that maintains compatibility with classical transformations.

Contribution

It introduces Enhanced Quantization as a method to overcome canonical quantization failures related to quantum coordinate transformations.

Findings

Enhanced Quantization aligns quantum operators with classical transformations.

It resolves issues where quantum canonical transformations fail.

The approach broadens the applicability of quantization methods.

Abstract

Following Dirac, the rules of canonical quantization include classical and quantum contact transformations of classical and quantum phase space variables. While arbitrary classical canonical coordinate transformations exist that is not the case for some analogous quantum canonical coordinate transformations. This failure is due to the rigid connection of quantum variables arising by promoting the corresponding classical variable from a $c$-number to a $q$-number. A different relationship of $c$-numbers and $q$-numbers in the procedures of Enhanced Quantization shows the compatibility of all quantum operators with all classical canonical coordinate transformations.