To what extent do the Classical Equations of Motion Determine the Quantization Scheme?

TL;DR

This paper demonstrates that classical equations of motion alone do not uniquely determine the quantization scheme, using the harmonic oscillator with two inequivalent Lagrangians as an example.

Contribution

It provides a detailed analysis showing that different Lagrangian formulations can lead to the same classical equations but different quantization schemes.

Findings

Classical equations of motion do not uniquely specify quantization.

Two inequivalent Lagrangians produce the same classical solutions.

Quantization depends on the chosen Lagrangian, not just the equations of motion.

Abstract

A simple example of one particle moving in a (1+1) space-time is considered. As an example we take the harmonic oscillator. We confirm the statement that the classical Equations of Motion do not determine at all the quantization scheme. To this aim we use two inequivalent Lagrange functions, yielding Euler-Lagrange Equations, having the same set of solutions. We present in detail the calculations of both cases to emphasize the differences occuring between them.