Born-Jordan and Weyl Quantizations of the 2D Anisotropic Harmonic Oscillator

TL;DR

This paper compares Born-Jordan and Weyl quantization methods applied to the 2D anisotropic harmonic oscillator, analyzing how each affects the algebra of constants of motion and the superintegrable structure.

Contribution

It provides a detailed comparison of Born-Jordan and Weyl quantizations on superintegrable systems, highlighting differences in preserving algebraic structures.

Findings

Weyl quantization preserves the superintegrable structure.

Born-Jordan quantization may alter the algebra of constants of motion.

Weyl's method consistently maintains the original system properties.

Abstract

We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator. Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures. In the examples considered, we have that the Weyl formula always preserves the original superintegrable structure of the system, while the Born-Jordan formula, when producing different operators than the Weyl's one, does not.