On the Weyl transform and the quantization of the hypersphere

TL;DR

This paper explores the challenges of quantizing the 2-sphere, reviewing classical and modern quantization methods, and discusses the known impossibility of a rigorous quantization in this context, with implications for quantum models like the hydrogen atom.

Contribution

It provides a detailed analysis of the limitations of monomial quantization on the 2-sphere and discusses the implications for quantum theory modeling.

Findings

Quantization of the 2-sphere is fundamentally impossible.

Classical and modern quantization methods are reviewed.

Implications for quantum models like the hydrogen atom are discussed.

Abstract

After having dealt with the classical Weyl quantization, the deformation quantization and the recently (but old) Born-Jordan quantization, the purpose of the article is a sort of ''monomial quantization'' of the $2$-sphere. The result of the impossibility of a rigorous quantization of the sphere is well known and treated in the literature, despite everything the case of the hydrogen atom remains one of the most interesting cases in the modeling of quantum theories.