On the Weyl transform and the quantization of the hypersphere

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.