Lie Algebra Quantization by the Star Product

TL;DR

This paper explores applying star product quantization to Lie algebras, introducing an x-dependent deformation parameter, and proposes a higher-dimensional unification approach similar to Kaluza-Klein theory.

Contribution

It extends star product quantization to Lie algebras with variable deformation parameters and suggests a higher-dimensional unification framework.

Findings

Star product quantization can be applied to Lie algebras.

Introducing x-dependent deformation parameters generalizes the quantization.

A higher-dimensional unification approach is proposed.

Abstract

We apply the star product quantization to the Lie algebra. The quantization in terms of the star product is well known and the commutation relation in this case is called the $\theta$-deformation where the constant $\theta$ appears as a parameter. In the application to the Lie algebra, we need to change the parameter $\theta$ to $x$-dependent $\theta(x)$. There is no essential difference between the quantization in the quantum mechanics and deriving quantum numbers in the Lie algebra from the viewpoint of the star product. We propose to unify them in higher dimensions, which may be analogous to the Kaluza-Klein theory in the classical theory.