Lie Algebra Quantization by the Star Product

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.