Star Products and Geometric Algebra

TL;DR

This paper explores how geometric algebra can be understood through a deformation of super analysis using star products, linking classical and quantum mechanics, especially for systems with spin.

Contribution

It introduces a formalism that unifies classical and quantum mechanics via fermionic and bosonic star products in geometric algebra, clarifying Grassmann and Clifford structures.

Findings

Connects geometric algebra with deformation quantization.

Provides a formalism for classical and quantum mechanics integration.

Clarifies the relation between Grassmann, Clifford, and quantum structures.

Abstract

The formalism of geometric algebra can be described as deformed super analysis. The deformation is done with a fermionic star product, that arises from deformation quantization of pseudoclassical mechanics. If one then extends the deformation to the bosonic coefficient part of superanalysis one obtains quantum mechanics for systems with spin. This approach clarifies on the one hand the relation between Grassmann and Clifford structures in geometric algebra and on the other hand the relation between classical mechanics and quantum mechanics. Moreover it gives a formalism that allows to handle classical and quantum mechanics in a consistent manner.