Geometric Algebra and Star Products on the Phase Space

TL;DR

This paper develops a unified geometric algebra framework using star products on phase space, enabling new formulations of Riemannian, symplectic, and Poisson geometries, with applications to quantum mechanics and classical mechanics structures.

Contribution

It introduces a novel formalism combining superanalysis, Clifford calculus, and deformation quantization, linking geometric algebra with phase space structures and mechanics.

Findings

Clifford calculus equivalent to superanalysis deformed by fermionic star product

Formulation of Riemannian and exterior calculus in this geometric algebra framework

Natural emergence of spin and BRST structures in phase space

Abstract

Superanalysis can be deformed with a fermionic star product into a Clifford calculus that is equivalent to geometric algebra. With this multivector formalism it is then possible to formulate Riemannian geometry and an inhomogeneous generalization of exterior calculus. Moreover it is shown here how symplectic and Poisson geometry fit in this context. The application of this formalism together with the bosonic star product formalism of deformation quantization leads then on space and space-time to a natural appearance of spin structures and on phase space to BRST structures that were found in the path integral formulation of classical mechanics. Furthermore it will be shown that Poincare and Lie-Poisson reduction can be formulated in this formalism.