Spin Description in the Star Product and the Path Integral Formalism

TL;DR

This paper extends the star product formalism to include spin by incorporating fermionic variables, establishing a connection between the star product, operator, and path integral approaches in quantum mechanics.

Contribution

It introduces a fermionic star product formalism for spin and relates it to existing operator and path integral methods, unifying different descriptions of spin.

Findings

Fermionic star product formalism is equivalent to the Clifford product of geometric algebra.

The formalism establishes a relation between star product and path integral approaches for spin.

The approach generalizes the bosonic star product to include fermionic degrees of freedom.

Abstract

The spin can be described in the star product formalism by extending the bosonic Moyal product in the fermionic sector. The fermionic star product is then the Clifford product of geometric algebra and it is possible to formulate the fermionic star product formalism in analogy to the bosonic star product formalism. For the fermionic star product description of spin, one can then establish the relation to other approaches that describe spin with fermionic variables, i.e. the operator formalism and the path integral formalism. It is shown that the fermionic star product formalism and the fermionic path integral formalism are related in analogy to their bosonic counterparts.