Covariance of noncommutative Grassmann star product

TL;DR

This paper introduces a noncommutative Grassmann star product using fermionic coherent states, studies its covariance under unitary transformations, and extends the concept to super star products within supersymmetric frameworks.

Contribution

It presents a novel noncommutative Grassmann star product and analyzes its covariance properties, extending the framework to supercoherent states for supersymmetric systems.

Findings

Star product is covariant under unitary transformations.

Super star product is formulated using supercoherent states.

Framework extends noncommutative geometry to fermionic and supersymmetric contexts.

Abstract

Using the Coherent states of many fermionic degrees of freedom labeled by Gra\ss mann variables, we introduce the noncommutative (precisely non anticommutative) Gra\ss mann star product. The covariance of star product under unitary transformations, particularly canonical ones, is studied. The super star product, based on supercoherent states of supersymmetric harmonic oscillator, is also considered.