The general form of the *-commutator on the Grassman algebra

TL;DR

This paper investigates the possible deformations of the Poisson bracket on Grassman algebra, revealing new forms of *-commutators beyond the well-known Moyal commutator, which cannot be simplified to it.

Contribution

It identifies additional deformations of the Poisson bracket on Grassman algebra that are not equivalent to the Moyal commutator through similarity transformations.

Findings

Existence of deformations beyond the Moyal commutator

Identification of even and odd n deformations

Deformations not reducible to Moyal form by similarity transformations

Abstract

We study the general form of the *-commutator treated as a deformation of the Poisson bracket on the Grassman algebra. We show that, up to a similarity transformation, there are other deformations of the Poisson bracket in addition to the Moyal commutator (one at even and one at odd n, n is the number of the generators of the Grassman algebra) which are not reduced to the Moyal commutator by a similarity transformation.