Dirac matrices as elements of superalgebraic matrix algebra

TL;DR

This paper explores a Clifford extension of Grassmann algebra, revealing a subalgebra isomorphic to matrix algebra that generalizes it and introduces superalgebraic operators for supersymmetric transformations.

Contribution

It demonstrates the existence of a matrix algebra-like subalgebra within a Clifford-extended Grassmann algebra, expanding the algebraic framework for supersymmetry.

Findings

Identifies a subalgebra isomorphic to matrix algebra within the Clifford extension.

Shows the algebra generalizes matrix algebra and includes superalgebraic operators.

Provides a foundation for supersymmetric transformations within this algebraic structure.

Abstract

The paper considers a Clifford extension of the Grassmann algebra, in which operators are built from Grassmann variables and by the derivatives with respect to them. It is shown that a subalgebra which is isomorphic to the usual matrix algebra exists in this algebra, the Clifford exten-sion of the Grassmann algebra is a generalization of the matrix algebra and contains superalgebraic operators expanding matrix algebra and produces supersymmetric transformations.