A Clifford Algebra Realization of Supersymmetry and its Polyvector Extension in Clifford Spaces

Abstract

It is shown explicitly how to construct a novel (to our knowledge) realization of the Poincaré superalgebra in 2D. These results can be extended to other dimensions and to (extended) superconformal and (anti) de Sitter superalgebras. There is a fundamental difference between the findings of this work with the other approaches to Supersymmetry (over the past four decades) using Grassmannian calculus and which is based on anti-commuting numbers. We provide an algebraic realization of the anticommutators and commutators of the 2D super-Poincaré algebra in terms of the generators of the tensor product ${Cl_{1,1}(R) \otimes \mathcal{A}}$ of a two-dim Clifford algebra and an internal algebra A whose generators can be represented in terms of powers of a 3 × 3 matrix ${\mathcal{Q}}$ , such that ${\mathcal{Q}^3 = 0}$ . Our realization differs from the standard realization of superalgebras in terms of differential operators in Superspace involving Grassmannian (anti-commuting) coordinates θ ^α and bosonic coordinates x ^μ. We conclude in the final section with an analysis of how to construct Polyvector-valued extensions of supersymmetry in Clifford Spaces involving spinor-tensorial supercharge generators ${{{\mathcal {Q}}_{{\alpha}}^{\mu_1\mu_2\ldots\mu_n}}}$ and momentum polyvectors ${P_{\mu_1\mu_2\ldots\mu_n}}$ . Clifford-Superspace is an extension of Clifford-space and whose symmetry transformations are generalized polyvector-valued supersymmetries.