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 Poincare 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-Poincare 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.