A Clifford algebra realization of Supersymmetry and its Polyvector extension in Clifford Spaces
by Carlos Castro
Center for Theoretical Studies of Physical Systems
Clark Atlanta University, Atlanta, GA. 30314; perelmanc_at_hotmail.com
June 2010, submitted to Foundations of Physics
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 Cl1,1(R) (x) 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 Q, such that Q^3 = 0. Our realization differs from the standard realization of superalgebras in terms of differential operators in Superspace involving Grassmannian (anticommuting) coordinates \theta^\alpha 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 Q^μ1μ2…..μn_\alpha and momentum polyvectors P_μ1μ2….μn. Clifford-Superspace is an extension of Clifford-space and whose symmetry transformations are generalized polyvector-valued supersymmetries.
Clifford algebras; Supersymmetry; Polyvector-supersymmetry; M, F theory superalgebras.
Source: Email by C. Castro, 2010/06/27 6:36, (czarlosromanov_at_yahoo.com)