## A Clifford Algebra Realization of Supersymmetry and Its Polyvector Extension in Clifford Spaces

**Authors:** Carlos Castro

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 x 3 matrix Q, such that Q3 = 0. Our realization
differs from the standard realization of superalgebras in terms
of differential operators in Superspace involving Grassmannian (anticommuting)
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 Qμ1μ2.....μn
and momentum polyvectors Pμ1μ2....μn.
Clifford-Superspace is an extension of Clifford-space and whose symmetry
transformations are generalized polyvector-valued supersymmetries.

**Comments:** 15 pages, submitted to Foundations of Physics.

**Download:** **PDF**

### Submission history

[v1] 26 Jun 2010

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