Quantum Gravity and String Theory


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 θα and bosonic coordinates xμ. 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.

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Submission history

[v1] 26 Jun 2010

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