${\mathbb{Z}}_2^n$-Graded extensions of supersymmetric quantum mechanics via Clifford algebras

Abstract

It is shown that the $\mathcal{N}=1$ supersymmetric quantum mechanics (SQM) can be extended to a ${\mathbb{Z}}_2^n$-graded superalgebra. This is done by presenting quantum mechanical models which realize, with the aid of Clifford gamma matrices, the ${\mathbb{Z}}_2^n$-graded Poincaré algebra in one-dimensional spacetime. Reflecting the fact that the ${\mathbb{Z}}_2^n$-graded Poincaré algebra has a number of central elements, a sequence of models defining the ${\mathbb{Z}}_2^n$-graded version of SQM are provided for a given value of $n.$ In a model of the sequence, the central elements having the same ${\mathbb{Z}}_2^n$-degree are realized as dependent or independent operators. It is observed that as use the Clifford algebra of lager dimension, more central elements are realized as independent operators.

19 pages, no figure