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

TL;DR

This paper extends supersymmetric quantum mechanics to a $ Z_2^n$-graded framework using Clifford algebras, introducing models that realize the algebra's central elements and exploring their independence.

Contribution

It constructs $ Z_2^n$-graded supersymmetric quantum mechanical models utilizing Clifford gamma matrices, expanding the algebraic structure beyond traditional supersymmetry.

Findings

Models realize $ Z_2^n$-graded Poincaré algebra in 1D

Central elements are represented as operators, with independence depending on Clifford algebra dimension

More independent central elements are achieved with higher-dimensional Clifford algebras

Abstract

It is shown that the ${\cal N}=1$ 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 Poincar\'e algebra in one-dimensional spacetime. Reflecting the fact that the $\mathbb{Z}_2^n$-graded Poincar\'e 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.