# Double-Graded Supersymmetric Quantum Mechanics

## TL;DR

This paper introduces a new quantum mechanical model with a $\,\mathbb{Z}_2 \times \mathbb{Z}_2$-graded structure, demonstrating the feasibility of higher gradings in non-relativistic quantum systems, expanding the landscape of supersymmetric models.

## Contribution

It presents a novel quantum mechanics model with higher grading structures, extending the concept of supersymmetry beyond traditional frameworks.

## Key findings

- The model realizes a $\,\mathbb{Z}_2 \times \mathbb{Z}_2$-graded algebra in quantum mechanics.
- It shares features with Witten and parasupersymmetric models but is distinct from both.
- The work shows higher gradings are possible in non-relativistic quantum systems.

## Abstract

A quantum mechanical model that realizes the $ \mathbb{Z}_2 \times \mathbb{Z}_2$-graded generalization of the one-dimensional supertranslation algebra is proposed. This model shares some features with the well-known Witten model and is related to parasupersymmetric quantum mechanics, though the model is not directly equivalent to either of these. The purpose of this paper is to show that novel "higher gradings" are possible in the context of non-relativistic quantum mechanics.