On the Classification of the L\'evy-Leblond Spinors

TL;DR

This paper classifies Levy-Leblond spinors similarly to relativistic spinors and explores their solutions with potentials, revealing new algebraic structures in non-relativistic quantum mechanics.

Contribution

It extends the classification of relativistic spinors to Levy-Leblond spinors and analyzes their solutions with potentials, introducing a new realization of the superalgebra.

Findings

Classification of Levy-Leblond spinors analogous to relativistic cases

Extension to include potential terms in the equations

New differential realization of the osp(1|2) superalgebra in 1+1 dimensions

Abstract

The first-order L\'evy-Leblond differential equations (LLEs) are the non-relativistic analogous of the Dirac equation: they are the "square roots" of the Schr\"odinger equation in ($1+d$) dimensions and admit spinor solutions. In this paper we show how to extend to the L\'evy-Leblond spinors the real/complex/quaternionic classification of the relativistic spinors (which leads to the notions of Dirac, Weyl, Majorana, Majorana-Weyl, Quaternionic spinors). Besides the free equations, we also consider the presence of potential terms. Applied to a conformal potential, the simplest $(1+1)$-dimensional LLE induces a new differential realization of the $osp(1|2)$ superalgebra in terms of differential operators depending on the time and space coordinates.