On the Classification of the Lévy-Leblond Spinors

Abstract The first-order Lévy-Leblond differential equations (LLEs) are the non-relativistic analogous of the Dirac equation: they are the “square roots” of the Schrödinger equation in (1 + d ) dimensions and admit spinor solutions. In this paper we show how to extend to the Lévy-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 first-order differential operators depending on the time and space coordinates.