${\mathbb Z}_2\times {\mathbb Z}_2$-graded Lie Symmetries of the L\'evy-Leblond Equations

Abstract

The first-order differential L\'evy-Leblond equations (LLE's) are the non-relativistic analogs of the Dirac equation, being square roots of ($1+d$)-dimensional Schr\"odinger or heat equations. Just like the Dirac equation, the LLE's possess a natural supersymmetry. In previous works it was shown that non supersymmetric PDE's (notably, the Schr\"odinger equations for free particles or in the presence of a harmonic potential), admit a natural ${\mathbb Z}_2$-graded Lie symmetry. In this paper we show that, for a certain class of supersymmetric PDE's, a natural ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie symmetry appears. In particular, we exhaustively investigate the symmetries of the $(1+1)$-dimensional L\'evy-Leblond Equations, both in the free case and for the harmonic potential. In the free case a ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie superalgebra, realized by first and second-order differential symmetry operators, is found. In the presence of a non-vanishing quadratic potential, the Schr\"odinger invariance is maintained, while the ${\mathbb Z}_2$- and ${\mathbb Z}_2\times{\mathbb Z}_2$- graded extensions are no longer allowed. The construction of the ${\mathbb Z}_2\times {\mathbb Z}_2$-graded Lie symmetry of the ($1+2$)-dimensional free heat LLE introduces a new feature, explaining the existence of first-order differential symmetry operators not entering the super Schr\"odinger algebra.