$\mathbb{Z}_2× \mathbb{Z}_2$-graded Lie symmetries of the Lévy-Leblond equations. (25th December 2016) - Aizawa, N.; Kuznetsova, Z.; Tanaka, H.; Toppan, F.
Abstract : The ${\mathbb Z}$ first-order differential Lévy-Leblond equations (LLEs) are the non-relativistic analogs of the Dirac equation, being square roots of ($1+d$ )-dimensional Schrödinger or heat equations. Just like the Dirac equation, the LLEs possess a natural supersymmetry. In previous works it was shown that non-supersymmetric partial differential equations (notably the Schrödinger 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 partial differential equation, a natural ${\mathbb Z}_2×{\mathbb Z}_2$ -graded Lie symmetry appears. In particular, we exhaustively investigate the symmetries of the $(1+1)$ -dimensional Lévy-Leblond equations, both in the free case and for the harmonic potential. In the free case a ${\mathbb Z}_2×{\mathbb Z}_2$ -graded Lie superalgebra, realized by first- and second-order differential symmetry operators, is found. In the presence of a non-vanishing quadratic potential, the Schrödinger invariance is maintained, while the ${\mathbb Z}_2$ - and ${\mathbb Z}_2×{\mathbb Z}_2$ -graded extensions are no longer allowed. The construction of the ${\mathbb Z}_2× {\mathbb Z}_2$ -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 Schrödinger algebra.