Z_2× Z_2-graded Lie symmetries of the Lévy-Leblond equations. (25th December 2016) - Aizawa, N.; Kuznetsova, Z.; Tanaka, H.; Toppan, F.
Abstract : The 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 Z_2 -graded Lie symmetry. In this paper we show that, for a certain class of supersymmetric partial differential equation, a natural Z_2×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 Z_2×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 Z_2 - and Z_2×Z_2 -graded extensions are no longer allowed. The construction of the Z_2× 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.