Nota de FísicaPublicado
-graded Lie Symmetries of the Lévy-Leblond Equations
Z. Kuznetsova; N. Aizawa; H. Tanaka
Dados da publicação
- Autores
- Z. Kuznetsova; N. Aizawa; H. Tanaka
- Tipo
- Nota de Física
- Status
- Publicado
- Data de publicação
- 29/09/2016
- Descrição
- 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.\par 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. \par 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.
- Ano
- 2016
- Número
- CBPF-NF-004/16
- Vínculo com tese
- Não
- Vínculo com bolsa
- Não