Ｚ２ｘＺ２-graded Lie symmetries of the Levy-Leblond equations
Ｚ２ｘＺ２-graded Lie symmetries of the Levy-Leblond equations
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Levy-Leblond 方程的 Z2xZ2 分级李对称性
DOI：
10.1093/ptep/ptw176
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发表时间：
2016
影响因子：
3.5
通讯作者：
H. Tanaka and F. Toppan
中科院分区：
文献类型：
--
作者：
N. Aizawa;Z. Kuznetsova;H. Tanaka and F. Toppan
一阶微分l<s:1> -莱布隆德方程（LLEs）是狄拉克方程的非相对论类似物，是（）维Schrödinger或热方程的平方根。就像狄拉克方程一样，lle具有天然的超对称性。在以前的工作中，证明了非超对称偏微分方程（特别是自由粒子或存在谐波势的Schrödinger方程）承认自然梯度李对称。本文证明了对于一类超对称偏微分方程，存在一个自然梯度李对称。特别地，我们详尽地研究了在自由情况下和谐波势下的维度l<s:1> - leblond方程的对称性。在自由情况下，得到了由一阶和二阶微分对称算子实现的a级李超代数。当存在不消失的二次势时，保持Schrödinger不变性，而不再允许-和分级扩展。（）维自由热LLE的梯度李对称的构造引入了一个新特征，解释了不进入超级Schrödinger代数的一阶微分对称算子的存在性。
Thefirst-order differential Lévy-Leblond equations (LLEs) are the non-relativistic analogs of the Dirac equation, being square roots of ()-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-graded Lie symmetry. In this paper we show that, for a certain class of supersymmetric partial differential equation, a natural-graded Lie symmetry appears. In particular, we exhaustively investigate the symmetries of the-dimensional Lévy-Leblond equations, both in the free case and for the harmonic potential. In the free case a-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- and-graded extensions are no longer allowed. The construction of the-graded Lie symmetry of the ()-dimensional free heat LLE introduces a new feature, explaining the existence of first-order differential symmetry operators not entering the super Schrödinger algebra.