--- title: "Z2× Z2-graded Lie Symmetries of the Lévy-Leblond Equations • CBPF Index" description: "CBPF INDEX is the system that tracks the scientific and technological output of CBPF. It lists the articles, books, technical reports and other publications of the whole CBPF community." canonical_url: "https://index.cbpf.br/en/publicacoes/6ad2a767639f62b11adbce88" locale: en meta: "og:description": "CBPF INDEX is the system that tracks the scientific and technological output of CBPF. It lists the articles, books, technical reports and other publications of the whole CBPF community." "og:image": "https://index.cbpf.br/_og/d/c_Item%2Ctitulo_~WjLDlyBaMi1ncmFkZWQgTGllIFN5bW1ldHJpZXMgb2YgdGhlIEzDqXZ5LUxlYmxvbmQgRXF1YXRpb25z%2Ctipo_Physics%2BNote%2Ccor_~I2E4MmMwMA%2Cautores_~Wi4gS3V6bmV0c292YTsgTi4gQWl6YXdhAzsgSC4gVGFuYWth%2Cdetalhe_~MjkvMDkvMjAxNg%2Csecao_Publications%2Cp_Ii9lbi9wdWJsaWNhY29lcy82YWQyYTc2NzYzOWY2MmIxMWFkYmNlODgi%2Cs_1fw4rBULfX4nmuTL.png" "og:image:height": "600" "og:image:type": image/png "og:image:width": "1200" "og:title": "Z2× Z2-graded Lie Symmetries of the Lévy-Leblond Equations • CBPF Index" "twitter:image": "https://index.cbpf.br/_og/d/c_Item%2Ctitulo_~WjLDlyBaMi1ncmFkZWQgTGllIFN5bW1ldHJpZXMgb2YgdGhlIEzDqXZ5LUxlYmxvbmQgRXF1YXRpb25z%2Ctipo_Physics%2BNote%2Ccor_~I2E4MmMwMA%2Cautores_~Wi4gS3V6bmV0c292YTsgTi4gQWl6YXdhAzsgSC4gVGFuYWth%2Cdetalhe_~MjkvMDkvMjAxNg%2Csecao_Publications%2Cp_Ii9lbi9wdWJsaWNhY29lcy82YWQyYTc2NzYzOWY2MmIxMWFkYmNlODgi%2Cs_1fw4rBULfX4nmuTL.png" --- > Index of every page on this site: [llms.txt](https://index.cbpf.br/llms.txt) **Physics Note****Published** # Z2×Z2{\\mathbb Z}\_2\\times {\\mathbb Z}\_2Z2​×Z2​**-graded Lie Symmetries of the Lévy-Leblond Equations** **Z. Kuznetsova; N. Aizawa; H. Tanaka** 09/29/2016 [**Download PDF **0.5 MB](https://index.cbpf.br/files/publicacao/6ad2a767639f62b11adbce88/pdf) [**Contact owner**](mailto:toppan@cbpf.br?subject=Z2%C3%97%20Z2-graded%20Lie%20Symmetries%20of%20the%20L%C3%A9vy-Leblond%20Equations) ## **Publication details**
Authors
Z. Kuznetsova; N. Aizawa; H. Tanaka
Type
Physics Note
Status
Published
Publication date
29/09/2016
Description
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.
Year
2016
Number
CBPF-NF-004/16
Linked to a thesis
No
Linked to a grant
No
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