Graded colour Lie superalgebras for solving Lévy-Leblond equations

Abstract

The Lévy-Leblond equation with free potential admits a symmetry algebra that is a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded colour Lie superalgebra (see arXiv:1609.08224). We extend this result in two directions by considering a time-independent version of the Lévy-Leblond equation. First, we construct a ${\mathbb{Z}}_2^3$-graded colour Lie superalgebra containing operators that leave the eigenspaces invariant and demonstrate the utility of this algebra in constructing general solutions for the free equation. Second, we find that the ladder operators for the harmonic oscillator generate a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded colour Lie superalgebra and we use the operators from this algebra to compute the spectrum. These results illustrate two points: the Lévy-Leblond equation admits colour Lie superalgebras with gradings higher than ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ and colour Lie superalgebras appear for potentials besides the free potential.

arXiv comment22 pages

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Cited by in corpus (4)

• Affine extensions of ${\mathbb{Z}}_2^2$-graded $osp(1|2)$ and Virasoro algebra
• On braid statistics versus parastatistics
• Universal weight systems from a minimal ${\mathbb{Z}}_2^2$-graded Lie algebra
• On the Classification of the Lévy-Leblond Spinors