Abstract

We present a generalized version of a quantum oscillator described by means of a ternary Heisenberg algebra. The model leads to a sixth-order Hamiltonian whose energy levels can be discretized using the Bohr–Sommerfeld quantization procedure. We note the similarity with the \(Z_3\)-extended version of Dirac’s equation applied to quark color dynamics, which also leads to sixth-order field equations. The paper also contains a comprehensive guide to \(Z_3\)-graded structures, including ternary algebras, which form a mathematical basis for the proposed generalization. The symmetry properties of the model are also discussed.

中文翻译：

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三元$$Z_3$$ -对称代数和广义量子振荡器

摘要

我们提出了通过三元海森堡代数描述的量子振荡器的广义版本。该模型得出六阶哈密顿量，其能级可以使用玻尔-索末菲量子化过程进行离散化。我们注意到与应用于夸克颜色动力学的狄拉克方程的\(Z_3\)扩展版本的相似性，这也导致了六阶场方程。该论文还包含对\(Z_3\)分级结构的综合指南，包括三元代数，它们构成了所提出的泛化的数学基础。还讨论了模型的对称性。