Ternary $Z_3$-symmetric algebra and generalized quantum oscillators

R. Kerner

Laboratoire de Physique Théorique de la Matière Condensée, Sorbonne Université, Paris, France

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.

Keywords: $Z_3$-graded algebraic structures, ternary algebras, cubic Heisenberg algebra, Bohr–Sommerfeld quantization, quantum oscillator.

Received: 06.06.2023
Revised: 06.06.2023

DOI: 10.4213/tmf10567

 English version: Theoretical and Mathematical Physics, 2024, 218:1, 87–105

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