Ternary $Z_3$-symmetric algebra and generalized quantum oscillators

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.