Ternary Z₃ -graded generalization of Heisenberg's algebra

We investigate a ternary, Z₃-graded generalization of the Heisenberg algebra. It turns out that introducing a non-trivial cubic root of unity, j = e ^2πi/3, one can define two types of creation operators instead of one, accompanying the usual annihilation operator. The two creation operators are non-hermitian, but they are mutually conjugate. Together, the three operators form a ternary algebra, and some of their cubic combinations generate the usual Heisenberg algebra.A cubic analogue of Hamiltonian operator is constructed by analogy with the usual harmonic oscillator. A set of eigenstates in coordinate representation is constructed in terms of functions satisfying linear differential equation of third order. (paper)