Generalized Heisenberg algebra: application to the harmonic oscillator

The deformed Poisson algebra recently introduced to investigate integrable systems (2003 J. Phys. A: Math. Gen.36 12181–203, 2005 J. Math. Phys.46 042702) is used to perform the transition from the phase space of classical observables (functions depending on positions and momentums) to the Hilbert space of physically well-defined Hermitian operators. A Hamiltonian operator for the harmonic oscillator system is constructed and the eigenvalue problem is solved. The generalization to an n-dimensional space shows that such an algebra does not break the rotational symmetry.