A perturbative approach to the relativistic harmonic oscillator and unitarity

A quantum realization of the relativistic Harmonic oscillator is achieved i n terms of the spatial variable x and -i (h) over bar d/dx (the minimal can onical representation). The Hamiltonian operator is found (at lower order) by using a perturbative expansion in the constant c(-1). Unlike the Foldy-W outhuysen version of the relativistic hydrogen atom, conventional perturbat ion theory cannot be applied and a perturbation of the scalar product itsel f is required to make the theory unitary.