The invariant Berry phase of the quantum relativistic four-dimensional harmonic oscillator

We review our previous work on the existence of Lorentz invariant Berry phases generated in the Stueckleberg-Horwitz-Piron manifestly covariant quantum theory (SHP), also reviewed here, by a perturbed four dimensional harmonic oscillator. These phases are associated with a fractional perturbation of the azimuthal symmetry of the oscillator. They are computed numerically by using time independent perturbation theory and the definition of the Berry phase generalized to the framework of the SHP relativistic quantum theory. In this paper, we give an explicit proof of the Lorentz invariance.