We show the existence of Lorentz invariant Berry phases generated, in the Stueckleberg-HorwitzPiron manifestly covariant quantum theory (SHP), 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 SHP relativistic quantum theory.

We study the transport of a small wave packet in the embedding of the Stueckelberg-Horwitz-Piron relativistic quantum theory into the manifold of general relativity around the Schwarzschild solution using a semiclassical approximation. We find that the parallel transport of the momentum leads to a geometrical (Berry type) phase.

Abstract.We show that there is nontrivial Berry relativistically covariant phase generated by a perturbed relativistic oscillator. This phase is associated with a fractional perturbation of the azimuthal symmetry of the oscillator.A manifestly covariant quantum mechanics was formulated by E. C. G. Stueckelberg [1] in 1941. He studied this theory for the case of a single particle in an external field. He considered the phenomenon of pair annihilation and creation as a manifestation of the development, in each case, of a single world line that curves in such a way that one part runs backwards in time, and above the turning point the line does not pass at all. This configuration was considered by Stueckelberg to represent pair annihilation. To describe such a curve, parametrization by the variable t is ineffective, since the trajectory is not single valued. He therefore introduced a parametric description, with parameter τ , along the world line. Hence one branch of the curve is generated by motion in the positive sense of t as a function of increasing τ , and the other branch by motion in the negative sense of t.The motion, in space-time, of the point generating the world line, which we shall call an event (and has properties of space-time position and energy momentum), is governed in the classical case by the Hamiltonian equations in space-timewhere x µ = (t, x), p µ = (E, p) [we take c = 1 and g µν = (−1, 1, 1, 1)] and the evolution generator K is a function of the canonical variables x µ , p µ . For the special case of free motion,where M is an intrinsic parameter assigned to the generic event, and hence

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