The geometric phase of a relativistically covariant four dimensional harmonic oscillator

Abstract

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