Relativistic Harmonic Oscillator and Superconvergent Amplitudes

Current commutation relations and Fubini sum rules are saturated in a model that is a relativistic version of the three-dimensional harmonic oscillator. The model is essentially determined by the requirement that a local nonderivative relativistic coupling of the oscillator to an external electromagnetic field give rise to form factors that reduce to the usual result in the nonrelativistic limit. The relativistic form factors decrease as a power of the invariant momentum transfer, although they fall off exponentially in the limit $c\to \infty$. Vertex functions and scattering amplitudes are investigated, and it is found that (i) the Compton scattering amplitudes for current-particle interactions satisfy Fubini sum rules. (ii) All strong-interaction amplitudes are superconvergent in the Born approximation, in which an infinite/equal-mass multiplet is either exchanged or forms a set of intermediary states. (iii) Scattering amplitudes can be arranged in a hierarchy of increasing convergence (e.g., no spin flip, single spin flip, double spin flip), as suggested by de Alfaro et al. Finally, the problem of introducing the mass spectrum in the one-particle propagator is discussed.