Regge-Pole Exchange in Classical Physics and in Nonrelativistic Quantum Mechanics

The Regge-trajectory parameter $\alpha \mathrm{}\left(t\right)\mathrm{}$ has a simple physical interpretation in classical relativistic scattering phenomena. The "Lorentz-pole parameter" $\sigma$ of Toller has the same interpretation and is equal to $\alpha \mathrm{}\left(0\right)\mathrm{}$. Regge-pole exchange has a precise analog in nonrelativistic potential scattering. The relativistic Regge-Joos representation is a decomposition of the amplitude according to the nature of the exchanged object, each term transforming irreducibly under the Poincaré group; in potential scattering this group is replaced by the Galilei group. There is a specious analog of the relativistic "daughter trajectories," which now form a continuous set. Examples are worked out in detail and show a phenomenon similar to the complicated singularity structure associated with the Pomeranchuk trajectory. In the classical case, the physical interpretation imposes the requirement that $\alpha (\alpha +1)<\mathrm{}0\mathrm{}$, which means that only unitary representations of the Poincaré group can be exchanged. Similarly, only unitary representations of the Galilei group can be exchanged in nonrelativistic potential scattering. It is tempting to speculate about the possibility that similar results can be obtained in relativistic quantum theory.