Regge Trajectories and Elementary Poles

Elementary poles are introduced as poles of physical partial wave amplitudes $F_l\mathrm{}\left(s\right)\mathrm{}$ which are not present in the analytic interpolating function $F(s,\lambda )$. It is shown that $F_l\mathrm{}\left(s\right)=F(s,l)\mathrm{}$ for $l>\mathrm{}1\mathrm{}$, and that all particles with spin larger than one must be members of Regge trajectories; only bosons are considered explicitly. Additional restrictions are discussed which would make it possible to eliminate elementary poles also for spin one and zero. The possibility that the physical $s$-wave amplitude $F_0\mathrm{}\left(s\right)\mathrm{}$ is not determined by the interpolation function $F_{+}\mathrm{}(s,0\mathrm{})\mathrm{}$ could be used to avoid the ghost associated with the vacuum trajectory. The problem of branch-point trajectories is discussed briefly.