46 Chapter 2: Motivation Matrix of on-shell asymptotic particles, as opposed to off-shell physics. General physical arguments of causality and nonnegative probabilities implied unitarity and maximal analyticity of the S matrix. Chew and Frautschi argued for the further requirement of maximal analyticity in the angular momentum. Then the partial wave amplitudes a$_{l}$ ( s ), become analytic functions a ( l, s ) of angular momentum and center of mass energy with isolated poles, Regge poles. Regge trajectories, l = α ( s ), then give the position of these poles. Flush with hadrons from the Bevatron, $_{Ags}$, and $_{Ps}$, plots of angular momentum vs. center-of-mass energy showed a stunning linear dependence with a common slope see Fig. 2.11 for one example. It was argued from the crossing symmetry of analytically continued scattering amplitudes that the t -channel exchange of Regge poles dominated the high-energy, fixed momentum transfer, asymptotic behavior of physical amplitudes. In this way The partial wave expansion of this amplitude revealed a tower of Breit-Wigner resonances at physical values of l , integer for mesons and half-integer for baryons. α ( s ) = α (0) + α $^{′}$s, α ′ ∼ 1 . 0 (GeV) − $^{2}$; (2.10) A ( s, t ) ∼ β ( t )( s/s$_{0}$ ) α ( s $^{)}$; s → ∞ , t < 0 . (2.11) In order to fully characterize the S Matrix theory Chew proposed the boot-