I am wondering something about the tree amplitude for exchange of a spin J resonance-- the basic calculation that exhibits the famous regge behavior. I see that one way to do the calculation is simply to note that the 4-point function for spin J exchange amongst massless particles arises from an interaction term with J derivatives, and hence J powers of momenta, from which we immediately infer that the t-channel amplitude is of the form s^J/(t-M^2) + ... . However another method appears in the literature, which seems to be a partial wave expansion, noting that the spin J term of the expansion for the scattering amplitude is of the form (2J+1) f(t) P_J(cos \theta), where P_J (\cos(\theta) is the Legendre polynomial of spin J and \theta is the angle between the incoming t-channel particles. The asymptotic large s behavior can then be obtained by noting that cos(\theta) --> s/4m^2, and then that P_J (\cos (theta)) --> (cos \theta)^J --> s^J. My question is why the two methods? Why does anyone use partial waves if it is so much simpler to get the result from field theory? And this is a very basic question, but what is the validity of a partial wave expansion in a field theory context?