In 1989 John Baez attempted to construct a scattering theory for Yang?Mills fields using conformal techniques, however ran into the problem that at the time the global well-posedness for merely finite energy data had not been proven. He therefore treated data with more derivatives, and provided a construction of a scattering operator corresponding to a distinguished element of the conformal group, but did not show invertibility of the wave operators. In the mid 90s Klainerman and Machedon were the first to obtain a finite energy well-posedness result in Minkowski space in the Coulomb gauge for, in the first instance, Maxwell?Klein?Gordon fields, and then for the Yang?Mills equations. Their construction relied crucially on the celebrated null structure in the nonlinearities. however even this theorem is not enough to obtain what Baez initially set out for. In an upcoming paper with J.-P. Nicolas we prove the finite energy global well-posedness of Maxwell?Klein?Gordon fields on the Einstein cylinder. I will discuss how this leads to the existence of finite energy scattering states on Minkowski space, completing one half of Baez?s goal, and mention the backward problem, which remains open.