Classical eikonal in relativistic scattering

The classical eikonal is defined to be the generator of all scattering observables in a scattering problem in classical mechanics. It was originally introduced as the log of the quantum S-matrix in the classical limit. But its classical nature calls for a definition and computational methods independent of quantum mechanics. In this paper, we formulate a classical interaction picture which serves as the foundation of the classical eikonal. Our emphasis is on generality. In perturbation theories, both Hamiltonian deformation and symplectic deformation are considered. Particles and fields are treated on a similar footing. The causality prescription of the propagator is essentially the same for non-relativistic and relativistic kinematics. For a probe particle in electromagnetic or gravitational background, we present all order formulas for the perturbative eikonal. In the electromagnetic setting, we also illustrate how the eikonal encodes the information on radiation of external fields.