Magnusian: relating the eikonal phase, the on-shell action, and the scattering generator

A bstract Two fundamentally distinct types of quantities are both called “eikonal” in present amplitudes literature. The unitarity of the S-matrix ensures it can be written as the exponential of a Hermitian operator. The eikonal generator or Magnusian , which is the classical limit of the expectation value of that operator, generates all scattering observables. The leading order classical behavior of the phase of an S-matrix element is called the classical eikonal phase, and it coincides with a classical on-shell action. We demonstrate that the eikonal generator (Magnusian) and the eikonal phase (classical on-shell action) are inequivalent and find the exact general relationship between them. That relationship explains the special case of integrable scattering in which the two do coincide up to a Legendre transformation and explains why such a correspondence fails in general when spin or radiation are included.