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Résumé : Any nontrivial scattering with massless fields in four spacetime dimensions will generically produce an out-state with memory. Scattering with any massless fields violates the standard assumption of asymptotic completeness—that all “in” and “out” states lie in the standard (zero-memory) Fock space—and therefore leads to infrared divergences in the standard S -matrix amplitudes. In this paper, we define an infrared finite scattering theory which assumes only (1) the existence of in-/out-algebras and (2) that Heisenberg evolution is an automorphism of these algebras. The resulting “superscattering” map $ allows for transitions between different in/out memory states and agrees with the standard S matrix when it is defined. We construct $ amplitudes by defining (3) a “generalized asymptotic completeness” which accommodates states with memory in the space of asymptotic states and (4) a complete basis of improper states that generalize the usual n -particle momentum basis to account for states with memory. Using only general properties of $ , we prove an analog of the Weinberg soft theorems in quantum gravity and QED which imply that all $ amplitudes are well defined in the infrared. We comment on how one must generalize this framework to consider $ amplitudes for theories with collinear divergences (e.g., massless QED and Yang-Mills theories). Published by the American Physical Society 2024