IR-finite Scattering theory in QFT

The traditional S-matrix amplitudes approach to scattering of massless fields in QFT is ill-defined due to IR-divergences, which are intimately tied to the classical memory effect. In the quantum theory, states with different memories live in different inequivalent Hilbert spaces. Any non-trivial scatterring process generates nontrivial memory and hence maps between in and out states living in these different Hilbert spaces leading to the IR divergences in the traditional approach. I will outline an approach to scattering theory in QFT, using the notion of algebraic states from QFT in curved spacetimes without reference to any preferred Hilbert space. In the case where no massless fields are present, this approach reporduces the traditional S-matrix formulation. For massless fields, I will consider the case of QED coupled to a fixed external classical source current. Even though the traditional approach gives IR-divergent amplitdues, the scattering process is well-defined and IR-finite in the sense of algebraic states. I discuss generalizations to QED with quantum sources and gravity.