The infrared problem in QED: A lesson from a model with Coulomb interaction and realistic photon emission

The scattering of photons and heavy classical Coulomb interacting particles, with realistic particle-photon interaction (without particle recoil) is studied adopting the Koopman formulation for the particles. The model is translation invariant and allows for a complete control of the Dollard strategy devised by Kulish-Faddeev and Rohrlich (KFR) for QED: in the adiabatic formulation, the M{\o}ller operators exist as strong limits and interpolate between the dynamics and a non-free asymptotic dynamics, which is a unitary group; the $S$-matrix is non-trivial and exhibits the factorization of all the infrared divergences. The implications of the KFR strategy on the open questions of the LSZ asymptotic limits in QED are derived in the field theory version of the model, with the charged particles described by second quantized fields: i) asymptotic limits of the charged fields, $\Psi_{out/in}(x)$, are obtained as strong limits of modified LSZ formulas, with corrections given by a Coulomb phase operator and an exponential of the photon field; ii) free asymptotic electromagnetic fields, $B_{out/in}(x)$, are given by the massless LSZ formula, as in Buchholz approach; iii) the asymptotic field algebras are a semidirect product of the canonical algebras generated by $B_{out/in}$, $\Psi_{out/in}$; iv) on the asymptotic spaces, the Hamiltonian is the sum of the free (commuting) Hamiltonians of $B_{out/in}$, $\Psi_{out/in}$ and the same holds for the generators of the space translations.