Dynamics of Dollard asymptotic variables. Asymptotic fields in Coulomb scattering

Abstract

Generalizing Dollard's strategy, we investigate the structure of the scattering theory associated to any large time reference dynamics $U_D(t)$ allowing for the existence of M{\o}ller operators. We show that (for each scattering channel) $U_D(t)$ uniquely identifies, for $t \to \pm \infty$, {\em asymptotic dynamics} $U_\pm(t)$; they are unitary {\em groups} acting on the scattering spaces, satisfy the M{\o}ller interpolation formulas and are interpolated by the $S$-matrix. In view of the application to field theory models, we extend the result to the adiabatic procedure. In the Heisenberg picture, asymptotic variables are obtained as LSZ-like limits of Heisenberg variables; their time evolution is induced by $U_\pm(t)$, which replace the usual free asymptotic dynamics. On the asymptotic states, (for each channel) the Hamiltonian can by written in terms of the asymptotic variables as $H = H_\pm (q_{out/in}, p_{out/in})$, $ H_\pm (q,p) $ the generator of the asymptotic dynamics. As an application, we obtain the asymptotic fields $\psi_{out/in}$ in repulsive Coulomb scattering by an LSZ modified formula; in this case, $U_\pm(t)= U_0(t)$, so that $\psi_{out/in}$ are \emph{free} canonical fields and $H = H_0(\psi_{out/in})$.