Dynamics of Dollard asymptotic variables. Asymptotic fields in Coulomb scattering

TL;DR

This paper generalizes Dollard's approach to Coulomb scattering, establishing the structure of asymptotic dynamics and fields, and providing explicit formulas for asymptotic variables and Hamiltonians in scattering theory.

Contribution

It introduces a generalized framework for asymptotic dynamics in scattering theory, including Coulomb interactions, and extends the LSZ formalism to define asymptotic fields.

Findings

Asymptotic dynamics are uniquely identified by large time reference dynamics.

Asymptotic variables are obtained as LSZ-like limits of Heisenberg variables.

In Coulomb scattering, asymptotic fields are free canonical fields with Hamiltonians expressed in terms of asymptotic variables.

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 $H = H_0(\psi_{out/in})$.