Effective dynamics for particles coupled to a quantized scalar field

We consider a system of N non-relativistic spinless quantum particles (``electrons'') interacting with a quantized scalar Bose field (whose excitations we call ``photons''). We examine the case when the velocity v of the electrons is small with respect to the one of the photons, denoted by c (v/c= epsilon << 1). We show that dressed particle states exist (particles surrounded by ``virtual photons''), which, up to terms of order (v/c)^3, follow Hamiltonian dynamics. The effective N-particle Hamiltonian contains the kinetic energies of the particles and Coulomb-like pair potentials at order (v/c)^0 and the velocity dependent Darwin interaction and a mass renormalization at order (v/c)^{2}. Beyond that order the effective dynamics are expected to be dissipative. The main mathematical tool we use is adiabatic perturbation theory. However, in the present case there is no eigenvalue which is separated by a gap from the rest of the spectrum, but its role is taken by the bottom of the absolutely continuous spectrum, which is not an eigenvalue. Nevertheless we construct approximate dressed electrons subspaces, which are adiabatically invariant for the dynamics up to order (v/c)\sqrt{\ln (v/c)^{-1}}. We also give an explicit expression for the non adiabatic transitions corresponding to emission of free photons. For the radiated energy we obtain the quantum analogue of the Larmor formula of classical electrodynamics.