  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.
