We consider a system of N nonrelativistic particles of spin 1/2 interacting with the quantized Maxwell field (mass zero and spin one) in the limit when the particles have a small velocity. Two ways to implement the limit are considered: c -> infinity with the velocity upsilon of the particles fixed, the case for which rigorous results have already been discussed in the literature, and upsilon -> 0 with c fixed. The second case can be rephrased as the limit of heavy particles, m(j) -> epsilon(-2) m(j), observed over a long time, t -> epsilon(-1)t, epsilon -> 0(+), with kinetic energy E-kin = O(1). Focusing on the second approach we construct subspaces which are invariant for the dynamics up to terms of order epsilon root log(epsilon(-1)) and describe effective dynamics, for the particles only, inside them. At the lowest order the particles interact through Coulomb potentials. At the second one, epsilon(2), the mass gets a correction of electromagnetic origin and a velocity dependent interaction, the Darwin term, appears. Moreover, we calculate the radiated piece of the wave function, i.e., the piece which leaks out of the almost invariant subspaces and calculate the corresponding radiated energy.
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