a theory one could learn a number of things that have not been explained rigorously. i) The decay of excited states in atoms. This problem has been investigated in some ultraviolet cutoff models in [@BFS] and in a massive photon model in [@OY]. See also the review of Hogreve [@H]. ii) Non-relativistic QED could be a playground for truly non-perturbative calculations and it could shed light on renormalization procedures. In fact, this was the route historically taken by Kramers that led to the renormalization program of Dyson, Feynman, Schwinger and Tomonaga. iii) Last but not least, one could formulate and answer the problems of stability of bulk matter interacting with the radiation field.

It has been proved in [@F],[@LLS] that stability of non-relativistic matter (with the Pauli Hamiltonian) interacting with classical magnetic fields holds provided that the fine-structure constant, $\alpha = e^2/\hbar c$, is small enough. It is certain, that the intricacies and difficulties of this classical field model will persist and presumably magnify in QED.

The same may be expected from a relativistic QED since replacing the Pauli Hamiltonian by a Dirac operator leads to a similar requirement on $\alpha$ [@LSS]. Indeed, stability of matter in this model (the 