Derivation of a Hamiltonian for photon decay in a cavity

We present a rigorous first principles derivation of the heuristic approach behind most of cavity QED: the quantization of the field in a high-Q cavit y as if the mirrors were perfect with losses modelled by a coupling with th e outside. We derive the Hamiltonian describing this coupling, thereby allo wing us to avoid the coarse-graining approximation implicit in the usual La ngevin or master equation treatments of cavity damping. We obtain an explic it expression for the coupling constant that describes the non-Markovian na ture of the reservoir in the regime beyond the validity of the coarse-grain ing approximation.