An infrared-finite algorithm for Rayleigh scattering amplitudes, and Bohr's frequency condition (Q2642290)

The problem of atoms interacting with the quantized electromagnetic field is reconsidered here for the purpose of finding an infrared-finite algorithm for computing \(S\)-matrix amplitudes. To simplify the analysis the authors study a nonrelativistic spinless electron bound to a static proton and interacting with the second-quantized transverse photon field according to the rules of QED. Also, an ultraviolet cutoff is imposed. The known infrared behavior prevents naive expansions of amplitudes into powers of \(\alpha\), the fine structure constant since there are always terms containing \(\log(1/\alpha)\). Still, it is shown here that there exists another mathematically controlled expansion such that the remainder term is of the order \(\alpha^N\) where \(N\) may be arbitrarily large. Also, Bohr's frequency condition is recovered in the following sense. When the exited atom relaxes to the groundstate, the transition amplitude differs significantly from zero only for photon energies close to the energy difference of the two atomic states.