ar X iv : m at h / 98 03 15 3 v 1 [ m at h . FA ] 3 1 M ar 1 99 8 An Adiabatic Theorem with Soft Photons

The adiabatic theorem of Quantum Mechanics disregards the soft photons of QED. Why, then, is it so successful? To understand the role of soft photons we prove an adiabatic theorem for the ground state of a simple model Hamiltonians that allows soft photons. We show that for weak electron-photon coupling, the adiabatic time scale is close to the time scale in a theory without the quantized radiation field up to a Lamb shift. The inclusion of photons also leads to a logarithmic correction in three dimensions coming from an infrared singularity characteristic of QED.