Gauge invariance in non–relativistic electrodynamics

Abstract

A novel Hamiltonian scheme for non–relativistic quantum electrodynamics in which the gauge arbitrariness of the field potential is kept explicit is used to study the gauge–dependence properties of various versions of perturbation theory when resonance and line–broadening effects are admitted. Time–dependent perturbation theory is shown to have severe gauge–dependence problems unless ad hoc modifications are made. A time–independent formulation of S –matrix theory is then studied. Far from resonance, the S –matrix is gauge invariant in all orders of perturbation theory due to a very precise cancellation of gauge–dependent terms which requires, among other things, complete sets of intermediate states; energy conservation also has a crucial role. However, an obvious separation of the S –matrix into a resonant (pole) and non–resonant background leads to incomplete cancellation of the gauge–dependent terms. The introduction of the Heitler damping matrix into an integral equation for the T –matrix leads to a gauge–invariant result. This provides the basis for a gauge invariant S –matrix theory of atoms and molecules interacting with electromagnetic radiation that encompasses resonance and damping effects.