Gauge invariant resolution of the A⃗ · p⃗ versus r⃗ · E⃗ controversy

We delve into the problem of the choice of the interaction Hamiltonian in quantum transitions by using the gauge invariant formalism. We demonstrate that the gauge invariant transition amplitudes for the bare-bare transitions, which is the case in the majority of experiments, are identical to the conventional results with the velocity form interaction Hamiltonian. Under the electric dipole approximation, the gauge invariant formalism derives the modified length form interaction Hamiltonian in any gauge. We propose a simple experiment that one can perform to verify our prediction. On the other hand, the gauge invariant transition amplitudes for the dressed-dressed transitions, which is the case in the Lamb shift experiment, are identical to the conventional results with the length form interaction Hamiltonian. There is no preferred gauge in the calculations of transition amplitudes. The present investigation resolves the long-standing A ⃗ · p ⃗ versus r ⃗ · E ⃗ controversy and clarifies the prevailing misconception in the literature.