Gauge invariance and relativistic radiative transitions

The gauge invariance of the matrix elements of electromagnetic interaction is a property that is usually taken for granted. The author describes a unified derivation in which the dependence of the matrix elements on the gauge in which the electromagnetic potentials are written is exhibited explicitly. The necessary and sufficient condition for the transition matrices for all multipoles to be gauge invariant is that the transition matrix for longitudinal photons should vanish identically. This imposes conditions on the initial and final wavefunctions which are automatically satisfied for a single-particle model but may not necessarily hold in, say, the Hartree-Fock model. For electric dipole transitions, it is shown that the Coulomb gauge leads to the dipole velocity form of the matrix element in the nonrelativistic limit and that a different choice is needed to give the dipole length form. Arguments are advanced suggesting that the dipole velocity form should be given a privileged position in approximate calculations of atomic and molecular transition probabilities.