Gauge-invariant resolution of the controversy over length versus velocity forms of the interaction with electric dipole radiation

The controversy over whether to use the length or the velocity form of the interaction of electric dipole radiation with atoms is resolved on the basis of gauge invariance. When the unperturbed Hamiltonian is chosen to be the atomic Hamiltonian, the condition that the probability amplitudes are gauge invariant implies that the interaction is of the length form, $-q\overrightarrow{\mathrm{E}}·\overrightarrow{\mathrm{r}}$, not the velocity form, $-\left(\frac{q}{\mathrm{mc}}\right)\overrightarrow{\mathrm{A}}·\overrightarrow{\mathrm{p}}$. Hartree-Fock theory is examined and shown to be form invariant under local gauge transformations, i.e., gauge invariant. A comparison is made between the length and velocity forms of the interaction and oscillator strengths. If the true Hamiltonian is nonlocal, only the length form of the dipole oscillator strength is valid. If the true Hamiltonian is local, the length form of the dipole oscillator strength may be transformed to the velocity form.