Vector Potential Versus Field Intensity

This paper deals with the apparent breakdown of gauge invariance in atomic-level-width formulas. Specifically, it is shown that in the analysis of the measurement od the Lamb shift the use of the $\frac{\overrightarrow{\mathrm{p}}·\overrightarrow{\mathrm{A}}}{\left(mc\right)}$ and the $\overrightarrow{\mathrm{E}}·\overrightarrow{\mathrm{r}}$ forms of the interaction yields almost identical results for the shape of the resonance curve and equal positions for the line center. This is achieved by adding the contribution of all intermediate states to the basic three-level resonance formula. While the contribution of these intermediate states in the $\overrightarrow{\mathrm{E}}·\overrightarrow{\mathrm{r}}$ language is negligible, it is nevertheless significant when the $\overrightarrow{\mathrm{p}}·\overrightarrow{\mathrm{A}}\left(mc\right)$ form of the interaction is used. Although this particular example would indicate that the use of the $\overrightarrow{\mathrm{E}}·\overrightarrow{\mathrm{r}}$ form of the interaction is computationally simpler, it is shown that no general statement to this effect is permissible. A simple counter example is presented within the context of elastic photon scattering off atomic targets.