is the gauge in which scalar binding influences are described by scalar potentials $\phi$ and laser fields are described by 3-vector potentials $\mathbf{A}$.

Consider the gauge transformation generated by the function [@hr79]$$\Lambda=-A^{\mu}\left(  \varphi\right)  x_{\mu}. \label{ag}$$ This leads to the transformed gauge$$\widetilde{A}^{\mu}=-k^{\mu}x^{\nu}\left(  \frac{d}{d\varphi}A_{\nu}\right)  ,
\label{ah}$$ which was introduced in Ref. [@hr79] in an attempt to base the Keldysh approximation [@keldysh] on plane-wave fields rather than on quasistatic electric fields. The transformed 4-potential can also be written as [@hr79]$$\widetilde{A}^{\mu}=-\frac{k^{\mu}}{\omega/c}\mathbf{r\cdot E}\left(
\varphi\right)  , \label{ai}$$ thus suggesting a relativistic generalization of the nonrelativistic *length gauge* used by Keldysh and widely employed within the AMO community. The problem with the $\widetilde{A}^{\mu}$ of Eq. (\[ah\]) or (\[ai\]) is that it violates the symmetry (\[ac\]) required of a propagating field. Nevertheless, this $\widetilde{A}^{\mu}$ satisfies the Lorenz condition (\[j\]) and the transversality condition (\[m\]); and the generating function of Eq. (\[ag\]) satisfies the homogeneous wave equation of Eq. (\[c\]) [@hr79]. That is, all the usual requirements for a gauge transformation are met even though the transformed 4-vector potential $\widetilde{A}^{\mu}$ of Eq. (\[ah\]) or (\[ai\]) violates the symmetry required of a propagating field like a laser field.

This violation of a basic requirement for a laser field has unphysical 