and hence unacceptable consequences. The most obvious is that the covariant statement of the all-important [@hrup] ponderomotive energy $U_{p}$ produces a null result since$$\widetilde{U}_{p}\sim\widetilde{A}^{\mu}\widetilde{A}_{\mu}=0 \label{aj}$$ as a consequence of the self-orthogonality of the propagation 4-vector $k^{\mu}$. The resemblance of Eq. (\[ai\]) to the length-gauge representation of a quasistatic electric field suggests a tunneling model for the relativistic case [@vspopov; @heidelberg], which is inappropriate for strong laser fields. Tunneling can occur only through interference between scalar potentials, and a strong laser field is inherently vector, not scalar.

The basic defect of the potentials (\[ah\]) or (\[ai\]) is violation of the Einstein condition of the constancy of the speed of light in all Lorentz frames, despite the validity of the gauge transformation leading to those potentials. The importance of the physical situation in which this occurs is robust evidence of the significance of the proper choice of potentials, since the electric and magnetic fields attained from the unacceptable potentials (\[ah\]) or (\[ai\]) are exactly the same as those that follow from potentials that satisfy properly the condition (\[ac\]).

Gauge transformations and unitarity
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Unitary transformations in quantum physics preserve the values of physical observables. It was shown above that not all gauge 