$$\Lambda^{\prime}=\frac{d}{d\varphi}\Lambda\left(  \varphi\right)  . \label{ae}$$ As is evident from Eq. (\[l\]), transversality is maintained by the gauge transformation (\[ad\]).

A further limitation arises if an electron is subjected to a scalar binding potential in addition to the vector potential associated with the laser field. A relativistic Hamiltonian function for a charged particle in a plane-wave field contains a term of the form$$\left(  i\hslash\partial^{\mu}-\frac{q}{c}A^{\mu}\right)  \left(
i\hslash\partial_{\mu}-\frac{q}{c}A_{\mu}\right)  . \label{ae1}$$ This occurs in the classical case, in the Klein-Gordon equation of quantum mechanics, and in the second-order Dirac equation of quantum mechanics [@feynmangm; @schweber]. The expansion of the expression in Eq. (\[ae1\]) contains the squared time part$$\left(  i\hslash\partial_{t}-\frac{q}{c}A^{0}\right)  ^{2}. \label{ae2}$$ If $A^{0}$ contains contributions from both a scalar potential and the time part of the plane-wave 4-vector potential, then executing the square in Eq. (\[ae2\]) would give a term containing the product of these two scalar potentials that is not physical; it does not occur in the reduction of relativistic equations of motion to their nonrelativistic counterparts [@hrgge]. This applies specifically to applications in AMO physics. That is, it must be true that [@hr79; @hrgge]$$A_{pw}^{0}=\phi_{pw}=0. \label{af}$$ This means that gauge freedom vanishes. Only the *radiation gauge* (also known as *Coulomb gauge*) is possible. This 