alternative potentials. The next example is quite different in that the improper choice of potentials is very attractive to a laser-physics community that is accustomed to the dipole approximation. The requisite propagation property never appears within the dipole approximation, and its violation is thereby invisible.

A preliminary step is to introduce the units and conventions employed in this article, and to add some general remarks about terminology.

Units and conventions
---------------------

Gaussian units are employed for all electromagnetic quantities. The expressions for the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ in terms of the scalar potential $\phi$ and the 3-vector potential $\mathbf{A}$ are$$\mathbf{E}=-\mathbf{\nabla}\phi\mathbf{-}\frac{1}{c}\partial_{t}\mathbf{A},\qquad\mathbf{B}=\mathbf{\nabla\times A.} \label{a}$$ A gauge transformation generated by the scalar function $\Lambda$ is$$\widetilde{\phi}=\phi+\frac{1}{c}\partial_{t}\Lambda,\qquad\widetilde
{\mathbf{A}}=\mathbf{A-\nabla}\Lambda, \label{b}$$ where $\Lambda$ must satisfy the homogeneous wave equation$$\left(  \frac{1}{c^{2}}\partial_{t}^{2}-\mathbf{\nabla}^{2}\right)
\Lambda=\partial^{\mu}\partial_{\mu}\Lambda=0. \label{c}$$ Relativistic quantities are expressed with the time-favoring Minkowski metric, with the signature $\left(  +---\right)  $, where the scalar product of two 4-vectors $a^{\mu}$ and $b^{\mu}$ is$$a\cdot b=a^{\mu}b_{\mu}=a^{0}b^{0}-\mathbf{a\cdot b}. \label{e}$$ The 4-vector potential $A^{\mu}$ incorporates the scalar and 3-vector potentials as$$A^{\mu}:\left(  \phi,\mathbf{A}\right)  . \label{g}$$ In 4-vector notation, the two gauge transformation expressions in Eq. (\[b\]) become the single expression$$\widetilde{A}^{\mu}=A^{\mu}+\partial^{\mu}\Lambda. \label{h}$$ Both the initial and gauge-transformed 4-vector potentials must satisfy the Lorenz condition$$\partial^{\mu}A_{\mu}=0,\qquad\partial^{\mu}\widetilde{A}_{\mu}=0. \label{j}$$ 