Gauge transformations

We can see that the scalar and vector potential are not unique, as the simultaneous transformation of the scalar and vector potentials \[\begin{align} \require{physics} & \phi'(\vb{r}, t) = \phi(\vb{r}, t) - \pdv{\chi(\vb{r}, t)}{t} \\ % & \vb{A}' (\vb{r}, t) = \vb{A}(\vb{r}, t) + \grad{\chi(\vb{r}, t)} \end{align}\] leaves the resulting electric and magnetic fields invariant: \[\begin{align} & \vb{E}'(\vb{r}, t) = \vb{E}(\vb{r}, t) \\ & \vb{B}'(\vb{r}, t) = \vb{B}(\vb{r}, t) \thinspace . \end{align}\] This simultaneous transformation is called a gauge transformation. We can therefore always choose a gauge (i.e. implicitly determining \(\chi(\vb{r}, t)\) such that a gauge-defining equation is met) so as to simplify any resulting equations.

Some common choices of gauge are Lorenz gauge and Coulomb gauge.