A different and more elegant way to express Maxwell's equations by using the:instead of the:
E.g. thinking about the electric potential alone, you could set the zero anywhere, and everything would remain be the same.
The Lorentz gauge is just one such choice. It is however a very popular one, because it is also manifestly Lorentz invariant.
Alternative to the Lorentz gauge, but less used in general as it is not as nice for relativity invariance.
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The four-potential combines the electric potential and magnetic vector potential as . The electromagnetic four-potential transforms underwhile the antisymmetric field tensor is unchanged.