The gauge symmetry of electromagnetism plus charges does not hold in a relativistic setting

Much of modern physics rests on the assumption that electromagnetism has a gauge symmetry, and that the electromagnetic field is a gauge field.

https://en.wikipedia.org/wiki/Introduction_to_gauge_theory

The very first gauge symmetry introduced in Wikipedia is the option to set the zero electric potential at any value we like.

Let us then consider the electromagnetic field with charges, say, electrons and positrons.

As long as the inertial mass of an electron does not change substantially with the potential, we can indeed fix V = 0 at any voltage we like.

However, this is not true if the potential, relative to far-away space, is significant compared to the rest mass 511 keV of the electron.

We can say that the electric potential of far-away space is a preferred electric potential. All potentials were not created equal.

The Klein-Gordon equation is derived from the energy-momentum relation

        E^2 = p^2 + m^2,

(where we have set c = 1).

The relation holds in vacuum. But it does not hold if the particle has significant potential energy.

If we in a potential write the relation as

       (E - V)^2 = p^2 + m^2,

that relation holds approximately if the potential |V| is small. It does not hold in a relativistic setting.

The "minimal coupling", which is used to introduce an electromagnetic vector potential A to the Klein-Gordon equation or the Dirac equation, assumes that |V| is small.

The error in the minimal coupling explains the Klein paradox of the Dirac equation.

https://meta-phys-thoughts.blogspot.com/2018/10/we-solved-klein-paradox.html

The minimal coupling is derived in various sources from an assumed gauge symmetry of electromagnetism. The symmetry does not hold in relativistic settings. That explains why the minimal coupling does not work reasonably with large potentials.

We have shown that relativistic electromagnetism with charges is not a gauge theory, at least in the sense which is traditionally assumed. This means that much of the standard model is based on a shaky assumption.

If the magnetic field is strong, is the minimal coupling formula wrong, in the same fashion as for the electric potential? No. The total energy of a charge is not affected by a strong magnetic field.