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It is said that : electrodynamics is a gauge theory for U(1) gauge group . what is its physical concept?

Mathematically it mean that the field is invariant under transformation under components of U(1) group, that we can show them with e[itex]^{i\theta}[/itex] and we can consider them as a phase angle . so does it mean that if we product a field with a phase angle , the physics of problem doesn't change ? if it is true so what about teleparallel gravity that consider gravity as a gauge theory for translation group ? is it mean that the physics of problem doesn't change with small translation of spacetime points ? nad if is it true so it should be a gauge group for electrodynamics too.

Thanks

Mathematically it mean that the field is invariant under transformation under components of U(1) group, that we can show them with e[itex]^{i\theta}[/itex] and we can consider them as a phase angle . so does it mean that if we product a field with a phase angle , the physics of problem doesn't change ? if it is true so what about teleparallel gravity that consider gravity as a gauge theory for translation group ? is it mean that the physics of problem doesn't change with small translation of spacetime points ? nad if is it true so it should be a gauge group for electrodynamics too.

Thanks

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