I know that we have a free parameter in the relativistic potential for electromagnetism. I also know that we can introduce this as a scalar field ψ which gives this free parameter. I understand that this can be related to U(1) by mapping ψ: ---> e^iψ, which is the U(1) group. It just seems a little forced. I mean, sure you can map every scalar field into U(1) but why do it. It seems to me that the symmetry here is more analagous to some one dimensional translational symmetry since at each point in spacetime I can choose any point on the real line with no change n the corresponding equation of motion. Am I missing something?