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What I don't understand is this: what does it mean for the wavefunction to have this local symmetry? How do we know that electrons / photons are described by this theory? What principal of nature says that the wavefunction has this symmetry?

I know this is three questions, but I am just trying to get a sense of what informs the choice of symmetry in these theories before I continue transforming and writing gauge invariant theories.

Thank you for any attention you may pay this question!