So, I have a basic/general question here. I understand that, for example, the QED Langrangian has U(1) gauge symmetry. I also understand that this means (when you have written the Lagrangian with the covariant derivative) that if you transform the wavefunction ([itex]\psi \rightarrow e^{i \theta (x)} \psi[/itex]) and the covariant derivative, this Lagrangian remains invariant. 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!
I just realized that the discussions related to this may contain some of what I am looking for. I would be interested in any additional discussion about this. Thanks!