Vanessa’s reply is excellent, and Steven clearly descibed the gauge transforms of EM that I was gesturing at. That said, if you want to see some examples of fibre bundles in physics other than those in non-relativistic QM, nLab has a great article on the topic.
Also, if you know some GR, the inability to have more than local charts in general spacetimes makes clear that you do need to user bundles other than trivial ones i.e. ones of the form M×T. In my view, GR makes the importance of between global structure = fibre bundle clearer than QM, but maybe you have different intuitions.
Vanessa’s reply is excellent, and Steven clearly descibed the gauge transforms of EM that I was gesturing at. That said, if you want to see some examples of fibre bundles in physics other than those in non-relativistic QM, nLab has a great article on the topic.
Also, if you know some GR, the inability to have more than local charts in general spacetimes makes clear that you do need to user bundles other than trivial ones i.e. ones of the form M×T. In my view, GR makes the importance of between global structure = fibre bundle clearer than QM, but maybe you have different intuitions.