New wholesome thread. In the context of electromagnetism i understand that the Dirac equation, and associated Dirac lagrangian, describe single electron/positron fields but without photons, thus it doesnt describe all of electromagnetism. So it is said that this lagrangian is invariant if wavefunctions change by a phase factor, and its the same factor for the whole wavefunction described in all spacetime. So in this sense the Lagrangian is globally invariant under U(1). This was common to the "state vectors" in non-relativistic quantum mechanics, multiplying the wavefunction by a constant complex number of norm 1 doesnt change anything.
Then *leap of logic* it is said that now that symmetry has to be local, you can multiply the wavefunction by a complex number of norm 1, but it can be a different complex number on each position of spacetime. The Dirac lagrangian is not invariant under this change, so then its modified, new terms are added on to make it symmetric, and these new terms describe the electromagnetic field and the way the EM field is coupled to the electron field. Thus is the birth of QED.
I dont understand *leap of logic*. Why must incidental global symmetries also be local? What is the justification for this?