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## Main Question or Discussion Point

In almost every QFT or particle textbook we learn that complex scalar fields or spinor fields (or even multiplets of spinor fields) have a phase symmetry (global gauge symmetry.) You can append to these fields an exponential with a complex phase in the Lagrangian and the dynamics remain the same. If we make the phase depend on spacetime and introduce a massless spin-1 field, we end up with local gauge symmetry or just gauge symmetry.

My question: can we also start with a pure spin-1 Lagrangian (massless or not) and just append an exponential with a complex phase to the spin-1 field? Since the dynamics are described by the square of the field tensor, I don't see how this could work. But what are the deeper reasons that complex scalar fields and spinors have phase symmetries and spin-1 fields have not? Or does it matter whether a field is complex or not?

thanks in advance for any anwers!

My question: can we also start with a pure spin-1 Lagrangian (massless or not) and just append an exponential with a complex phase to the spin-1 field? Since the dynamics are described by the square of the field tensor, I don't see how this could work. But what are the deeper reasons that complex scalar fields and spinors have phase symmetries and spin-1 fields have not? Or does it matter whether a field is complex or not?

thanks in advance for any anwers!