Consider a complex scalar field ϕ with the Lagrangian:
L=∂μϕ†∂μϕ−m2ϕ†ϕ.
Consider also two real scalar fields ϕ1 and ϕ2 with the Lagrangian:
L=12∂μϕ1∂μϕ1−12m2ϕ21+12∂μϕ2∂μϕ2−12m2ϕ22.
Are these two systems essentially the same? If not -- what is the difference?
Answer
There are some kind of silly answers here, except for QGR who correctly says they are identical. The two Lagrangians are isomorphic, the fields have just been relabeled. So anything you can do with one you can do with the other. The first has manifest U(1) global symmetry, the second manifest SO(2) but these two Lie algebras are isomorphic. If you want to gauge either global symmetry you can do it in the obvious way. You can use a complex scalar to represent a single charged field, but you could also use it to represent two real neutral fields. If you don't couple to some other fields in a way that allows you to measure the charge there is no difference.
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