Consider a complex scalar field $\phi$ with the Lagrangian:
$$L = \partial_\mu\phi^\dagger\partial^\mu\phi - m^2 \phi^\dagger\phi.$$
Consider also two real scalar fields $\phi_1$ and $\phi_2$ with the Lagrangian:
$$L = \frac12\partial_\mu\phi_1\partial^\mu\phi_1 - \frac12m^2 \phi_1^2
+\frac12\partial_\mu\phi_2\partial^\mu\phi_2 - \frac12m^2 \phi_2^2.$$
Are these two systems essentially the same? If not -- what is the difference?
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.
They're identical. Typically, we use complex fields if we have a $U(1)$ symmetry, or some more complicated gauge group with complex representations.
Incidentally, the same comment applies to whether we use Majorana spinors or Weyl spinors.
A complex scalar field represents a single charged particle whereas two real scalar fields may represent two independent neutral particles. The difference is easy to note while imposing physical initial, boundary and/or normalization conditions which essentially depend on what you are describing - one charged or two different neutral particles. Two independent neutral scalars do not obey a superposition principle, one cannot mix them in one field.
they are equivalent from a physics point of view and can be mapped into each other.
I think the free Lagrangian alone does not give the physical content. We can also alternatively represent $\phi = \phi_0 \exp(i \theta)$. Then we have $$ L = { 1 \over 2} \partial^\mu \phi_0 \partial_\mu \phi_0 + m^2 \phi_0^2 + {1 \over 2} \partial^\mu \theta \partial_\mu \theta $$ Here we can also ask whether we have one charged massive field or one massive neutral field and one massless one. In order to decide the field content, one must couple the scalar field with the vector field or spinor field. The complex scalar representation can have a coupling with the vector gauge field, while the two real scalar representation does not have one. Now I have another question: If we want to couple a complex scalar field with a Dirac spinor, how can we choose from the two following $$ L_1 = \bar \psi (\phi^\dagger + \phi) \psi $$ or alternatively $$ L_2 = i \bar \psi (\phi^\dagger - \phi) \psi $$ And what is the physical meaning of the above two interactions?
