# Charge in field theory [duplicate]

Why is it that a real scalar field is charge neutral whereas a complex scalar field (or a spinor field like Dirac theory) describe charged fields?

Charge is a measure of how strongly a field couples to the electromagnetic field. The electromagnetic field is a gauge field generated by $U(1)$ gauge symmetry1, so to couple to it the field needs to be a non-trivial representation of the $U(1)$ group. $U(1)$ is the symmetry group of points on a circle, so we need to look for mathematical objects that have a similar structure.
The trivial representation is the singlet representation, naively speaking a "point", a real scalar field. This is a singlet, meaning that it doesn't change as you act on it with $U(1)$ operators (there is no way to tell afterwards whether the point has been rotated around itself or not2). Since it does not respond to the $U(1)$ field, a real scalar field does not couple to the electromagnetic field: it is chargeless.
The simplest non-trivial representation is a complex field. This should be clear if you imagine the complex phases $e^{i\theta}$ on the complex plane, which have the same topology as points on a circle. The complex phase does change as you act on it with $U(1)$ operators: the phase angle changes. So this couples to the electromagnetic field: it is charged.
2 Using the "value" of the field at the point to tell how much it has been rotated does not work, because it doesn't have the topology of a circle: you could tell the difference between, say $0$ and $2\pi$. A real value by itself is not a representation of the $U(1)$ group.