I am taking a course on classical field theory, and am not entirely sure as to what motivates the for of the Lagrangian density for a complex scalar field. In my lecture notes, this is first introduced in the context of the Klein-Gordon equation, and the following statements are made:
$\psi$ can be decomposed into $\psi = \psi_1 +i\psi_2$ where $\psi_1$ and $\psi_2$ are independent. Thus $\psi$ and $\psi^*$ are independent.
The Lagrangian density $L=L[\psi_1]+L[\psi_2]$ can be written $L=\frac{\partial \psi^*}{\partial t}\frac{\partial \psi}{\partial t}-\frac{\partial \psi^*}{\partial x}\frac{\partial \psi}{\partial x} - m^2\psi^*\psi$.
Now I am okay with the consistency of point 2 with itself. I.e. I see that The form of the Klein-Gordan Lagrangian density given does indeed give $L=L[\psi_1]+L[\psi_2]$, and conversely writing the Klein Gordon Lagrangian as $L=L[\psi_1]+L[\psi_2]=L[\psi + \psi^*]+L[\psi-\psi^*]$ gives the explicit form form.
However I am not at all sure why the Lagrangian density is made to be additive in $\psi_1$ and $\psi_2$. I do not see the motivation for this, other than that it reproduces the Euler-Lagrange equations for each of the independent scalar fields separately. Is there any better explanation?
