# Lagrangian density for a complex scalar field

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:

1. $$\psi$$ can be decomposed into $$\psi = \psi_1 +i\psi_2$$ where $$\psi_1$$ and $$\psi_2$$ are independent. Thus $$\psi$$ and $$\psi*$$ are independent.

2. 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?

• – Qmechanic Nov 10 '18 at 21:25
• Can you clarify your question? I'm not really sure what you're asking. – J. Murray Nov 10 '18 at 21:56