(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the equations of motion for the Lagrangian below:

$$ L=\partial_\mu \phi^* \partial^\mu \phi - V( \phi,\phi^* ) $$

Where :

$$ V( \phi,\phi^* )= m^2 \phi^* \phi + \lambda (\phi^* \phi)^2 $$

2. Relevant equations

Euler Lagrange equation:

$$ \partial_\mu \dfrac {\partial L} {\partial (\partial_\mu \phi)} -\dfrac {\partial L} {\partial \phi} =0 $$

3. The attempt at a solution

So I have calculated the equations of motion for each field but I'm surprised to find they're not independant of each other so i'm wondering if I've made a mistake somewhere? Here are my workings:

$$ \dfrac {\partial L} {\partial \phi} =m^2 \phi^* +2\lambda (\phi^*)^2 \phi $$

$$\dfrac {\partial L} {\partial (\partial_\mu \phi)} = \partial_\mu \phi^* $$

So then the equations of motion are:

$$\Box \phi^* -m^2 \phi^* +2\lambda (\phi^*)^2 \phi =0$$

$$\Box \phi -m^2 \phi +2\lambda (\phi)^2 \phi^* =0$$

Any suggestions would be appreciated :)

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# Homework Help: EOM for a complex scalar field

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