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Homework Help: EOM for a complex scalar field

  1. Nov 5, 2017 #1
    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 :)
  2. jcsd
  3. Nov 5, 2017 #2


    User Avatar
    Homework Helper
    Gold Member
    2017 Award

    Looks like a sign error in one of the terms on the left. Otherwise, I think it's OK.

    You are right that the fields are not independent. They are interacting with one another.
  4. Nov 5, 2017 #3
    Ahh yeah I see the issue, thanks :) Yeah I just thought it was odd as one of the tutorial helpers said they were independent... but they were clearly incorrect. Thanks for your insight :)
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