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Current of Complex scalar field

  1. Nov 1, 2014 #1
    I was trying to derive current for Complex Scalar Field and I ran into the following:

    So we know that the Lagrangian is:

    $$L = (\partial_\mu \phi)(\partial^\mu \phi^*) - m^2 \phi^* \phi$$
    The Lagrangian is invariant under the transformation:
    $$\phi \rightarrow e^{-i\Lambda} \phi $$ and $$\phi^* \rightarrow e^{i\Lambda} \phi^* $$

    Infinitesimal Transformation:
    $$\delta \phi = -i\Lambda \phi$$ and $$\delta \phi^* = i\Lambda \phi^*$$

    So, applying Noether's Theorem and using the Lagrangian above,

    I get

    $$J^{\mu} = \frac{\partial L}{\partial (\partial_\mu \phi} (-i\Lambda \phi) + \frac{\partial L}{\partial (\partial_\mu \phi^*} (i\Lambda \phi^*) =$$
    $$\partial ^\mu \phi^* (-i\Lambda \phi) + \partial _\mu \phi (i\Lambda \phi) = $$
    $$ i(\Lambda \phi^* \partial _\mu \phi - \Lambda \phi \partial ^\mu \phi^*)$$

    but as I googled it there is no $$\Lambda$$ in the final equation of the current. What did I do wrong?
  2. jcsd
  3. Nov 1, 2014 #2


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    You did nothing wrong. The current is simply defined as the expression proportional to the infinitesimal group parameter.
  4. Nov 1, 2014 #3
    Ah phew! Thanks!!
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