- #1
PhyAmateur
- 103
- 2
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?
$$L = (\partial_\mu \phi)(\partial^\mu \phi^*) - m^2 \phi^* \phi$$
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?