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I'd like to ask about the calculation of Noether current.

On page16 of David Tong's lecture note(http://www.damtp.cam.ac.uk/user/tong/qft.html), there is a topic about Noether current and Lorentz transformation.

I want to derive ##\delta \mathcal{L}##, but during my calculation, I encountered this:

\begin{align}

\delta \mathcal{L}&=\dfrac{\partial \mathcal{L}}{\partial \phi}\delta \phi+\dfrac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}\partial_\mu (\delta \phi)\\

&=\dfrac{\partial \mathcal{L}}{\partial \phi}(-\omega^\rho_{~\sigma}x^\sigma \partial_\rho \phi)+

\dfrac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}\partial_\mu

(-\omega^\rho_{~\sigma}x^\sigma \partial_\rho \phi).

\end{align}

The second term,

$$

\partial_\mu(-\omega^\rho_{~\sigma}x^\sigma \partial_\rho \phi)

$$

is a troubling term for me. Since there is ##x^\sigma## and ##\partial_\mu##, I thought I have to derivate ##x^\sigma## like ##\partial_\mu x^\sigma##. But if I do so, it doesn't match with the result in the textbook.

Am I supposed not to derivate ##x##? or am I missing something?

Thanks.

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# I Lorentz transformation and its Noether current

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