- #1
Ken Gallock
- 30
- 0
Hi.
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.
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.