Basics: The Lagrangian in field theory was written as $$\frac{\partial \mathfrak{L}}{\partial \varphi}=\partial_\mu(\frac{\partial\mathfrak{L}}{\partial(\partial_\mu\varphi)})$$.

Question 1:

Is $\partial_\mu\varphi=(\partial_t+\partial_x)\varphi$ now a vector/tensor, or still a single variable?

Question 2:

Further, is there any difference between $\displaystyle \frac{\partial (\partial_\mu\varphi) }{\partial (\partial_\mu \varphi)}$ (which equals to $1$ I suppose) and $\displaystyle \frac{\partial (\partial^\mu\varphi) }{\partial (\partial_\mu \varphi)}$?

Question 3:

How to take derivative with respect to complex field? i.e. $\displaystyle \frac{\partial (\partial_\mu\varphi^\dagger) }{\partial (\partial_\mu \varphi)}$ , $\displaystyle \frac{\partial (\partial^\mu\varphi^\dagger) }{\partial (\partial_\mu \varphi)}$ and $\displaystyle \frac{\partial \varphi^\dagger}{\partial\varphi}$?


Correct me if I'm wrong on this, guys, and take this answer with a grain of salt until verified by someone more experienced.

For a scalar field $\phi$ in flat space,

Question 1: The derivative of a scalar field is a vector, which we can see because we pick up an index $\mu$ when we operate $\partial_\mu$ on $\phi$. This is like the gradient of a scalar field in vector calculus, which is a vector.

Question 2: $$\frac{\partial(\partial^\mu\phi)}{\partial(\partial_\mu\phi)}= \frac{\partial}{\partial(\partial_\mu\phi)}(\partial^\mu\phi)=\frac{\partial}{\partial(\partial_\mu\phi)}(g^{\mu\nu}\partial_\nu\phi)=g^{\mu\nu}\frac{\partial(\partial_\nu\phi)}{\partial(\partial_\mu\phi)}=g^{\mu\nu}\delta_\nu^\mu=g^{\mu\mu}$$

Question 3: Treat the conjugate field as independent from the field. That is, the partial derivatives you have provided as examples are all zero.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.