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- Why is the inverse transformation used for derivative and the position, while for a vector field the normal transformation is used?

I'm currently watching lecture videos on QFT by David Tong. He is going over lorentz invariance and classical field theory. In his lecture notes he has,

$$(\partial_\mu\phi)(x) \rightarrow (\Lambda^{-1})^\nu_\mu(\partial_\nu \phi)(y)$$, where ##y = \Lambda^{-1}x##.

He mentions he uses active transformation. So I understand the inverse in the y. For the derivative I tried

$$\partial'_\mu \phi=(\partial_\nu \phi )\partial'_\mu (x^\nu)$$

So then I need to show that ##\partial'_\mu (x^\nu) = (\Lambda^{-1})^\nu_\mu##.

$$\partial'_\mu (x^\nu)=\partial'_\mu (\Lambda^\nu_\sigma x'^\sigma) = \Lambda^\nu_\sigma \delta^\sigma_\mu = \Lambda^\nu_\mu$$.

This is how I thought to proceed, but clearly there should be a ##\Lambda^{-1}##, but I thought it would be that ##x' = \Lambda^{-1} x##. Why is it that although it is an active transformation ##x' = \Lambda x## instead? Did I make a mistake somewhere else? And if it is that the transformation should not be the inverse, why is it that the x transforms to y, which has the inverse?

Also, for a vector field he makes the claim that $$A^\mu \rightarrow A'^\mu = \Lambda^\mu_\nu A_\nu(\Lambda^{-1}x)$$ Why is it not the inverse transformation on the A but an inverse on x?

Thank you in advance.

$$(\partial_\mu\phi)(x) \rightarrow (\Lambda^{-1})^\nu_\mu(\partial_\nu \phi)(y)$$, where ##y = \Lambda^{-1}x##.

He mentions he uses active transformation. So I understand the inverse in the y. For the derivative I tried

$$\partial'_\mu \phi=(\partial_\nu \phi )\partial'_\mu (x^\nu)$$

So then I need to show that ##\partial'_\mu (x^\nu) = (\Lambda^{-1})^\nu_\mu##.

$$\partial'_\mu (x^\nu)=\partial'_\mu (\Lambda^\nu_\sigma x'^\sigma) = \Lambda^\nu_\sigma \delta^\sigma_\mu = \Lambda^\nu_\mu$$.

This is how I thought to proceed, but clearly there should be a ##\Lambda^{-1}##, but I thought it would be that ##x' = \Lambda^{-1} x##. Why is it that although it is an active transformation ##x' = \Lambda x## instead? Did I make a mistake somewhere else? And if it is that the transformation should not be the inverse, why is it that the x transforms to y, which has the inverse?

Also, for a vector field he makes the claim that $$A^\mu \rightarrow A'^\mu = \Lambda^\mu_\nu A_\nu(\Lambda^{-1}x)$$ Why is it not the inverse transformation on the A but an inverse on x?

Thank you in advance.