A question about Lorentz invariance for Klein-Gordon field

[Comanche]

Homework Statement

Hi everyone, in Peskin & Schroeder, P36, the derivative part of KG field is transformed as eqn (3.3). But why does the partial derivative itself not transform?

Homework Equations

$$\partial_{\mu} \phi (x) \rightarrow \partial_{\mu} ( \phi ( \Lambda^{-1} x) ) = ( \Lambda^{-1})^{\nu}_{\mu} (\partial_{\nu} \phi ) (\Lambda^{-1} x)$$

The Attempt at a Solution

I thought the transformation was

$$\partial_{\mu} \phi (x) \rightarrow \Lambda^{\nu}_{\mu} \partial_{\nu} ( \phi ( \Lambda^{-1} x) ) = \Lambda^{\nu}_{\mu} ( \Lambda^{-1})^{\lambda}_{\nu} (\partial_{\lambda} \phi ) (\Lambda^{-1} x) ?$$

Thank you~

 

[WithinReason]

Peskin & Schroeder are using an active transformation (not passive) which is why the transformed field is $\phi ( \Lambda^{-1} x)$ and not $\phi ( \Lambda x)$.
Only the field is transformed - not the coordinate system. This means the Lorentz transformation you introduce should not be there. I assume you put that in there because you believed the the partial derivative should transform.

Hope this helps!