Lorentz invariance of KG equation

Question

I am confused with the equation 3.3 in Peskin & Schroeder:

Given a Lorentz transformation $$x^\mu\rightarrow x'^\mu=\Lambda^\mu_\nu x^\nu,$$ the field transforms as$$\phi(x)\rightarrow\phi'(x)=\phi(\Lambda^{-1}x).$$ I'm trying to reproduce the following equation: $$\partial_\mu\phi\rightarrow\partial_\mu(\phi(\Lambda^{-1}x))=(\Lambda^{-1})^\nu_{\;\;\mu}\partial_\nu\phi(\Lambda^{-1}x),$$ but Im doing:$$\partial_\mu\phi(\partial_\mu((\Lambda^{-1})^\alpha_{\;\;\beta}x^\beta))\rightarrow\partial_\mu\phi((\Lambda^{-1})^\alpha_{\;\;\mu})$$ So, what Im doing wrong?

Answer 1

The equation that you're trying to reproduce is just an application of the chain rule. I think the "$\rightarrow$" maybe confuses, but the equality on the right of the arrow is just calculus.

Looking at your equation, I'm not able to understand what you thought the starting point for this calculation should be. For sure what you have on the left side of your equation cannot be right because the function $\phi$ here takes a vector (1-index object), and you have given it some sort of two-index object with indices $\mu$ and $\alpha$. (The index $\beta$ in your expression is summed inside the argument to $\phi$. The index $\mu$ is summed in the overall expression you gave, but not entirely within the argument to $\phi$ itself.)