I am currently studying the behavior of a scalar field $\phi$ under a Lorentz transformation $\Lambda$. However I am having trouble understanding why the following holds true:
$$\partial_{\mu}\left(\phi(\Lambda^{-1}x)\right) = (\Lambda^{-1})^{\nu}_{~~\mu}~(\partial_{\nu}\phi)\left(\Lambda^{-1}x\right).$$
This should should probably follow from the chain rule with $\Lambda$ being a linear transformation but I can't quite figure out why this is.
Best Answer
This is a simple application of the chain rule $$ \frac{\partial}{\partial x^\mu} \phi \big( \Lambda_\nu{}^\mu x^\nu \big) = \frac{\partial}{\partial x^\mu} \big( \Lambda_\rho{}^\nu x^\rho \big) \left[ \frac{ \partial }{ \partial y^\nu } \phi (y) \right]_{y^\mu = \Lambda_\nu{}^\mu x^\nu} = \Lambda_\mu{}^\nu \left[ \frac{ \partial }{ \partial y^\nu } \phi (y) \right]_{y^\mu = \Lambda_\nu{}^\mu x^\nu} ~. $$