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## Homework Statement

1. Show directly that if ##\varphi(x)## satisfies the Klein-Gordon equation, then ##\varphi(\Lambda^{-1}x)## also satisfies this equation for any Lorentz transformation ##\Lambda##.

2. Show that ##\mathcal{L}_{Maxwell}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}## is invariant under the Lorentz transformation ##x \rightarrow \Lambda x##.

## Homework Equations

## The Attempt at a Solution

The Klein-Gordon equation is ##\partial^{\mu}\partial_{\mu}\varphi(x) + m^{2}\varphi(x)=0##.

Under a Lorentz tranformation ##x \rightarrow \Lambda x##, the Klein-Gordon equation becomes

##{(\Lambda^{-1})_{\rho}}^{\mu}\partial^{\rho}{(\Lambda^{-1})_{\sigma}}^{\mu}\partial^{\sigma}\varphi(\Lambda^{-1}x)+m^{2}\varphi(\Lambda^{-1}x)=0##

##\implies {(\Lambda^{-1})_{\rho}}^{\mu}{(\Lambda^{-1})_{\sigma}}^{\mu}\partial^{\rho}\partial^{\sigma}\varphi(\Lambda^{-1}x)+m^{2}\varphi(\Lambda^{-1}x)=0##

##\implies {(\Lambda^{-1})_{\rho}}^{\mu}{\Lambda^{\mu}}_{\sigma}\partial^{\rho}\partial^{\sigma}\varphi(\Lambda^{-1}x)+m^{2}\varphi(\Lambda^{-1}x)=0##

Am I correct so far?