I would like to prove the Lorentz invariance of the Klein-Gordon equation by proving the invariance of the action ##\mathcal{S} = \int d^{4}x\ \mathcal{L}_{KG}## under a Lorentz tranformation.(adsbygoogle = window.adsbygoogle || []).push({});

I would like to do this by first proving the Lorentz invariance of the ##\mathcal{L}_{KG}## and then by proving the Lorentz invariance of the spacetime volume element ##d^{4}x##.

Firstly, under a Lorentz transformation ##\Lambda##,

##\mathcal{L}_{KG} = \frac{1}{2}(\partial_{\mu}\phi)(x)(\partial^{\mu}\phi)(x)-\frac{1}{2}m^{2}\phi^{2}(x) = \frac{1}{2}\eta^{\mu\nu}(\partial_{\mu}\phi)(x)(\partial_{\nu}\phi)(x)-\frac{1}{2}m^{2}\phi^{2}(x)##

##\qquad \rightarrow \frac{1}{2}\eta^{\mu\nu}{(\Lambda^{-1})^{\rho}}_{\mu}{(\Lambda^{-1})^{\sigma}}_{\nu}(\partial_{\rho}\phi)(\Lambda^{-1}x)(\partial_{\sigma}\phi)(\Lambda^{-1}x)-\frac{1}{2}m^{2}\phi^{2}(\Lambda^{-1}x)##

##\qquad = \frac{1}{2}{(\Lambda^{-1})^{\rho}}_{\mu}{(\Lambda^{-1})^{\sigma}}_{\nu}\eta^{\nu\mu}(\partial_{\rho}\phi)(\Lambda^{-1}x)(\partial_{\sigma}\phi)(\Lambda^{-1}x)-\frac{1}{2}m^{2}\phi^{2}(\Lambda^{-1}x)##

##\qquad = \frac{1}{2}{(\Lambda^{-1})^{\rho}}_{\mu}(\Lambda^{-1})^{\sigma\mu}(\partial_{\rho}\phi)(\Lambda^{-1}x)(\partial_{\sigma}\phi)(\Lambda^{-1}x)-\frac{1}{2}m^{2}\phi^{2}(\Lambda^{-1}x)##

##\qquad = \frac{1}{2}{(\Lambda^{-1})^{\rho}}_{\mu}(\Lambda)^{\mu\sigma}(\partial_{\rho}\phi)(\Lambda^{-1}x)(\partial_{\sigma}\phi)(\Lambda^{-1}x)-\frac{1}{2}m^{2}\phi^{2}(\Lambda^{-1}x)##

##\qquad = \frac{1}{2}\eta^{\rho\sigma}(\partial_{\rho}\phi)(\Lambda^{-1}x)(\partial_{\sigma}\phi)(\Lambda^{-1}x)-\frac{1}{2}m^{2}\phi^{2}(\Lambda^{-1}x)##

##\qquad = \frac{1}{2}\eta^{\sigma\rho}(\partial_{\rho}\phi)(\Lambda^{-1}x)(\partial_{\sigma}\phi)(\Lambda^{-1}x)-\frac{1}{2}m^{2}\phi^{2}(\Lambda^{-1}x)##

##\qquad = \frac{1}{2}(\partial^{\sigma}\phi)(\Lambda^{-1}x)(\partial_{\sigma}\phi)(\Lambda^{-1}x)-\frac{1}{2}m^{2}\phi^{2}(\Lambda^{-1}x)##

Are my steps to show that the Klein-Gordon Lagrangian ##\mathcal{L}_{KG}## is Lorentz invariant all correct?

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# A Proof of Lorentz invariance of Klein-Gordon equation

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