Proof of Lorentz invariance of Klein-Gordon equation

[spaghetti3451]

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.

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?

 

[spaghetti3451]

There's nothing new to add to my post.

If someone checks my working and gives me a thumbs up, I will proceed to show that the spacetime volume element $d^{4}x$ is also Lorentz invariant, thus proving that the Klein-Gordon equation is Lorentz invariant (since the Klein-Gordon equation is derived from the variation of a soon-to-be-proved Lorentz invariant Klein-Gordon action).

I'm new to quantum field theory and general relativity, and so I'm practicing my technical skills and, in the process, learning the subjects.

 

[spaghetti3451]

As my query falls under the subject of classical (relativistic) field theory, I initially thought that the 'Classical Physics' subforum was better suited for this thread.

 

[Paul Colby]

Your steps look correct to me. The transformation $\phi \rightarrow \phi(\Lambda^{-1} x)$ permutes the points of $\mathbb{R}^4$ and since you're summing over them all the action is invariant. The invariance of $dx^4$ I believe follows from $\det(\Lambda)=1$.

 

[spaghetti3451]

Thanks!

Let me now prove the Lorentz invariance of the spacetime volume element $d^{4}x$.

Under a Lorentz transformation $\Lambda$,

$d^{4}x \rightarrow d^{4}x \Big|\frac{\partial x^{\mu}}{\partial x^{\mu'}}\Big|$, where $\Big|\frac{\partial x^{\mu}}{\partial x^{\mu'}}\Big|$ is the Jacobian of the Lorentz transformation.

The Jacobian of the Lorentz transformation = 1, so the spacetime volume element $d^{4}x$ is Lorentz invariant.