Equação de Klein-Gordon

A densidade de Lagrangiana

$$ {\cal L} = \frac{1}{2} {\partial}_{\mu}\phi\partial^{\mu}\phi - \frac{1}{2}m^2\phi^2 $$
$$
\frac{\partial {\cal L}}{\partial \phi} - \partial_\mu\frac{\partial {\cal L}}{\partial\partial_{\mu}\phi} = 0
$$
$$
\frac{\partial {\cal L}}{\partial \phi} = - m^2\phi
$$
$$
\frac{\partial {\cal L}}{\partial\partial_{\mu}\phi} = \frac{\partial}{\partial\partial_{\alpha}\phi} \left( \frac{1}{2}\partial_\mu \phi\partial^\mu \phi \right)
$$
$$
\frac{\partial {\cal L}}{\partial\partial_{\alpha}\phi} = \frac{\partial}{\partial\partial_\alpha \phi}\left( \frac{1}{2}\partial_\mu \phi g^{\mu\theta}\partial_{\theta}\phi \right)
$$
$$
\frac{\partial {\cal L}}{\partial\partial_{\alpha}\phi} = \frac{g^{\mu\theta}}{2} \frac{\partial}{\partial\partial_\alpha \phi}\left( \partial_\mu \phi \partial_\theta \phi \right)
$$
$$
= \frac{g^{\mu\theta}}{2}\left( \frac{\partial\partial_\mu \phi}{\partial\partial_\alpha\phi}\partial_\theta\phi + \partial_\mu\phi\frac{\partial\partial_\theta \phi}{\partial\partial_\alpha\phi} \right)
$$
$$
= \frac{g^{\mu\theta}}{2}\left( \delta^\mu_\alpha\partial_\theta\phi + \partial_\mu\phi\delta^\theta_\mu \right)
$$
$$
= \frac{1}{2}\left( g^{\alpha\theta}\partial_\theta\phi + \partial_\mu\phi g^{\mu\alpha} \right)
$$
$$
= \frac{1}{2}\left( \partial^\alpha\phi + \partial^\alpha\phi \right) = \partial^\alpha\phi
$$
$$
\partial_\mu\frac{\partial{\cal L}}{\partial\partial_\mu\phi} = \partial_\mu\partial^\mu\phi
$$
Substituindo na eq. de Euler-Lagrange:

$$
- m^2\phi - \partial_\mu\partial^\mu\phi = 0
$$
$$
(\partial_\mu\partial^\mu + m^2)\phi = 0
$$