Intuitive

Concrete

The Klein-Gordon equation can be derived from the Lagrangian

\begin{equation} \mathscr{L}= \frac{1}{2}( \partial _{\mu} \Phi \partial ^{\mu} \Phi -m^2 \Phi^2) \end{equation}

using the Euler-Lagrange equations.

• A nice discussion can be found in chapter 3 of Klauber's Student Friendly QFT book
• For an elementary derivation of the Klein-Gordon equation see Physics from Symmetry by Schwichtenberg

Abstract

Why is it interesting?

The Klein-Gordon equation is the correct equation of motion that describes free spin $1$ particles.

Definitions

• $\partial _{\mu} $ denotes the partial derivative and $\partial _{\mu} \partial ^{\mu}$ stands for a sum using the Einstein sum convention, i.e. $\partial _{\mu} \partial ^{\mu} = \partial _0 \partial^0 - \partial _1 \partial^1 -\partial _2 \partial^2 -\partial _3 \partial^3$,
• $m$ denotes the mass of the particle,
• $\Phi$ is either the wave function of the spin $0$ particle if we use the Klein-Gordon equation in a particle theory, or describes the spin $0$ field if we work in a field theory.