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equations:klein-gordon_equation

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$( \partial _{\mu} \partial ^{\mu}+m^2)\Phi = 0$

# Klein-Gordon Equation

## Intuitive

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

## Concrete

The Klein-Gordon equation can be derived from the Lagrangian

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

using the Euler-Lagrange equations.

## Abstract

The motto in this section is: the higher the level of abstraction, the better.

## 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.