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

Klein-Gordon Equation


The Klein-Gordon equation describes how the state of a relativistic (= fast moving) quantum system without spin changes in time.

The analogous equation for systems with half-integer spin is the Dirac Equation.

If the system only moves slowly, the Klein-Gordon equation becomes the Schrödinger Equation.


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.


The most general solution of the Klein-Gordon equation is\begin{equation}\label{KGsol} \Phi(x)= \int \mathrm{d }k^3 \frac{1}{(2\pi)^3 2\omega_k} \left( a(k){\mathrm{e }}^{ -i(k x)} + a^\dagger(k) {\mathrm{e }}^{ i(kx)}\right) .\end{equation}

Graphical Summary

The diagram below shows the Klein-Gordon equation and its Lagrangian in various forms. For a more detailed explanation see Fun with Symmetry.



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 $0$ particles. For a spin-1 generalization see the Duffin-Kemmer-Petiau equation.


  • $\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,
  • $\Phi$ describes the spin $0$ field if we work in a field theory.
  • Note: $\Phi$ cannot be interpreted as a wavefunction because it is a real valued field; it is its own anti-particle like the Majorana fermion. Only in the case that it is the U(1)-charged (requires 2 independent real Klein Gordon fields that are symmetry transform into each other) is a naive wavefunction interpretation possible. Basically, you get a relativistic scalar superfluid field. Nevertheless, there are single particle wavefunctions lurking in the single real Klein-Gordon theory. But you need to use the coherent state representation to see the 1st quantized operators from the complex annihilation and creation operators. Essentially undoing the second quantization.

Contributing authors:

Eduard Sackinger
equations/klein-gordon_equation.txt · Last modified: 2023/04/02 03:24 by edi