$ ( \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 [[equations:dirac_equation|]]. If the system only moves slowly, the Klein-Gordon equation becomes the [[equations:schroedinger_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 [[equations:euler_lagrange_equations|Euler-Lagrange equations]]. ---- **Solutions** 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} ---- * A nice discussion can be found in [[http://www.quantumfieldtheory.info/website_Chap03.pdf |chapter 3 of Klauber's Student Friendly QFT book]] * For an elementary derivation of the Klein-Gordon equation see Physics from Symmetry by Schwichtenberg ---- **Graphical Summary** The diagram below shows the Klein-Gordon equation and its Lagrangian in various forms. For a more detailed explanation see [[https://esackinger.wordpress.com/blog/lie-groups-and-their-representations/#klein_gordon|Fun with Symmetry]]. {{:equations:klein_gordon.jpg?nolink}} The motto in this section is: //the higher the level of abstraction, the better//. The Klein-Gordon equation is the correct equation of motion that describes free [[basic_notions:spin|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.