The classical Klein-Gordon equation is ##(\partial^{2}+m^{2})\varphi(t,\vec{x})=0##.(adsbygoogle = window.adsbygoogle || []).push({});

To solve this equation, we need to Fourier transform ##\varphi(t,\vec{x})## with respect to its space coordinates to obtain

##\varphi(t,\vec{x}) = \int \frac{d^{3}\vec{k}}{(2\pi)^{3}}e^{i\vec{k}\cdot{\vec{x}}}\tilde{\varphi}(t,\vec{k})##.

Plugging ##\varphi(t,\vec{x})## into Klein-Gordon equation is supposed to give us the solution

##\tilde{\phi}(t,\vec{k})=A(\vec{k})e^{-iE_{\vec{k}}t}+B(\vec{k})e^{iE_{\vec{k}}t}##.

I am stuck in getting the solution. If I plug the Fourier transform of ##\varphi(t,\vec{x})## into the Klein-Gordon equation, I get ##(-\vec{k}^{2}+m^{2})\tilde{\phi}(t,\vec{k})=0##. I'm not sure how to proceed from there onwards. Can you help you me out?

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# A Solution of the classical Klein-Gordon equation

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