= Klein-Gordon plane-wave normalization
{c}
{title2=$\psi_{\mathbf p}=e^{-ip^0t+i\mathbf p\cdot\mathbf x}/[(2\pi)^{3/2}\sqrt{2p^0}]$}
For $p^0=\sqrt{\mathbf p^2+m^2}>0$ in <Minkowski spacetime>, these modes obey $i\partial_t\psi_{\mathbf p}=p^0\psi_{\mathbf p}$ and have <Klein-Gordon inner product> $(\psi_{\mathbf p},\psi_{\mathbf q})=\delta^3(\mathbf p-\mathbf q)$. Their conjugates have the opposite norm and are orthogonal to the positive-frequency modes. Plane waves are a generalized, delta-normalized basis; normalizable wave packets have positive norm $\int |a(\mathbf p)|^2d^3p$. Completeness follows by applying the spatial <Fourier transform> to the <Klein-Gordon equation>, which gives the two branches $e^{\mp ip^0t}$ for each momentum.
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