= Scalar field oscillator inversion
{title2=$a_{\mathbf p}=e^{iEt}\int e^{-i\mathbf p\cdot\mathbf x}(\sqrt{E/2}\,\phi+i\pi/\sqrt{2E})\,d^3x$}
For the <real scalar field> expansion $\phi=\int d^3p\,[a_{\mathbf p}e^{-ipx}+a^\dagger_{\mathbf p}e^{ipx}]/((2\pi)^3\sqrt{2E_{\mathbf p}})$, the equal-time fields extract $a_{\mathbf p}=e^{iEt}\int d^3x\,e^{-i\mathbf p\cdot\mathbf x}[\sqrt{E/2}\,\phi+i\pi/\sqrt{2E}]$. The two mixed <canonical commutation relation> terms yield $[a_{\mathbf p},a^\dagger_{\mathbf q}]=(2\pi)^3\delta^3(\mathbf p-\mathbf q)$, while the other oscillator <commutators> vanish. This verifies the normalization of the mode expansion directly from the canonical fields.
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