= Invariant-normalized scalar mode extraction
{title2=$a(\mathbf p)=i\int d^3x\,f_p^*\overleftrightarrow{\partial_t}\phi$}
For a free <real scalar field> expanded with measure $d^3p/[(2\pi)^3 2E_p]$ and $f_p=e^{-ip\cdot x}$, the displayed <Klein-Gordon inner product> extracts its <annihilation operator>. Indeed $(f_p,f_q)=(2\pi)^3 2E_p\delta^3(\mathbf p-\mathbf q)$ and $(f_p,f_q^*)=0$; spatial <Fourier inversion> removes the measure. Taking the <Hermitian adjoint> gives $a^\dagger(\mathbf p)=-i\int d^3x\,f_p\overleftrightarrow{\partial_t}\phi$. The equal-time <canonical commutation relation> then gives $[a(\mathbf p),a^\dagger(\mathbf q)]=(2\pi)^3 2E_p\delta^3(\mathbf p-\mathbf q)$.
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