= Complex scalar mode inversion
{title2=$a(p)=\int d^3x\,e^{ip\cdot x}(E_p\phi+i\dot\phi)$}
With the invariant measure $d\Pi_p=d^3p/[(2\pi)^3 2E_p]$, a free <complex scalar field> has particle and <antiparticle> modes. At any fixed time the displayed inverse and $b^\dagger(p)=\int d^3x\,e^{-ip\cdot x}(E_p\phi-i\dot\phi)$ project onto its positive- and negative-frequency components. The equal-time <canonical commutation relations> then give $[a(p),a^\dagger(q)]=[b(p),b^\dagger(q)]=(2\pi)^3 2E_p\delta^3(\mathbf p-\mathbf q)$; cross terms vanish because their <coefficient> contains $E_p-E_q$ on the <momentum> delta function.
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