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The fields satisfy the wave equation. We can therefore write \begin{equation} \begin{split} \phi(x) = \int \frac{ d^3 p}{ (2\pi)^3} \frac{1}{2 \omega_{\bf p} } \left[ a({\bf p}) e^{i p \cdot x} + b^\dagger({\bf p} ) e^{- i p \cdot x} \right] \\ \phi^\dagger (x) = \int \frac{ d^3 p}{ (2\pi)^3} \frac{1}{2 \omega_{\bf p} } \left[ b({\bf p}) e^{i p \cdot x} + ...


I am also surprised that the answerer apparently claims that the part before the equals sign, without taking the real part is equal to the part after the equals sign, with taking the real part. Why is this? The statement $$\tilde{\Gamma}(t) = \int_{-\infty}^\infty e^{2\pi ift} G(f) \,df=2\,\mathrm{Re}\left(\int_{f_\text{min}}^{f_\text{max}} e^{2\pi ...

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