QFT: Srednicki Eq. 7.5 I am studying QFT from Srednicki's book. I would appreciate if anyone could offer how he derived Eq. 7.5 from the textbook. He is considering the quantum harmonic oscillator and writes:
$$\langle 0 | 0 \rangle_{f} = \int Dq \exp i \int_{-\infty}^{+\infty} dt \left[(1/2)(1+i \epsilon) \dot{q}^2 - (1/2)(1-i\epsilon) \omega^2 q^2 + fq\right],\tag{7.3}$$
where $m = 1$ for notational convenience. Then, he introduces Fourier-transformed variables:
$$q(t) = \int_{-\infty}^{+\infty} \frac{dE}{2\pi} e^{-iEt}\tilde{q}(E).\tag{7.4}$$
He then says, the expression in the square brackets becomes: 
$$\frac{1}{2}\int_{-\infty}^{+\infty} \frac{dE}{2\pi}\frac{dE'}{2\pi} e^{-i(E + E')t }$$
$$\times \left[ \left(-(1+i\epsilon)EE' - (1-i\epsilon)\omega^2\right)\tilde{q}(E)\tilde{q}(E') + \tilde{f}(E)\tilde{q}(E') + \tilde{f}(E')\tilde{q}(E)\right].\tag{7.5}$$
However, he offers no derivation as to how he gets this. Perhaps, it is simple, but I cannot see it. Can someone fill in the details? 
As an example, if I compute $\dot{q}(t)$, I get:
$\dot{q}(t) = \frac{-i}{2\pi}\int_{-\infty}^{\infty} Ee^{-iEt}\tilde{q}(E) dE$,
but I don't see how squaring this gives the $\tilde{q}(E) \tilde{q}(E')$ terms.
 A: Consider the Fourier transformed variables. 
$$\tilde{q}(E) = \int dt ~ e^{i E t} q(t) $$
$$q(t) = \int \frac{dE}{2 \pi} ~ e^{-i E t} \tilde{q}(E) $$
Re-write the $dt$-integral in the $<0|0>$ expression using the above as 
$$\frac{1}{2} (1 + i \epsilon) \dot{q}^2 - \frac{1}{2} (1 - i \epsilon) \omega^2 q^2 + f q = \int \int \frac{(1+ i \epsilon)}{2} \frac{d}{dt} ~ \Big(\frac{dE}{2 \pi} e^{- i E t} \tilde{q}(E)\Big) \frac{d}{dt}\Big( \frac{dE'}{2 \pi} e^{- i E' t} \tilde{q}(E') \Big) - \frac{(1- i \epsilon) \omega^2}{2} \frac{dE}{2 \pi} e^{-i E t} \tilde{q}(E) \frac{dE'}{2 \pi} e^{- i E t'} \tilde{q}(E') + \frac{dE}{2 \pi} \frac{d E'}{2 \pi} e^{- i (E + E')t} \tilde{q}(E) \tilde{f}(E')$$
Find the common factors, perform a bit of algebra to simplify it a bit and obtain. 
$$\frac{1}{2} \int \int \frac{dE}{2 \pi} \frac{d E'}{2 \pi} e^{- i (E + E')t} \Bigg((-(1+i\epsilon)E E' - (1-i \epsilon) \omega^2) \tilde{q}(E) \tilde{q}(E') + \tilde{f}(E)\tilde{q}(E') + \tilde{q}(E) \tilde{f}(E') \Bigg)$$
Finally, calculate the action. 
Cheers!!!
Edit: The integrals are of course from $-\infty$ to $\infty$
