How to derive Eq. 6.21 in <PERSON>?
I'm reviewing <PERSON>'s chapter on path integrals and am having trouble understanding how he arrives at formula 6.21:
$$\left<0|0\right>{f,h}= \int \mathcal{D}q \,\mathcal{D}p \, \exp \left[i \int{-\infty}^{+\infty} dt \Big( p\dot{q}-(1-i\epsilon)H+fq+hp \Big) \right]. \qquad (6.21)$$
He gives a wordy explanation that sounded like it made sense at first before I thought about it too hard, but on further thought, I just don't see his logic. 916
He begins with the following equation, which I can accept:
$$\left<0|0\right>{f,h}=\lim\limits{t'\to-\infty}\lim\limits_{t''\to+\infty}\int dq'' dq' \psi_0^*(q'') \left"_{f,h} \,\,\psi_0(q')\,, \qquad (6.19)$$"
where the $\psi_0$'s represent the ground state of the Hamiltonian (without the $-i\epsilon$). Then he introduces the trick of changing $H$ to $(1-i\epsilon)H$, which I am also willing to accept. With this change, he shows that
$$\lim\limits_{t'\to-\infty} \left| \, q',t'\right> = \psi_0^*(q') \left|0\right>$$ $$\lim\limits_{t''\to+\infty} \left< \, q'',t''\right| = \psi_0(q'') \left<0\right|.$$
This is the end of his mathematical derivation—he then uses two paragraphs to explain how this all means that "we can be cavalier about the boundary conditions on the endpoints of the path" and jump to Eq. (6.21).
Can someone please fill in the mathematical gaps here? The issue I'm having is that the above fact seems to only bring us in a circle:
$$\left<0|0\right>{f,h}=\lim\limits{t'\to-\infty}\lim\limits_{t''\to+\infty}\int dq'' dq' \psi_0^*(q'') \left"_{f,h} \,\,\psi_0(q') \qquad (6.19)$$"
$$ =\int dq'' dq' \psi_0^(q'') \bigg[ \psi_0(q'') \left<0\right| \bigg] \bigg[ \psi_0^(q') \left|0\right> \bigg] \,\,\psi_0(q') \qquad$$
$$ =\left<0|0\right> \int dq'' \left|\psi_0(q'')\right|^2 \int dq' \left|\psi_0(q')\right|^2 \qquad\qquad\qquad\quad$$
$$=\left<0|0\right>\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,\,\,$$ 916