I am reading the Srednicki's quantum field theory book and stuck at some statement. In the book p.46, the author worte that :
"Now consider modifying the lagrangian of our theory by including external force acting on the particle: $$H(p,q) \to H(p,q) - f(t)q(t) -h(t)p(t), \tag{6.15}$$ where $f(t)$ and $h(t)$ are specified functions. In this case, we will write $$ \langle q'',t''|q',t'\rangle_{f,h}=\int\mathcal{D}q \mathcal{D}q \exp[ i\int_{t'}^{t''}dt(p\dot{q}-H+fq+hp)] \tag{6.16}$$ where $H$ is the original Hamiltonian."
And in the final paragraph in the p.46 he wrote:
"Suppose we are also interested in initial and final states other than position eigenstates. Then we must multiply by the wave functions for these states, and integrate. We will be interested, in particular, in the ground state as both the initial and final state. Also, we will take the limits $t'\to -\infty$ and $t''\to +\infty$. The object of our attention is then $$\langle 0|0 \rangle_{f,h}=\lim_{t'\to -\infty, t''\to +\infty}\int dq''dq' \psi^{*}_0(q'') \langle q'', t''|q',t'\rangle_{f,h} \psi_0(q'), \tag{6.19}$$
Q. Why this formula holds?
where $\psi_0(q)=\langle q|0 \rangle $ is the ground-state wave function. Eq. $(6.19)$ is a rather cumbersome formula, however. We will therefore, employ a trick to simplify it."
And intermediate argument is as follows :
I am trying to understand these argument, in particular the underlined statement. What mechanism he is teaching? I understand some of his argument as, $\lim_{t'\to -\infty} |q' ,t' \rangle =\psi_0^{*}(q')|0 \rangle $ and $\lim_{t''\to \infty} \langle q'', t''| = \psi_0(q'') \langle 0| $ with $H$ replaced by $(1-i\epsilon )H$. Am I following well?
Anyway, next, he continue to argue as follows:
"What all this means is that if we use $(1-i \epsilon)H$ instead of $H$, we can be cavalier about the boundary conditions on the endpoints of the path. Any reasonable boundary conditions will result in the ground state as both the initial and final state. Thus we have $$\langle 0 | 0\rangle_{f,h} = \int \mathcal{D}p \mathcal{D}q \exp[i \int_{-\infty}^{\infty} dt (p \dot{q}-(1- i \epsilon)H + fq + hp)]. " \tag{6.21}$$
Q. My main question is, I don't understand why this final equality $(6.21)$ is true. Why, where did the measures $dq'', dq'$ and integrands $\psi_0^{*}(q'')$, $\psi_0(q')$ in the $(6.19)$ above are disappear? I don't understand his overall argument which seems to be rather analytic.
Can anyone help?
