It is often stated that the path integral defines a state in the Hilbert space of the theory. I've seen this in low dimensional examples, with specific boundary conditions (for example paragraph 9.2 or last para of page 4, but it should be valid more in general. Is there a good reference discussing this for a generic QFT and with examples?
I understand that loosely speaking one can think of the path integral with one set of boundary conditions and an 'open cut' as a quantum state, namely
$$ |\phi_1 \rangle = \int_{\phi(t=0)=\phi_1} d\phi \, e^{s[\phi]}$$
which 'looks like' a functional of the second boundary condition at $\phi(t=\beta)$, in the sense that it turns field data $\phi_2$ into complex numbers
$$ \langle \phi_2|\phi_1\rangle= \int_{\phi(t=0)=\phi_1}^{\phi(t=\beta)=\phi_2} d\phi \, e^{s[\phi]}$$
but can this be made more precise? or is it just formal?
Edit1: The general principle is that for any QFT (not necessarily conformally invariant) performing the path integral on $M$ with varying boundary conditions on $\partial M$, one gets a functional of the boundary values, and thus a vector in $\mathcal H_{\partial M}$.
