# Path integral defines a vector in Hilbert space

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

$$|\Psi \rangle = \int_{\phi(t=0)=\phi_1} d\phi \, e^{s[\phi]}$$

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

$$\langle \phi_2|\Psi\rangle= \int_{\phi(t=0)=\phi_1}^{\phi(t=\beta)=\phi_2} d\phi \, e^{s[\phi]}$$

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 $\mathcal H_{\partial M}$.

So the question could be: are there any nice books or lectures (besides the ones already mentioned) that discuss this basic principle?

• are you perhaps referring to the state-operator map of CFT's? – AccidentalFourierTransform Mar 7 '17 at 20:07
• @AccidentalFourierTransform I don't think the theory needs to be conformal to talk about the above. – jj_p Mar 7 '17 at 20:15
• See these notes by Tom Hartman for some nice discussion of how this is applied in practice by physicists. If by "made precise" you mean has it been turned into mathematics, generally speaking only for topological qft. See e.g. Dijkraaf's Les Houches lectures for a nice introduction along these lines. – Elliot Schneider Mar 7 '17 at 21:11
• @user81003 Thanks, this points in the right direction. – jj_p Mar 7 '17 at 21:30