What is the connection between Hilbert Space and path integrals?

I have looked at this. Which seems to give more of a clue.

The ground state can be written as:

$$<0|\phi> = \int\limits^{\eta_0=\phi}e^{i S[\eta]} D\eta $$

The transition function can be written as:

$$<\phi|U(t,t')|\psi> = \int\limits^{\eta_t=\phi}_{\eta_{t'}=\psi}e^{i S[\eta]} D\eta $$

So:

$$D(x-y) = <0|\phi_0(x)\phi_t(y)|0> = <0|\phi_t>\phi_t(x)<\phi_t|U(t,t')|\phi_{t'}>\phi(y)<\phi|0> $$

$$= \int \left(\int\limits^{\eta_t=\phi_t}e^{i S[\eta]} D\eta \phi_t(x) \int\limits^{\eta_t=\phi_t}_{\eta_{t'}=\phi_{t'}}e^{i S[\eta]} D\eta \phi_{t'}(y)\int\limits_{\psi_t=\phi_t}e^{i S[\eta]} D\eta \right) D\phi D\psi $$

$$= \int \phi_t(x)\phi_{t'}(y) e^{iS[\phi]} D\phi$$

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