The definition of the vacuum state of quantum field by path-integral In the review Entanglement entropy of black holes by Sergey Solodukhin (arXiv:1104.3712, equation 13), I see a definition of vacuum state of quantum field by path integral over half of the total Euclidean spacetime defined as $\tau\leq 0$ such that the quantum field satisfies the fixed boundary condition $\psi(\tau=0,x,z)=\psi_{0}(x,z)$ on the boundary of the half-space:
$$\Psi[\psi_{0}(x,z)]=\int_{\psi(X)|_{\tau=0}=\psi_{0}(x,z)}D\psi e^{-W[\psi]}\tag{13},$$ where $W$ is the action. 
I can't understand what does this mean. I learned that in canonical quantization, a vacuum state is defined by Hamiltonian of the quantum field, which has the lowest energy. But here in path integral. I want to know the meaning of the above definition of vacuum state, why should we define it in this way?
 A: I) Let us here phrase the problem in the context of some position operator $\hat{q}$ of QM for simplicity. The generalization to QFT can formally be achieved by replacing the position operator $\hat{q}$ with a quantum field $\hat{\psi}({\bf x})$.
We know that the overlap with Minkowski (M) signature is given as a path integral
$$K(q_f,t_f;q_i,t_i)~\equiv~\langle q_f,t_f\mid q_i,t_i\rangle
~=~\langle q_f,t_i\mid \exp\left[ -\frac{i}{\hbar}\hat{H} \Delta t \right] \mid q_i,t_i\rangle$$
$$~=~\sum_{n}\langle q_f,t_i\mid n \rangle \exp\left[ -\frac{i}{\hbar}E_n \Delta t \right]\langle n\mid q_i,t_i\rangle \tag{O}$$
$$~=~\int_{q(t_i)=q_i}^{q(t_f)=q_f} \! {\cal D}q~\exp\left[\frac{i}{\hbar}S_M[q]\right],  $$
where $\Delta t:=t_f-t_i$ and $\mid n \rangle$ is an energy basis for the Hamiltonian $\hat{H}$.
II) For fixed initial values $(q_i,t_i)$, we may identify the overlap (O) with the the wave function(al) $\Psi(q_f,t_f)$. Alternatively, for fixed final values $(q_f,t_f)$, we may identify the overlap (O) with the complex conjugate of the wave function(al) $\Psi^{\ast}(q_i,t_i)$. Solodukhin is doing the latter.
III) Next Wick rotate to Euclidean variables $S_M=iS_E$, $\tau=it$, $\tau_i=it_i$, $\tau_f=it_f$, etc. Notice that the ground state (=vacuum) contribution $E_0$ in the overlap (1) becomes completely dominant in the large time limit $ \tau_f \to \infty$, cf. comment by Meng Cheng. In that limit, the wave function(al) $\Psi^{\ast}(q_i,t_i)$ becomes the vacuum wave function(al). 
The translation to Solodukhin's formula (13) is straightforward.
