Equivalents definitions of string field One can define the Tachyon field in SFT via State-Operator correspondence by $\Phi(0)|0\rangle = c(0)e^{ikX(0)}|0\rangle = c_1|0;k\rangle$ with $|0;k\rangle = e^{ikX(0)}|0\rangle$. I'm trying to realize how to construct this field using another approach that makes clear the second quantized character.
To do this, another way to think of a field is by path integral, following Witten or Rastelli for example. Let be a semi-infinit open worldsheet - or even the semi-disk at UHP (as you know, diference is just a conf transf between space-time and $\mathbb{C}$). Boundary condition at ends $\sigma=0$ or $\pi$ - or at real axis part of the disk - is already fixed in order to construct the fields of the string (I guess). But boundary at the top of the worldsheet - or at the contour of the semi-disk - is not a priori fixed. The Idea is that the field can be seen as providing boundary conditions at this region, after we insert the corresponding vertex operator at infinity past - origin of UHP - and finally performing the path integral. This is the field. I cannot see the way this definition is equivalent to the latter simple one I gave on first paragraph.
Can someone help me to see this equivalence?
Thanks
 A: Let us work with the semi-disk $D$ with radial time flowing from the origin. Let us make the time $\tau=1$ be the contour boundary of the semi-disk while $\tau=0$ be the origin. Also, for simplicity let us neglect $b$ and $c$ for a while. An arbitrary first-quantized state of the string at $\tau=1$ is given by a functional $\Psi(X|_{\tau=1}(\sigma))$. A local operator will be a functional $V(X|_{\tau=0})$. Note that $\tau=0$ implies $\sigma=0$, i.e. a single point, while $\tau=1$ is a line with $\sigma$ running from $0$ to $\pi$.
Now, inserting a local operator $V(X|_{\tau=0}))$ at the origin of the semi-disk and perform the path integral will define precisely a functional of the form  $\Psi_{V}(X|_{\tau=1}(\sigma)))$, i.e. a first-quantized state at $\tau=1$.
$$
\Psi_{V}(X|_{\tau=1}(\sigma)))=\int_{D,\,X|_{\tau=1}(\sigma)}\mathcal{D}X(\tau,\sigma)\,V(X|_{\tau=0}) e^{-S}
$$
This means that any local operator insertion at the origin will define a first-quantized state at $\tau=1$ via this path integral.
The other way around is also true. Any first-quantized state at $\tau =1$ will define a local operator at the origin of the semi-disk, i.e. there is a one-to-one map between them. In order to see that just start with
$$
V_{\Psi}(X|_{\tau=0}))=\int_{\tau=1} \mathcal{D}X|_{\tau=1}(\sigma)\Psi(X|_{\tau=1}(\sigma))\int_{D/\{0\},\,X|_{\tau=1}(\sigma)} \mathcal{D}X(\tau,\sigma) e^{-S}
$$
where $D/\{0\}$ is the semi-disk with the origin removed. This will define a functional of the form $V_{\Psi}(X|_{\tau=0}))$, i.e. a local operator at the origin.
This means that there are two ways of represent a first-quantized state, by a functional of the contour boundary, or a local operator insertion at the origin.
