Equivalents definitions of string field

Question

One can define the Tachyon field in SFT via State-Operator correspondence by $Phi(0)|0rangle = c(0)e^{ikX(0)}|0rangle = c_1|0;krangle$ with $|0;krangle = e^{ikX(0)}|0rangle$. 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

Nogueira · Accepted Answer

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.

Equivalents definitions of string field

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