Operator-state correspondence in CFT: computing operator for given state

Question

In 2D CFT there is a bijection between states and operators. In one direction it is easy: if $\phi(z)$ is a primary field then $$|\phi \rangle:=\lim_{z \to 0} \phi(z)|0\rangle$$ is a highest weight state. Now, suppose that a highest weight vector $|\phi\rangle$ is given, how one can compute the corresponding operator $\phi(z)$? In other words, how to compute action of this operator on any state?

The Hilbert space is obtained by acting raising operators $L_n$, $n<0$ on highest weight states and we looking for a primary operator that satisfy $$ [L_n,\phi(z)]=z^n(z\partial +(n+1)h) \phi(z), $$ thus, it is enough to determine action on all highest weight states.

We also know action of such operator on the vacuum $$ \phi(z) |0\rangle = e^{z L_{-1}} |\phi\rangle, $$ and therefore we know the action of $\phi(z)$ on all descendants of the vacuum.

How $\phi(z)$ acts on other highest weight states in the Hilbert space?

Answer 1

Suppose $\left|\psi\right\rangle =\psi\left(0\right)\left|0\right\rangle $ is another primary state, created by the primary operator $\psi\left(z\right)$. Now you want to calculate

\begin{align*} \phi\left(z\right)\left|\psi\right\rangle & =\phi\left(z\right)\psi\left(0\right)\left|0\right\rangle \\ & =\sum_{p}C_{\phi\psi p}O_{p}\left(0\right)\left|0\right\rangle \end{align*}

where in the second line I expand $\phi\left(z\right)\psi\left(0\right)$ at $0$, which is called the operator product expansion. $O_{p}\left(0\right)$ represents the contribution to the OPE from a primary operator and its decedents. $C_{\phi\psi p}$ are the OPE coefficients.