Below, I paraphrase the path integral derivation of the state-operator correspondence in David Tong's notes on CFT (see pdf here). This is my interpretation of the text in that pdf, so please correct me if I'm wrong
He starts with the standard formula for time-evolution of the wave-function in the path integral formalism
$$
\psi[ \phi_f(x),t_f] = \int [d\phi_i(x)] \int\limits_{\phi(x,t_i) = \phi_i(x)}^{\phi(x,t_f) = \phi_f(x) }[d\phi(x,t)] \exp \left[ \frac{i}{\hbar } \int_{t_i}^{t_f} dt' L \right] \psi[ \phi_i(x) , t_i ]
$$
Now, we consider a radially quantized CFT, where the time direction is radial. Further, we take $t_i = 0$ in the equation above. Since this corresponds to the origin of the radial plane, the apriori function $\phi_i(x)$ reduces to a number $\phi_i$. The path integral then reduces to
$$
\psi[ \phi_f(x),t_f] = \int d\phi_i \int\limits_{\phi(0) = \phi_i}^{\phi(x,t_f) = \phi_f(x) }[d\phi(x,t)] \exp \left[ \frac{i}{\hbar } \int_{0}^{t_f} dt' L \right] \psi(\phi_i , 0)
$$
Next, he says and I quote
The only effect of the initial state is now to change the weighting of the
path integral at the point $z = 0$. But that’s exactly what we mean by a local operator
inserted at that point.
Can anyone help me understand why this is what we mean by a local operator inserted at that point? I feel like I understand the statement, in principal, but I would like a more precise description. In other words, what I would really like is an explicit construction of the operator whose insertion in a certain path integral would reproduce the equation above.
Best Answer
Inserting a local operator means multiplying the integrand of the path integral by an operator with fixed position. This way, only the value of the operator at this position contributes to the path integral. If you now assume that the operator is an insertion at the position $z=0$, which in the present context of radial quantization corresponds to the initial point in time, it simply plays the role of a weight factor. The concept is understandable from the formulae you wrote down: in the first one, you have the general form where $t_i$ is left arbitrary, and in the second one you restrict the operator to a certain position $t_i=0$, therefore "localizing" it.
Regarding a reference I can recommend you chapter two of Polchinski: it discusses insertions both in a general context and in their application to radial quantization and the operator/state correspondence.