hilbert space - State-operator map, and scalar fields

Up so far, i have been studied state-operator correspondence, $i.e$, i have been questioned https://physics.stackexchange.com/q/215060/ which was wrong question. By studing Ginsparg's applied conformal field theory now I become familiar with the concept of operator state map. Which indicates that between the state in $R\times S^1$, cylinder and operator in $R^2$, plane, there is a one-to-one map. $i.e$, following conformal map we can make one to one map between them. \begin{align} \xi = t+ix, \quad z = \exp[\xi]=\exp[t+ix] \end{align} here $\xi$ is a cylinder's complex coordinate, and $z$ is a plane's complex coordinate.

Now i am curious about the field between them. For example, for scalar field $\phi(t,x)$ in cylinder after conformal map how this changes in plane? $i.e$ From the conformal map, combination of $t,x$ maps to specific value of $z$, and scalar field is dependent of $t,x$ thus it should be function of $z$ in the other side. I want to know how this works in detail.

Answer

Under conformal mapping z=>w(z) and $\bar{z}$=>$\bar{w}(\bar{z})$ a field of conformal dimension(h,$\bar{h}$) transforms as $\tilde{\phi}(w,\bar{w})=(\frac{\partial w}{\partial{z}})^{-h}(\frac{\partial \bar{w}}{\partial\bar{z}})^{-\bar{h}}\phi(z,\bar{z})$..