State-operator map, and scalar fields

Question

Up so far, i have been studied state-operator correspondence, $i.e$, i have been questioned [State operator corrponding $i.e$ $S\times S$ to $R^2$ 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.

This post imported from StackExchange Physics at 2015-11-11 08:32 (UTC), posted by SE-user phy_math