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