Korybut

I know the answer In QFT when we create a particle of a certain momenta we create it everywhere, this procedure is highly non-local in coordinate space. So when we create our |in> and |out> states in QFT we need to take the particles to infinity in order to make them free and to neglect the interaction at this level. But this story is a bit different in CFT In CFT we have Vertex Operators, which also create strings but they are LOCAL. They have nothing to do with non local Fourier transform which is always present in QFT calculations. We can create a free strings at a point with certain momenta. Due to locality all of them 100% free(because there is no way to interract. Strings simply do not "feel" each other). This definition of S-matrix is by far better then in usual QFT

Thanks for response anyway As soon I discover the answer I'll post it here

Hi there! S-matrix is Path Integral with Vertex Operators inserted. I know how to compute Shapiro-Virasoro amplitude. So I don't have problems with calculations but with understanding. In this calculations formalism of 2-dimensional CFT is used. But there is no S-matrix in CFT, only correlators (N-point functions). I can treat embedding of world sheet into Minkowski space like scalar conformal fields with color indices. In this sense it is pure CFT where again no S-matrix is available. In QFT we have assymptoticaly free particles, but due to scale invariance we can't build such states in CFT. What I actually compute when I compute Polyakov's path integral with vertex operators?