Beginners questions concerning Conformal Field Theory

I started reading about Conformal Field Theory a few weeks ago. I'm from a more mathematical background. I do know Quantum Mechanics/Classical Mechanics, but I'm not really an expert when it comes to Quantum Field Theory.

So while reading through several papers, some questions appeared and I would be more than happy if someone could answer them:

1. First I'm refering to http://www.phys.ethz.ch/~mrg/CFT.pdf (which is a german document) Chapter 3.2.1 (starting at page 29). The main topic here is the Free Boson as an example of a CFT. In this chapter he tries to proof the conformal invariance of the quantized theory. He's doing this by constructing the generators of conformal symmetry. Now here comes the first question: Why does this show conformal invariance? In the classical case you show conformal invariance by showing that the integrand of the action functional doesn't change under a conformal group action (which seems reasonable). But why does constructing generators of conformal symmetry imply conformal invariance in the quantum case?

Next I'm refering to equation (3.2.26) in the same chapter. Here he states that the equation (3.2.26) $[L_{m},\phi]=z^{m+1}\partial_{z}\phi(z,\bar{z})$ proofs that the operators L_{m} actually do implement conformal transformations of the type $L_{n}=-z^{n+1} \partial_{z}$. Why is that so? Why does this proof that the L_{m} actually implement these conformal transformations $L_{n}=-z^{n+1} \partial_{z}$? Or better: What does he mean by "implementing a transformation"? (What's the defintion if you want). The equation $[L_{m},\phi]=z^{m+1}\partial_{z}\phi(z,\bar{z})$ kind of looks like an eigenvalue-equation (which of course it isnt). But if you think that there's no Lie-Bracket. It looks like $z^{m+1}\partial_{z}$ being an "eigenvalue" of $L_{m}$ of the eigenvector $\phi$.

1. I'm now refering to the paper of David Tong: http://www.damtp.cam.ac.uk/user/tong/string/four.pdf (which is english). In Chapter 4.5.1 (on page 92) he adresses radial quantization. So he has a Theory living on a cylinder and maps this theory with the exponential map to the complex plane. So first of all, this seems a bit restrictive to me. I mean why should we only consider theories living on a cylinder as the general case? What I've heard is that that since you have a finite spatial direction on the cylinder you avoid singularities. But since I don't know Quantum Field Theory well enough I'm not so sure about this. N*ext it's not so obvious where the actual quantization appears in this process.* For me quantization means imposing certain commutation relations on operators/fields. And I don't see where this is happening here? Maybe it happens implicitly when you transform the energy momentum tensor from the cylinder to the plane by using the Schwartzian derivative??

I'd really be more than happy if someone can provide me with some detailed explanations. (because I'm really not an expert yet in this subject)

Thanks in advance!!