This question is more a request for explanations. I'm reading now the Di Francesco book in attempt to understand how the free field representations of 2d CFTs are constructed. The first steps in constructing the Wakimoto representation seem natural to me:
What authors say later about $\beta_0$ and $\gamma_0$ does not surprise me as well - indeed, it seems plausible to replace them with Laurent polynomials in order for their zero modes to obey the original Lie algebra, while the polynomials themselves - the corresponding vertex algebra.
However, the part about $p$ and $q$ confuses me a lot - I would never guess these steps myself:
Why do we need the vacuum of the Fock space to be the eigenvector of the operator $p$? Why don't we just start with some $|0\rangle$?
And why does $p$ have to take its values in the weight lattice of $su(2)$?? (confuses me most...)
Basically, why do we need the operators / fields $p$ and $q$ in order to construct the representation? Why not just $\beta$ and $\gamma$, while keeping $j$ a number?