I have a question regarding equation (2.22) in Ginsparg's lecture notes on CFTs. Equation (2.22) is $$ \langle T(z) \phi_1(w_1, {\bar w}_1) \cdots \rangle = \sum_{i=1}^n \left( \frac{h_i}{(zw_i)^2} + \frac{1}{zw_i} \frac{\partial}{ \partial w_i} \right) \langle \phi_1(w_1, {\bar w}_1) \cdots \rangle $$ Here, $T(z)$ is the stress tensor of the CFT and $\phi_i$ is a primary operator of weight $(h_i,0)$ which transforms under conformal transformations as $$ \delta_\epsilon \phi_i = \left( h_i \partial \epsilon + \epsilon \partial \right) \phi_i $$ He derives (2.22) from (2.21) which reads $$ \langle \oint \frac{dz}{2\pi i} \epsilon(z) T(z)\phi_1(w_1, {\bar w}_1) \cdots \rangle = \sum_{i=1}^n \langle \phi_1(w_1, {\bar w}_1) \cdots \delta_\epsilon\phi_i(w_i, {\bar w}_i) \cdots \rangle $$ by setting $\epsilon(x) = \frac{1}{xz}$.
My question is  Is (2.22) correct?
Here are my reasons to believe that it is not 

I believe he derives (2.22) from (2.21) by setting $\epsilon(x) = \frac{1}{xz}$ in (2.21). (2.22) is then derived if the following holds $$ \langle \oint \frac{dx}{2\pi i} \frac{T(x)}{xz} \phi_1(w_1, {\bar w}_1) \cdots \rangle = \langle T(z)\phi_1(w_1, {\bar w}_1) \cdots \rangle $$ This would be true if the integrand on the LHS had only a pole at $xz$. However, it has also has poles at each $x = w_i$, but those contributions aren't considered.

I can try and derive (2.22) in a different way  namely via contractions. I start with the LHS of (2.22) and contract $T(z)$ with each $\phi_i$. Each contraction is replaced with the operator product $$ T(z) \phi_i(w_i {\bar w}_i) = \frac{h_i \phi_i(w_i {\bar w}_i) }{ ( z  w_i )^2 } + \frac{ \partial \phi_i(w_i {\bar w}_i) }{ z  w_i } + : T(z) \phi_i(w_i {\bar w}_i) : $$ Again, if I only consider the singular terms, I reproduce the RHS of (2.22). But what about $: T(z) \phi_i(w_i {\bar w}_i) :$?? In a general CFT, conformal normal ordering $:~:$ is not equivalent to creationannihilation normal ordering ${}^\circ_\circ~{}^\circ_\circ$. The latter would vanish in a correlation function, but not the former. So, I believe in general there would be extra terms on the right of (2.22).
What am I misunderstanding?
This post imported from StackExchange Physics at 20150115 14:03 (UTC), posted by SEuser Prahar