- #1

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- Homework Statement
- I can't understand the line of reasoning used by David Tong (on its lectures of CFT).

- Relevant Equations
- .'

Where

##:## really means normal ordered, in the sense that ##:A(w)B(z): = \lim_{w \to z} \left ( A(w)B(z) - \langle A(w)B(z) \rangle \right )##

##\partial X(z) = \frac{\partial X(z)}{\partial z}##

How do we go form the first line to the second one?? I am not understanding it!

it seems to me that we start with

$$\partial X(z) : X(w)^n : = \partial X(z) : X(w)^{n-1} X(w) :$$

Then, for some reason

$$\partial X(z) : X(w)^{n-1} X(w) : \rightarrow n X(w)^{n-1} :\partial X(z) X(w): $$

Since

$$: \partial X(z) X(w) = \frac{-\alpha'}{2 (z-w)} $$

We got the answer, but how?