Proving that e^{ikx} is primary with weight (h=\hbar = \alpha k^2/4)

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In summary, the first line represents the normal ordering of the operators A and B at the point z. The second line uses the Wick theorem to expand the product of A and B in terms of creation and annihilation operators. The third line applies the operator relation to get the final result.
  • #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
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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?
 
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  • #2
You must use Wick theorem, which is the same as in ordinary QFT.
 
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