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Proving that e^{ikx} is primary with weight (h=\hbar = \alpha k^2/4)
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[QUOTE="LCSphysicist, post: 6854911, member: 675151"] [B]Homework Statement:[/B] I can't understand the line of reasoning used by David Tong (on its lectures of CFT). [B]Relevant Equations:[/B] .' [ATTACH type="full" alt="1676395218266.png"]322263[/ATTACH] 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? [/QUOTE]
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Proving that e^{ikx} is primary with weight (h=\hbar = \alpha k^2/4)
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