I'm having a little trouble with correlation functions wick theorem and ordering in the context of OPE and CFT, for string theory.
(1) My first question, the propagator is:
$$<X(z) X(w)> = \frac{\alpha}{2} \ln(z-w).$$
In the context of primary operators it's easy to see that $X$ it's not a good conformal field. But $\partial X$ yes, so I need to get:
$$<\partial X(z) \partial X(w) >$$ which I can get from the propagator of $X$ by taking two derivatives, if I take the first one:
$$\partial < X(z) X(w) > = <\partial X(z) X(w) > + <X(z) \partial X(w)>$$
But this seem to get the wrong result. So I guess that the derivative is:
$$\partial <X(z) X(w) > = <\partial X(z) X(w) >$$
If I want to take the second derivative the result seems to be:
$$\partial <\partial X(z) X(w)> = <\partial X(z) \partial X(w). $$
But I don't understand why I should want that derivative and not:
$$\partial <\partial X(z) X(w)> = <\partial^2 X(z) X(w)>.$$
(2) Regarding normal ordering and Wick's theorem, I have the following definition of normal ordering:
$$T = \frac{-1}{\alpha} :\partial X \partial X: = \frac{-1}{\alpha} \lim_{z \to w} (\partial X(z) \partial X(w) – <\partial X(z)\partial X(w)>)$$
And the condition:
$$<T> = 0$$
But what happens if I want to compute this:
$$T(z) T(w) = \frac{1}{\alpha^2} : \partial X(z) \partial X(z) : :\partial X(w) \partial X(w): $$
What's the meaning of product of normal ordered operators?
Best Answer
You want to take the derivative with respect to both z and w.
Take $${X^\mu }\left( z \right){X^\nu }\left( w \right) \sim - {1 \over 4}{\eta ^{\mu \nu }}\ln \left( {z - w} \right)$$
and use the following derivative $${{{\partial ^2}} \over {\partial w\partial z}}\left[ {{X^\mu }\left( z \right){X^\nu }\left( w \right)} \right] = {\partial \over {\partial w}}\left[ {{X^\nu }\left( w \right){\partial \over {\partial z}}{X^\mu }\left( z \right)} \right] = {\partial \over {\partial z}}{X^\mu }\left( z \right){\partial \over {\partial w}}{X^\nu }\left( w \right)$$
If we do the same thing to the OPE... $${{{\partial ^2}} \over {\partial w\partial z}}\left[ { - {1 \over 4}{\eta ^{\mu \nu }}\ln \left( {z - w} \right)} \right] = {\partial \over {\partial w}}\left[ { - {1 \over 4}{\eta ^{\mu \nu }}{1 \over {z - w}}} \right] = - {1 \over 4}{\eta ^{\mu \nu }}{1 \over {{{\left( {z - w} \right)}^2}}}$$
Which gives us the correct result $$\partial {X^\mu }\left( z \right)\partial {X^\nu }\left( w \right) \sim - {1 \over 4}{\eta ^{\mu \nu }}{1 \over {{{\left( {z - w} \right)}^2}}}$$
This result can be verified using the relevant mode expansions. See Ex. 3.1 BBS ST & MT. http://www.nucleares.unam.mx/~alberto/apuntes/bbs.pdf
As far as the product of the normal ordered operators goes... EDIT: See user2309840's answer, better than what I had written.