In bosonic string theory, energy-momentum tensor in complex coordinates is $$T(z)=-2:\partial X(z).\partial X(z): $$ In order to obtain the conformal anomaly $c$ we need to compute the OPE of $T$ with itself, i.e., $$T(z)T(w)=4:\partial X(z).\partial X(z)::\partial X(w).\partial X(w):$$ On the other hand, we know that $$\langle \partial X(z).\partial X(w)\rangle =-\frac{1}{4}\frac{1}{(z-w)^2}$$ but my main problem is that I don't know how to use Wick's theorem in expressions like above. I would really really appreciate if someone could explain me how to apply Wick's theorem in expressions like $:AB::CD:$ and $:AB::CDE:$ (where $A$,...,$E$ are field operators) or something like these or even more general in order to get the OPE of different fields. I really get stuck in the way we use Wick's thm. Unfortunately I have'nt spent QFT course yet but I have string theory this term. I would be grateful if someone could help me. I really want to learn this theorem and how to use it
Asked
Active
Viewed 70 times
