# Wick theorem and OPE

I'm trying to work out in detail how the Wick theorem is used for constructing OPEs in CFT. One of the first things which bothers me is the difference in definitions of normal ordered product and contraction in usual QFT and in CFT. Namely, wiki and Peskin tell us that: $$X(z) Y(w) = \langle X(z) Y(w) \rangle+ \mathcal{N}\{X(z) Y(w)\} \quad.$$ On the contrary, di Francesco tells us: $$\begin{gathered} X(z) Y(w) = \langle X(z) Y(w) \rangle + \operatorname{Reg}\{X(z) Y(w) \} \\ = \underbrace{\sum \limits_{n=1}^{N} \dfrac{\{XY\}_n(w)}{(z-w)^n}}_{\text{contraction}} + \underbrace{\sum \limits_{n=-\infty}^0 \dfrac{\{XY\}_n(w)}{(z-w)^n}}_{\text{regular terms}} \quad. \end{gathered}$$

The Wick theorem for standard definitions reads as: \begin{alignedat}{9} ABCD\ldots = \mathcal{N}\{ABCD\ldots\} &+ \sum (\text{single contraction})\times (\text{normal ordered terms})\\ &+\sum (\text{double contraction})\times (\text{normal ordered terms})\\ &+\ldots\\ & +\sum (\text{products of all possible contractions}) \end{alignedat}

I would like to adopt Wick's theorem for the CFT definitions of contraction and normal ordering. Will it be correct to simply replace everywhere normal ordering with the 'regular terms' (which are obtained via the Taylor series expansion)? Then, the OPE will take form: \begin{alignedat}{9} ABCD\ldots = \operatorname{Reg}\{ABCD\ldots\} &+ \sum (\text{single contraction})\times (\text{regular terms})\\ &+\sum (\text{double contraction})\times (\text{regular terms})\\ &+\ldots\\ & +\sum (\text{products of all possible contractions}) \end{alignedat}

The whole point of this effort is to understand how to compute the non-singular terms in the OPEs (I believe, for the most singular terms one can simply use the usual Wick theorem).

• The generalized formula from Francesco is necessary for interacting fields. In appendix 6B of Francescos book ("The Generalized Wick Theorem") he states that he only wants to discuss a special case (contraction with a normal ordered product) and claims "Such a relation cannot be generalized to interacting fields". If I interpret that correctly, this means that the relation you are talking about doesn't exist in general. Dec 31, 2023 at 16:46