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I am studying CFT on my own and have some confusion regarding applications of Wick's Theorem to simplify normal ordered products to time ordered products. Wick's theorem is fairly straightforward, which says that \begin{equation} \mathcal{T}(\text{string of operators}) = :\sum_\text{contractions} \text{string of operators}: \end{equation} where $\mathcal{T}(\cdots)$ denotes time ordering and $:\cdots:$ denotes normal ordering. So far so good.

My confusion starts here. For reference, I am using this set of notes. Consider the stress energy tensor of the free boson \begin{equation} T(z) = -\frac{1}{\alpha'}:\partial X(z)\partial X(z):\tag{4.25} \end{equation} To calculate the OPE of this stress energy tensor with the field $\partial X$ we write \begin{equation} T(z) \partial X(w) = -\frac{1}{\alpha'}:\partial X(z) \partial X(z):\partial X(w)\tag{p.80} \end{equation} The LHS is a time ordered product whereas the RHS is a product of two normal ordered products. To rewrite the RHS as a time ordered product, Tong writes that we make us of contractions, i.e. "replace the pair by the propagator" \begin{equation} \partial X(z) \partial X(w) = -\frac{\alpha'}{2}\frac{1}{(z-w)^2}+\text{non-singular}.\tag{4.23} \end{equation} I don't understand how this follows from Wick's theorem, since the statement of Wick's theorem is a very different one. Can someone please explain, it will be of great help.

Qmechanic
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QFTheorist
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  • The "string of operators: that you write on the LHS of Wick's theorem is a product of normal-ordered operators. This is the same as in (p.80). – Prahar Dec 23 '23 at 09:10

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Eq. (4.23) is Wick's theorem between the (implicitly written) radial ordering ${\cal R}$ and the normal ordering $::$ $$ {\cal R}[\partial X(z) \partial X(w)] ~=~ \underbrace{-\frac{\alpha^{\prime}}{2}\frac{1}{(z-w)^2}}_{\text{contraction}}~+~ \underbrace{:\partial X(z) \partial X(w):}_{\text{non-singular terms}}\tag{4.23} $$ cf. e.g. this Phys.SE post.

Qmechanic
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