Because <normal ordering> removes contractions within either exponential, <Wick theorem> sums all cross-contractions into
$$
\begin{aligned}
G_{k,-k}
&=\exp\left[(ik)(-ik)
\langle X(z,\bar z)X(w,\bar w)\rangle\right]\\
&=\exp\left[-\frac{k^2}{2}\log|z-w|^2\right]
=|z-w|^{-k^2}.
\end{aligned}
$$
Under $z\mapsto\lambda z$, this scales as $\lambda^{-k^2}$. Each operator has scaling dimension $\Delta=h+\widetilde h=k^2/2$, so a two-point function of two such primaries must scale as $\lambda^{-2\Delta}=\lambda^{-k^2}$. The two calculations agree.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 3 / c / ii / Solution

{wiki}

Normal ordering removes self-contractions from a composite operator. For free fields it places annihilation modes to the right of creation modes.


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