# Wick Order and Radial Ordering in CFT

I am not so much familiar with the computations tools of conformal field theory, and I just run into an exercise asking to demonstrate the following formula (related to the bosonic field case):

$$\cal{R}j(z_1)j(z_2)~=~\frac{1}{(z_1-z_2)^2}~+~:j(z_1)j(z_2):$$

with $j$ defined as

$$j(z)~=~\sum_k \alpha_k z^{-k-1}.$$

My question is should I start the calculation form the Wick ordered term and make the two others appear, because starting from the left side, I don't see how could I develop some calculus?

Here we will outline a strategy to prove the sought-for operator identity $$(4)$$ from the following definitions of what the commutator and the normal order of two mode operators $$\alpha_m$$ and $$\alpha_n$$ mean:

$$[\alpha_m, \alpha_n]~=~ \hbar m~\delta_{m+n}^0, \tag{1}$$ \begin{align} :\alpha_m \alpha_n:~=~&\Theta(n-m) \alpha_m \alpha_n \cr~+~& \Theta(m-n) \alpha_n \alpha_m,\end{align}\tag{2}

where $$\Theta$$ denote the Heaviside step function.

1. Note that the current $$j(z)~=~j_{-}(z) + j_{+}(z)$$ is a sum of a creation part $$j_{-}(z)$$ and an annihilation part $$j_{+}(z)$$.

2. Recall that the radial order $${\cal R}$$ is defined as \begin{align}{\cal R}(j(z)j(w)) ~=~&\Theta(|z|-|w|) j(z)j(w)\cr ~+~& \Theta(|w|-|z|)j(w)j(z).\end{align}\tag{3}

3. Rewrite the sought-for operator identity as \begin{align}\cal{R}(j(z)j(w))~-~&:j(z)j(w):\cr ~=~&\frac{\hbar}{(z-w)^2}.\end{align}\tag{4}

4. Notice that each of the three terms in eq. $$(4)$$ are invariant under $$z\leftrightarrow w$$ symmetry. So we may assume from now on that $$|z|<|w|$$.

5. Show that \begin{align}j(w)j(z)~-~&:j(z)j(w):\cr ~=~&[j_{+}(w),j_{-}(z)].\end{align}\tag{5}

6. Show (under the assumption $$|z|<|w|$$) that \begin{align}j(w)j(z)~-~~&R(j(z)j(w))\cr ~\stackrel{|z|<|w|}{=}&0.\end{align}\tag{6}

7. Subtract eq. (6) from eq. (5):

\begin{align}\cal{R}(j(z)j(w))~-~~&:j(z)j(w):\cr ~\stackrel{|z|<|w|}{=}&[j_{+}(w),j_{-}(z)].\end{align} \tag{7}

1. Evaluate rhs. of eq. (7):

\begin{align} [j_{+}(w),j_{-}(z)]~=~&\ldots\cr ~=~&\frac{\hbar}{w^2} \sum_{n=1}^{\infty}n \left(\frac{z}{w}\right)^{n-1}\cr ~=~&\ldots ~=~ \frac{\hbar}{(z-w)^2}.\end{align} \tag{8} In the last step we will use that the sum is convergent under the assumption $$|z|<|w|$$. $$\Box$$

• Thanks a lot Qmechanic, I haven't done all the steps of you are pointing out in detail yet, but I clearly that this is just an extension of the traditional Wick theorem in QFT, for 2-points function: $T(\phi_{x_1}\phi_{x_2}) = :\phi_{x_1}\phi_{x_2}: + [\phi_{x_1},\phi_{x_2}]$ I am sorry it wasn't obvious for me earlier, now I will redo the calculations in details, trying to familiarize with it better this time. Thanks again ;)
– toot
Commented Mar 25, 2012 at 14:26
• Yes, the method is similar to, e.g., my Phys.SE answer here. Commented Mar 25, 2012 at 14:54