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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?

Qmechanic
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toot
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1 Answers1

6

Here, we will outline a strategy to prove the sought-for operator identity $\eqref{eq:4}$ 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 $$\begin{align}\cal{R}(j(z)j(w))~-~&:j(z)j(w):\cr ~=~&\frac{\hbar}{(z-w)^2}.\end{align}\tag{4}\label{eq:4}$$

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

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

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

  7. Subtract eq. $\eqref{eq:6}$ from eq. $\eqref{eq:5}$: $$\begin{align}\cal{R}(j(z)j(w))~-~~&:j(z)j(w):\cr ~\stackrel{|z|<|w|}{=}&[j_{+}(w),j_{-}(z)].\end{align} \tag{7}\label{eq:7}$$

  8. Evaluate rhs. of eq. $\eqref{eq:7}$: $$\begin{align} [j_{+}(w),j_{-}(z)]~=~&\ldots\cr ~=~&\frac{\hbar}{w^2} \sum_{n=1}^{\infty}n \left(\frac{z}{w}\right)^{n-1}\cr ~=~&\ldots ~=~ \frac{\hbar}{(z-w)^2}.\end{align} \tag{8} $$ In the last step, we will use that the sum is convergent under the assumption $|z|<|w|$. $\Box$

Qmechanic
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