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From the polchinski page 238, given \begin{equation} [x_L,p_L] =[x_R,p_R]=i\tag{8.2.14} \end{equation} and the mode expansion $$\begin{equation} \begin{split} X_L(z) = x_L -i\frac{\alpha'}{2}p_L \ln z +i(\frac{\alpha'}{2})^{1/2} \sum_{m\neq 0} \frac{\alpha_m}{mz^m} \\ X_R(\bar z) = x_R -i\frac{\alpha'}{2}p_R \ln \bar z +i(\frac{\alpha'}{2})^{1/2} \sum_{m\neq 0} \frac{\tilde \alpha_m}{m \bar z^m} \\ \end{split}\tag{8.2.16} \end{equation}$$

The OPE ought to be $$\begin{equation} \begin{split} X_L(z_1) X_L(z_2) \sim -\frac{\alpha'}{2} \ln z_{12} \\ X_R(\bar z_1) X_R(\bar z_2) \sim -\frac{\alpha'}{2} \ln \bar z_{12} \\ X_L(z_1) X_R(\bar z_2) \sim 0 \end{split}\tag{8.2.17} \end{equation}$$

However, when I tried to compute $X_L(z_1) X_L(z_2)$ OPE with the mode expansion directly, the expansion $\ln(z_1)\approx \ln z_2 +\frac{z_1-z_2}{z_2}$ and $p_Lx_L$ could not be removed.

How to calculate the OPE of the $X_L(z_1)X_L(z_2)$ in the free boson theory from the mode expansion?

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    $\begingroup$ Polchinski's chapter 8 is on compactification. In what way would this calculation differ from the uncompactified case, that I am sure can be found in most textbooks on strings and/or cft. In any case, it would be useful to explain in detail where your calculations seems to be going wrong. $\endgroup$ Apr 18, 2022 at 8:34
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    $\begingroup$ @Oбжорoв Quote:" It is easy to guess what the operator products should be", the yellow book also simply wrote down the expression.(Their argument was based on the kernel of the action in the QFT case, not the mode OPE expansion.) Computing the OPE of the $X_L(z_1)X_L(z_1)$ should provide Eq. 8.2.17 but not sure how it could be done. $\endgroup$ Apr 18, 2022 at 9:05

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Let us concentrate on the very first relation in eq. (8.2.17). A more precise version of eq. (8.2.17) is$^1$ $$\begin{align} {\cal R}(X_L(z_1) X_L(z_2)) ~-~& :X_L(z_1) X_L(z_2): \cr ~=~& -\frac{\alpha'}{2} {\bf 1}~{\rm Ln}(z_1-z_2) \end{align}\tag{8.2.17'}$$ To prove eq. (8.2.17') note that we can assume w.l.o.g. that $|z_1|>|z_2|$. Then we can remove the radial order ${\cal R}$ on the LHS. Next by choosing

  1. the mode expansion (8.2.16),

  2. a pertinent notion of normal order $::$, and

  3. commutation relations (like eq. (8.2.14)),

it is in principle straightforward to verify (8.2.17'). $\Box$

For more details, see also this & this related Phys.SE posts.

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$^1$ Note that the branch cut of the complex logarithm ${\rm Ln}$ creates jumps on the RHS.

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