Quick Calculation 13.1

A First Course in String Theory

Verify that

$\displaystyle{\left[ \bar L_m^\perp, x_0^I \right] = - i \sqrt{\frac{\alpha'}{2}} \bar \alpha^I_m}$ ,

$\displaystyle{\left[ L_m^\perp, x_0^I \right] = - i \sqrt{\frac{\alpha'}{2}} \alpha^I_m}$ .

~~~

Equation (13.37):

$\displaystyle{\bar L_m^\perp = \frac{1}{2} \sum_{p \in \mathbf{Z}} \bar \alpha_p^J \bar \alpha_{n-p}^J}$ , $\displaystyle{~~~L_m^\perp = \frac{1}{2} \sum_{p \in \mathbf{Z}} \alpha_p^J \alpha_{n-p}^J}$ .

$\displaystyle{ \begin{aligned} \left[ \bar L_m^\perp, x_0^I \right] &= \frac{1}{2} \sum_{p \in \mathbb{Z}} \left[ \bar \alpha^J_p \bar \alpha^J_{m-p}, x_0^I \right] \\ &= \frac{1}{2} \sum_{p \in \mathbb{Z}} \bar \alpha^I_p \left[ \bar \alpha^J_{m-p}, x_0^I \right] + \frac{1}{2} \sum_{p \in \mathbb{Z}} \left[ \bar \alpha^J_p, x_0^I \right] \bar \alpha^I_{m-p} \\ &= \frac{1}{2} \bar \alpha^J_m \left[ \bar \alpha^J_{0}, x_0^I \right] + \frac{1}{2} \left[ \bar \alpha^J_0, x_0^I \right] \bar \alpha^J_{m} \\ \end{aligned} }$

By Equation (13.33):

$\displaystyle{ \begin{aligned} \left[ \bar L_m^\perp, x_0^I \right] &= - \frac{1}{2} \bar \alpha^J_m \left[ i \sqrt{\frac{\alpha'}{2}} \eta^{IJ} \right] - \frac{1}{2} \left[ i \sqrt{\frac{\alpha'}{2}} \eta^{IJ} \right] \bar \alpha^J_{m} \\ &= - i \sqrt{\frac{\alpha'}{2}} \eta^{IJ} \bar \alpha^I_m \\ \end{aligned} }$