Quick Calculation 13.3

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A First Course in String Theory

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Justify UHU^{-1} = H directly from the oscillator expansion of H = L_0^\perp + \bar L_0^\perp -2 .

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Eq. (13.37):

\displaystyle{ \begin{aligned} \bar L_n^\perp &= \frac{1}{2} \sum_{p \in \mathbf{Z}} \bar \alpha_p^I \bar \alpha_{n-p}^I\text{,} ~~~L_n^\perp = \frac{1}{2} \sum_{p \in \mathbf{Z}} \alpha_p^I \alpha_{n-p}^I\text{.} \\ \end{aligned} }

So

\displaystyle{ \begin{aligned} \bar L_0^\perp &= \frac{1}{2} \sum_{p \in \mathbf{Z}} \bar \alpha_p^I \bar \alpha_{-p}^I\text{,} ~~~L_0^\perp = \frac{1}{2} \sum_{p \in \mathbf{Z}} \alpha_p^I \alpha_{-p}^I\text{.} \\ \end{aligned} }

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Eq. (13.93):

\displaystyle{ \begin{aligned} U x_0 U^{-1} &= -x_0 \text{,}~~~ U p U^{-1} = -p \text{,} \\ U \alpha_n U^{-1} &= -\alpha_n \text{,}~~~ U \bar \alpha_n U^{-1} = - \bar \alpha_n \text{,} \end{aligned} }

where n \ne 0.

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Eq. (13.22):

\displaystyle{ \begin{aligned} \alpha_0^\mu &= \sqrt{\frac{\alpha'}{2}} p^\mu \\ \end{aligned} }

So

\displaystyle{ \begin{aligned} U \alpha_0^\mu U^{-1} &= - \alpha_0^\mu \\ \end{aligned} }

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Eq. (13.17)

\displaystyle{ \begin{aligned} \bar \alpha_0^\mu &= \alpha_0^\mu \\ \end{aligned} }

So

\displaystyle{ \begin{aligned} U \bar \alpha_0^\mu U^{-1} &= - \bar \alpha_0^\mu \\ \end{aligned} }

— Me@2024-10-03 06:42:21 PM

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2024.10.03 Thursday (c) All rights reserved by ACHK