Lightlike compatification

A First Course in String Theory

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2.9 Lightlike compatification

(a) Rewrite this identification using light-cone coordinates.

\begin{aligned}  \begin{bmatrix}   x \\   ct   \end{bmatrix}   &\sim   \begin{bmatrix}   x \\   ct   \end{bmatrix}   +   2 \pi  \begin{bmatrix}   R \\   -R   \end{bmatrix}   \end{aligned}

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\begin{aligned}  \begin{bmatrix} x^+ \\ x^- \end{bmatrix}   &= \frac{1}{\sqrt{2}}      \begin{bmatrix}         1 & 1 \\         1 & -1 \\      \end{bmatrix}      \begin{bmatrix} x^0 \\ x^1 \end{bmatrix} \\    \end{aligned}

\begin{aligned}   \frac{1}{\sqrt{2}}      \begin{bmatrix}         1 & 1 \\         1 & -1 \\      \end{bmatrix}      \begin{bmatrix}   x^0 \\  x^1 \\   \end{bmatrix}   &\sim    \frac{1}{\sqrt{2}}      \begin{bmatrix}         1 & 1 \\         1 & -1 \\      \end{bmatrix}   \begin{bmatrix}   x^0 \\   x^1 \\   \end{bmatrix}   +    \frac{1}{\sqrt{2}}      \begin{bmatrix}         1 & 1 \\         1 & -1 \\      \end{bmatrix}   \begin{bmatrix}   - 2 \pi R \\     2 \pi R \\   \end{bmatrix}   \\  \end{aligned}

\begin{aligned}  \begin{bmatrix}   x^+ \\  x^- \\   \end{bmatrix}   &\sim   \begin{bmatrix}   x^+ \\   x^- \\   \end{bmatrix}   +    \frac{1}{\sqrt{2}}      \begin{bmatrix}         0 \\         - 4 \pi R \\      \end{bmatrix} \\  \end{aligned}

— Me@2021-03-22 06:06:10 PM

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2021.03.23 Tuesday (c) All rights reserved by ACHK