Lightlike compatification

Posted on by

A First Course in String Theory

.

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}

~~~

\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

.

.

2021.03.23 Tuesday (c) All rights reserved by ACHK