Problem 2.2b

A First Course in String Theory

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2.2 Lorentz transformations for light-cone coordinates.

Consider coordinates \displaystyle{x^\mu = ( x^0, x^1, x^2, x^3 )} and the associated light-cone coordinates \displaystyle{x^\mu = ( x^+, x^-, x^2, x^3 )}. Write the following Lorentz transformations in terms of the light-cone coordinates.

(b) A rotation with angle \displaystyle{\theta} in the \displaystyle{x^1, x^2} plane.

\displaystyle{ \begin{aligned} \begin{bmatrix} c t' \\ x' \\ y' \\ z' \end{bmatrix}  &= \begin{bmatrix}       1 & 0 & 0 & 0 \\       0 & \cos \theta & -\sin \theta & 0 \\       0 & \sin \theta & \cos \theta & 0 \\       0 & 0 & 0 & 1 \\     \end{bmatrix}     \begin{bmatrix} c\,t \\ x \\ y \\ z \end{bmatrix} \\   \end{aligned} }

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\displaystyle{ \begin{aligned}    \begin{bmatrix}        \frac{1}{\sqrt{2}}  & \frac{1}{\sqrt{2}} & 0 & 0 \\        \frac{1}{\sqrt{2}}  & -\frac{1}{\sqrt{2}} & 0 & 0 \\                         0  & 0 & 1 & 0 \\                         0  & 0 & 0 & 1 \\     \end{bmatrix}  \begin{bmatrix} (x^+)' \\ (x^-)' \\ y' \\ z' \end{bmatrix}  &=   \begin{bmatrix}       1 & 0 & 0 & 0 \\       0 & \cos \theta & -\sin \theta & 0 \\       0 & \sin \theta & \cos \theta & 0 \\       0 & 0 & 0 & 1 \\     \end{bmatrix}    \begin{bmatrix}        \frac{1}{\sqrt{2}}  & \frac{1}{\sqrt{2}} & 0 & 0 \\        \frac{1}{\sqrt{2}}  & -\frac{1}{\sqrt{2}} & 0 & 0 \\                         0  & 0 & 1 & 0 \\                         0  & 0 & 0 & 1 \\     \end{bmatrix}  \begin{bmatrix} x^+ \\ x^- \\ y \\ z \end{bmatrix} \\  \begin{bmatrix} (x^+)' \\ (x^-)' \\ y' \\ z' \end{bmatrix}  &= \frac{1}{2}  \begin{bmatrix} \cos\theta + 1 & 1 - \cos\theta & -\sqrt{2} \sin\theta & 0 \\  1 - \cos\theta & \cos\theta + 1 &  \sqrt{2} \sin\theta & 0 \\  \sqrt{2} \sin{\theta} & -\sqrt{2} \sin{\theta} & 2 \cos{\theta} & 0 \\ 0 & 0 & 0 & 2 \\ \end{bmatrix} \begin{bmatrix} x^+ \\ x^- \\ y \\ z \end{bmatrix}  \end{aligned} }

— Me@2020-03-22 10:16:09 PM

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2020.03.23 Monday (c) All rights reserved by ACHK