Problem 2.3a

A First Course in String Theory

2.2 Lorentz transformations, derivatives, and quantum operators.

(a) Give the Lorentz transformations for the components $\displaystyle{a_{\mu}}$ of a vector under a boost along the $\displaystyle{x^1}$ axis.

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$\displaystyle{\begin{aligned} \begin{bmatrix} c t' \\ z' \\ x' \\ y' \end{bmatrix} &= \begin{bmatrix} \gamma & -\beta \gamma & 0 & 0 \\ -\beta \gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} c\,t \\ z \\ x \\ y \end{bmatrix} \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} (x')^\mu &= L^\mu_{~\nu} x^\nu \\ a_\mu &= a^\nu \eta_{\mu \nu} \\ (a')_\mu &= L_\mu^{~\nu} a_\nu \\ [(a')_\mu] &= [a_\nu] [L^\mu_{~\nu}]^{-1} \\ [L^\mu_{~\nu}]^{-1} &= \begin{bmatrix} \gamma & \beta \gamma & 0 & 0 \\ \beta \gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \\ \end{aligned}}$