# 伽利略变换

## 平移

${\displaystyle x'=x-vt\,}$
${\displaystyle y'=y\,}$
${\displaystyle z'=z\,}$
${\displaystyle t'=t\,}$

${\displaystyle (x',t')=(x,t){\begin{pmatrix}1&0\\-v&1\end{pmatrix}}.}$

## 伽利略群的中心擴張

${\displaystyle [H,P_{i}]=0\,\!}$
${\displaystyle [P_{i},P_{j}]=0\,\!}$
${\displaystyle [L_{ij},H]=0\,\!}$
${\displaystyle [C_{i},C_{j}]=0\,\!}$
${\displaystyle [L_{ij},L_{kl}]=i[\delta _{ik}L_{jl}-\delta _{il}L_{jk}-\delta _{jk}L_{il}+\delta _{jl}L_{ik}]\,\!}$
${\displaystyle [L_{ij},P_{k}]=i[\delta _{ik}P_{j}-\delta _{jk}P_{i}]\,\!}$
${\displaystyle [L_{ij},C_{k}]=i[\delta _{ik}C_{j}-\delta _{jk}C_{i}]\,\!}$
${\displaystyle [C_{i},H]=iP_{i}\,\!}$
${\displaystyle [C_{i},P_{j}]=0\,\!.}$

H為時間平移的生成元（哈密顿算符），Pi為平移的生成元（動量算符），Ci為伽利略變換的生成元，而Lij為旋轉的生成元（角動量算符）。

${\displaystyle [H',P'_{i}]=0\,\!}$
${\displaystyle [P'_{i},P'_{j}]=0\,\!}$
${\displaystyle [L'_{ij},H']=0\,\!}$
${\displaystyle [C'_{i},C'_{j}]=0\,\!}$
${\displaystyle [L'_{ij},L'_{kl}]=i[\delta _{ik}L'_{jl}-\delta _{il}L'_{jk}-\delta _{jk}L'_{il}+\delta _{jl}L'_{ik}]\,\!}$
${\displaystyle [L'_{ij},P'_{k}]=i[\delta _{ik}P'_{j}-\delta _{jk}P'_{i}]\,\!}$
${\displaystyle [L'_{ij},C'_{k}]=i[\delta _{ik}C'_{j}-\delta _{jk}C'_{i}]\,\!}$
${\displaystyle [C'_{i},H']=iP'_{i}\,\!}$
${\displaystyle [C'_{i},P'_{j}]=iM\delta _{ij}\,\!}$

## 備註

1. ^ 伽利略變換. [2021-06-26]. （原始内容存档于2021-06-28）.
2. ^ Arthur Beiser; Kok Wai Cheah. Concepts of modern physics. McGraw-Hill. 2015: 第6頁. ISBN 9789814595261.
3. ^ Galileo 1638 Discorsi e Dimostrazioni Matematiche, intorno á due nuoue scienze 191 - 196, published by Lowys Elzevir (Louis Elsevier), Leiden, or Two New Sciences, English translation by Henry Crew and Alfonso de Salvio 1914, reprinted on pages 515-520 of On the Shoulders of Giants: The Great Works of Physics and Astronomy. Stephen Hawking, ed. 2002 ISBN 978-0-7624-1348-5
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5. ^ Lerner, Lawrence S., Physics for Scientists and Engineers, Volume 2, Jones and Bertlett Publishers, Inc, 1996, ISBN 0-7637-0460-1, Chapter 38 §38.2, p. 1046,1047页面存档备份，存于互联网档案馆
6. ^ Serway, Raymond A.; Jewett, John W., Principles of Physics: A Calculus-based Text, Fourth Edition, Brooks/Cole - Thomson Learning, 2006, ISBN 0-534-49143-X, Chapter 9 §9.1, p. 261页面存档备份，存于互联网档案馆
7. ^ Hoffmann, Banesh, Relativity and Its Roots, Scientific American Books, 1983, ISBN 0-486-40676-8, Chapter 5, p. 83页面存档备份，存于互联网档案馆
8. Arnold, V. I. Mathematical Methods of Classical Mechanics 2. Springer-Verlag. 1989: 6 [2013-01-30]. ISBN 0-387-96890-3. （原始内容存档于2013-03-12）.