# 伽利略变换

## 平移

$x'=x-vt\,$
$y'=y \,$
$z'=z \,$
$t'=t \,$

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

## 伽利略群的中心擴張

$[H,P_i]=0 \,\!$
$[P_i,P_j]=0 \,\!$
$[L_{ij},H]=0 \,\!$
$[C_i,C_j]=0 \,\!$
$[L_{ij},L_{kl}]=i [\delta_{ik}L_{jl}-\delta_{il}L_{jk}-\delta_{jk}L_{il}+\delta_{jl}L_{ik}] \,\!$
$[L_{ij},P_k]=i[\delta_{ik}P_j-\delta_{jk}P_i] \,\!$
$[L_{ij},C_k]=i[\delta_{ik}C_j-\delta_{jk}C_i] \,\!$
$[C_i,H]=i P_i \,\!$
$[C_i,P_j]=0 \,\!.$

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）。

$[H',P'_i]=0 \,\!$
$[P'_i,P'_j]=0 \,\!$
$[L'_{ij},H']=0 \,\!$
$[C'_i,C'_j]=0 \,\!$
$[L'_{ij},L'_{kl}]=i [\delta_{ik}L'_{jl}-\delta_{il}L'_{jk}-\delta_{jk}L'_{il}+\delta_{jl}L'_{ik}] \,\!$
$[L'_{ij},P'_k]=i[\delta_{ik}P'_j-\delta_{jk}P'_i] \,\!$
$[L'_{ij},C'_k]=i[\delta_{ik}C'_j-\delta_{jk}C'_i] \,\!$
$[C'_i,H']=i P'_i \,\!$
$[C'_i,P'_j]=i M\delta_{ij} \,\!$

## 備註

1. ^ 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 0-7624-1348-4
2. ^ Mould, Richard A., Basic relativity, Springer-Verla. 2002, ISBN 0-387-95210-1 , Chapter 2 §2.6, p. 42
3. ^ 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
4. ^ 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
5. ^ Hoffmann, Banesh, Relativity and Its Roots, Scientific American Books. 1983, ISBN 0-486-40676-8 , Chapter 5, p. 83
6. ^ 6.0 6.1 6.2 Arnold, V. I. Mathematical Methods of Classical Mechanics 2. Springer-Verlag. 1989. 6. ISBN 0-387-96890-3.