Galilean group - Definition and Overview

The Galilean transformation is used to transform between the coordinates of two coordinate systems in constant relative motion in Newtonian physics. This is the passive transformation point of view. The equations below, although apparently obvious, break down at speeds that approach the speed of light.

Unlike the Galilean transformation, the relativistic Lorentz transformation can be shown to apply at all velocities so far measured, and the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation.

<math>t^'=t<math>

<math>x^'=x-ut<math>

<math>y^'=y<math>

<math>z^'=z<math>

Under the Erlangen program, the space-time (no longer spacetime) of nonrelativistic physics is described by the symmetry group generated by Galilean transformations, spatial and time translations and rotations.

The Galilean symmetries (interpreted as active transformations)

Spatial translations:

<math>t\rightarrow t<math>

<math>\vec{x}\rightarrow \vec{x}+\vec{a}<math>

Time translations:

<math>t\rightarrow t+\tau<math>

<math>\vec{x}\rightarrow \vec{x}<math>

Boosts:

<math>t\rightarrow t<math>

<math>\vec{x}\rightarrow \vec{x}+\vec{v}t<math>

Rotations:

<math>t\rightarrow t<math>

<math>\vec{x}\rightarrow \mathbf{R}\vec{x}<math>

where R is an orthogonal matrix.

Central extension of the Galilean group

1. The Galilean group: Here, we will only look at its Lie algebra. It's easy to extend the results to the Lie group. The Lie algebra of L is spanned by E, P_i, C_i and L_ij (antisymmetric tensor) subject to

<math>[E,P_i]=0<math>

<math>[P_i,P_j]=0<math>

<math>[L_{ij},E]=0<math>

<math>[C_i,C_j]=0<math>

<math>[L_{ij},L_{kl}]=i\hbar [\delta_{ik}L_{jl}-\delta_{il}L_{jk}-\delta_{jk}L_{il}+\delta_{jl}L_{ik}]<math>

<math>[L_{ij},P_k]=i\hbar[\delta_{ik}P_j-\delta_{jk}P_i]<math>

<math>[L_{ij},C_k]=i\hbar[\delta_{ik}C_j-\delta_{jk}C_i]<math>

<math>[C_i,E]=i\hbar P_i<math>

<math>[C_i,P_j]=0<math>

We can now give it a central extension into the Lie algebra spanned by E', P'_i, C'_i, L'_ij (antisymmetric tensor), M such that M commutes with everything (i.e. lies in the center, that's why it's called a central extension) and

<math>[E',P'_i]=0<math>

<math>[P'_i,P'_j]=0<math>

<math>[L'_{ij},E']=0<math>

<math>[C'_i,C'_j]=0<math>

<math>[L'_{ij},L'_{kl}]=i\hbar [\delta_{ik}L'_{jl}-\delta_{il}L'_{jk}-\delta_{jk}L'_{il}+\delta_{jl}L'_{ik}]<math>

<math>[L'_{ij},P'_k]=i\hbar[\delta_{ik}P'_j-\delta_{jk}P'_i]<math>

<math>[L'_{ij},C'_k]=i\hbar[\delta_{ik}C'_j-\delta_{jk}C'_i]<math>

<math>[C'_i,E']=i\hbar P'_i<math>

<math>[C'_i,P'_j]=i\hbar M\delta_{ij}<math>

See also representation theory of the Galilean group, Poincaré group