Recent Articles

Galilean transformations

Newtonian physics. The equations below, although apparently obvious, break down at speeds that approach the speed of light.

Unlike the Galilean transformation, the relativistic Lorentz transformations can be shown to apply at all velocities so far measured, and the Galilean transformation can be regarded as low-velocity approximations to the Lorenz 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, spacial and time translations and rotations.

Central extension of the Galilean group

1. The 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>