Editing Galilean transformation (section)

==Origin in group contraction==
The [[Lie algebra]] of the [[Representation theory of the Galilean group|Galilean group]] is [[linear span|spanned]] by {{math|''H'', ''P<sub>i</sub>'', ''C<sub>i</sub>''}} and {{math|''L<sub>ij</sub>''}} (an [[antisymmetric tensor]]), subject to [[commutator|commutation relations]], where
:<math>[H,P_i]=0 </math>
:<math>[P_i,P_j]=0 </math>
:<math>[L_{ij},H]=0 </math>
:<math>[C_i,C_j]=0 </math>
:<math>[L_{ij},L_{kl}]=i [\delta_{ik}L_{jl}-\delta_{il}L_{jk}-\delta_{jk}L_{il}+\delta_{jl}L_{ik}] </math>
:<math>[L_{ij},P_k]=i[\delta_{ik}P_j-\delta_{jk}P_i] </math>
:<math>[L_{ij},C_k]=i[\delta_{ik}C_j-\delta_{jk}C_i] </math>
:<math>[C_i,H]=i P_i \,\!</math>
:<math>[C_i,P_j]=0 ~.</math>
{{mvar|H}} is the generator of time translations ([[Hamiltonian (quantum mechanics)|Hamiltonian]]), {{math|''P<sub>i</sub>''}} is the generator of translations ([[momentum operator]]), {{math|''C<sub>i</sub>''}} is the generator of rotationless Galilean transformations (Galileian boosts),<ref>{{cite book |title=Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces |edition=illustrated |first1=A. A. |last1=Ungar |publisher=Springer Science & Business Media |year=2006 |isbn=978-0-306-47134-6 |page=336 |url=https://books.google.com/books?id=MTTaBwAAQBAJ}} [https://books.google.com/books?id=MTTaBwAAQBAJ&pg=PA336 Extract of page 336]</ref> and {{math|''L<sub>ij</sub>''}} stands for a generator of rotations ([[angular momentum operator]]).

This Lie Algebra is seen to be a special [[classical limit]] of the algebra of the [[Poincaré group#Technical explanation|Poincaré group]], in the limit {{math|''c'' → ∞}}. Technically, the Galilean group is a celebrated [[group contraction]] of the Poincaré group (which, in turn, is a [[group contraction]] of the de Sitter group {{math|SO(1,4)}}).<ref>{{harvnb|Gilmore|2006}}</ref>
Formally, renaming the generators of momentum and boost of the latter as in
:{{math|''P''<sub>0</sub> ↦ ''H'' / ''c''}} 
:{{math|''K<sub>i</sub>'' ↦ ''c'' ⋅ ''C<sub>i</sub>''}}, 
where {{math|''c''}} is the speed of light (or any unbounded function thereof), the commutation relations (structure constants) in the limit {{math|''c'' → ∞}} take on the relations of the former. 
Generators of time translations and rotations are identified. Also note the group invariants {{math|''L''<sub>''mn''</sub> ''L''<sup>''mn''</sup>}} and  {{math|''P''<sub>''i''</sub> ''P''{{i sup|''i''}}}}.

In matrix form, for {{math|1=''d'' = 3}}, one may consider the ''regular representation'' (embedded in {{math|GL(5; '''R''')}}, from which it could be derived by a single group contraction, bypassing the Poincaré group), 
:
<math>
 iH=   \left( {\begin{array}{ccccc}
   0 &  0 &  0 & 0 &  0\\
 0 &  0 &  0 &  0 &  0\\
 0 &  0 &  0 &  0 &  0\\
   0 & 0 & 0 & 0 & 1\\
   0 & 0 & 0 & 0 & 0\\
  \end{array} } \right)    , \qquad            
   </math>
<math>
  i\vec{a}\cdot\vec{P}= 
  \left( {\begin{array}{ccccc}
    0&0&0&0 & a_1\\
   0&0&0&0  & a_2\\
   0&0&0&0  & a_3\\
   0 & 0 & 0 & 0& 0\\
   0 & 0 & 0 & 0 & 0\\
  \end{array} } \right), \qquad
  </math>
<math>
  i\vec{v}\cdot\vec{C}= 
  \left( {\begin{array}{ccccc}
  0 & 0 & 0 & v_1 & 0\\
  0 & 0 & 0 & v_2 & 0\\
  0 & 0 & 0 & v_3 & 0\\
   0 & 0 & 0 & 0 & 0\\
   0 & 0 & 0 & 0 & 0\\
  \end{array} } \right), \qquad
  </math>
<math>
i \theta_i \epsilon^{ijk} L_{jk} = 
  \left( {\begin{array}{ccccc}
   0& \theta_3 & -\theta_2 & 0 & 0\\
   -\theta_3 & 0 &  \theta_1& 0 & 0\\
   \theta_2 & -\theta_1 & 0 & 0 & 0\\
   0 & 0 & 0 & 0 & 0\\
   0 & 0 & 0 & 0 & 0\\
  \end{array} } \right )  ~.  </math>

The infinitesimal group element is then 
::<math>
G(R,\vec{v},\vec{a},s)=1\!\!1_5   +       \left( {\begin{array}{ccccc}
   0& \theta_3 & -\theta_2 & v_1& a_1\\      -\theta_3 & 0 &  \theta_1& v_2 & a_2\\
   \theta_2 & -\theta_1 & 0 & v_3 & a_3\\      0 & 0 & 0 & 0 & s\\
   0 & 0 & 0 & 0 & 0\\     \end{array} } \right )  +\ ... ~.  </math>