Editing Lorentz transformation (section)

====The Lie group SO{{sup|+}}(3,1)====

The set of transformations
<math display="block"> \{ B(\boldsymbol{\zeta}), R(\boldsymbol{\theta}), \Lambda(\boldsymbol{\zeta}, \boldsymbol{\theta}) \} </math>
with matrix multiplication as the operation of composition forms a group, called the "restricted Lorentz group", and is the [[special indefinite orthogonal group]] SO{{sup|+}}(3,1). (The plus sign indicates that it preserves the orientation of the temporal dimension).

For simplicity, look at the infinitesimal Lorentz boost in the {{mvar|x}} direction (examining a boost in any other direction, or rotation about any axis, follows an identical procedure). The infinitesimal boost is a small boost away from the identity, obtained by the [[Taylor expansion]] of the boost matrix to first order about {{math|1=''ζ'' = 0}},
<math display="block"> B_x = I + \zeta \left. \frac{\partial B_x}{\partial \zeta } \right|_{\zeta=0} + \cdots </math>
where the higher order terms not shown are negligible because {{mvar|ζ}} is small, and {{math|''B''{{sub|''x''}}}} is simply the boost matrix in the ''x'' direction. The [[matrix calculus|derivative of the matrix]] is the matrix of derivatives (of the entries, with respect to the same variable), and it is understood the derivatives are found first then evaluated at {{math|1=''ζ'' = 0}},
<math display="block"> \left. \frac{\partial B_x }{\partial \zeta } \right|_{\zeta=0} = - K_x \,. </math>

For now, {{math|''K''{{sub|''x''}}}} is defined by this result (its significance will be explained shortly). In the limit of an infinite number of infinitely small steps, the finite boost transformation in the form of a [[matrix exponential]] is obtained
<math display="block"> B_x =\lim_{N\to\infty}\left(I-\frac{\zeta }{N}K_x\right)^{N} = e^{-\zeta K_x} </math>
where the [[Exponential function#Formal definition|limit definition of the exponential]] has been used (see also [[characterizations of the exponential function]]). More generally<ref group="nb">Explicitly,
<math display="block"> \boldsymbol{\zeta} \cdot\mathbf{K} = \zeta_x K_x + \zeta_y K_y + \zeta_z K_z </math>
<math display="block"> \boldsymbol{\theta} \cdot\mathbf{J} = \theta_x J_x + \theta_y J_y + \theta_z J_z </math>
</ref>

<math display="block">B(\boldsymbol{\zeta}) = e^{-\boldsymbol{\zeta}\cdot\mathbf{K}} \, , \quad R(\boldsymbol{\theta}) = e^{\boldsymbol{\theta}\cdot\mathbf{J}} \,. </math>

The axis-angle vector {{math|'''θ'''}} and rapidity vector {{math|'''ζ'''}} are altogether six continuous variables which make up the group parameters (in this particular representation), and the generators of the group are {{math|1='''K''' = (''K''{{sub|''x''}}, ''K''{{sub|''y''}}, ''K''{{sub|''z''}})}} and {{math|1='''J''' = (''J''{{sub|''x''}}, ''J''{{sub|''y''}}, ''J''{{sub|''z''}})}}, each vectors of matrices with the explicit forms<ref group=nb>In [[quantum mechanics]], [[relativistic quantum mechanics]], and [[quantum field theory]], a different convention is used for these matrices; the right hand sides are all multiplied by a factor of the imaginary unit {{math|''i'' {{=}} {{sqrt|−1}}}}.</ref>

<math display="block">\begin{alignat}{3}

K_x &= \begin{bmatrix}
  0 & 1 & 0 & 0 \\
  1 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 \\
  \end{bmatrix}\,,\quad &
K_y &= \begin{bmatrix}
  0 & 0 & 1 & 0\\
  0 & 0 & 0 & 0\\
  1 & 0 & 0 & 0\\
  0 & 0 & 0 & 0
  \end{bmatrix}\,,\quad &
K_z &= \begin{bmatrix}
  0 & 0 & 0 & 1\\
  0 & 0 & 0 & 0\\
  0 & 0 & 0 & 0\\
  1 & 0 & 0 & 0
\end{bmatrix}

\\[10mu]

J_x &= \begin{bmatrix}
  0 & 0 & 0 &  0 \\
  0 & 0 & 0 &  0 \\
  0 & 0 & 0 & -1 \\
  0 & 0 & 1 &  0 \\
  \end{bmatrix}\,,\quad &
J_y &= \begin{bmatrix}
  0 &  0 & 0 & 0 \\
  0 &  0 & 0 & 1 \\
  0 &  0 & 0 & 0 \\
  0 & -1 & 0 & 0
  \end{bmatrix}\,,\quad &
J_z &= \begin{bmatrix}
  0 & 0 &  0 & 0 \\
  0 & 0 & -1 & 0 \\
  0 & 1 &  0 & 0 \\
  0 & 0 &  0 & 0
  \end{bmatrix}

\end{alignat}</math>

These are all defined in an analogous way to {{math|''K''{{sub|''x''}}}} above, although the minus signs in the boost generators are conventional. Physically, the generators of the Lorentz group correspond to important symmetries in spacetime: {{math|'''J'''}} are the ''rotation generators'' which correspond to [[angular momentum]], and {{math|'''K'''}} are the ''boost generators'' which correspond to the motion of the system in spacetime. The derivative of any smooth curve {{math|''C''(''t'')}} with {{math|1=''C''(0) = ''I''}} in the group depending on some group parameter {{mvar|t}} with respect to that group parameter, evaluated at {{math|1=''t'' = 0}}, serves as a definition of a corresponding group generator {{mvar|G}}, and this reflects an infinitesimal transformation away from the identity. The smooth curve can always be taken as an exponential as the exponential will always map {{mvar|G}} smoothly back into the group via {{math|''t'' → exp(''tG'')}} for all {{mvar|t}}; this curve will yield {{mvar|G}} again when differentiated at {{math|1=''t'' = 0}}.

Expanding the exponentials in their Taylor series obtains
<math display="block"> B({\boldsymbol {\zeta }})=I-\sinh \zeta (\mathbf {n} \cdot \mathbf {K} )+(\cosh \zeta -1)(\mathbf {n} \cdot \mathbf {K} )^2</math>
<math display="block">R(\boldsymbol {\theta })=I+\sin \theta (\mathbf {e} \cdot \mathbf {J} )+(1-\cos \theta )(\mathbf {e} \cdot \mathbf {J} )^2\,.</math>
which compactly reproduce the boost and rotation matrices as given in the previous section.

It has been stated that the general proper Lorentz transformation is a product of a boost and rotation. At the ''infinitesimal'' level the product
<math display="block"> \begin{align}
\Lambda
&= (I - \boldsymbol {\zeta } \cdot \mathbf {K} + \cdots )(I + \boldsymbol {\theta } \cdot \mathbf {J} + \cdots ) \\
&= (I + \boldsymbol {\theta } \cdot \mathbf {J} + \cdots )(I - \boldsymbol {\zeta } \cdot \mathbf {K} + \cdots ) \\
&= I - \boldsymbol {\zeta } \cdot \mathbf {K}  + \boldsymbol {\theta } \cdot \mathbf {J} + \cdots  
\end{align} </math>
is commutative because only linear terms are required (products like {{math|('''θ'''·'''J''')('''ζ'''·'''K''')}} and {{math|('''ζ'''·'''K''')('''θ'''·'''J''')}} count as higher order terms and are negligible). Taking the limit as before leads to the finite transformation in the form of an exponential
<math display="block">\Lambda (\boldsymbol{\zeta}, \boldsymbol{\theta}) = e^{-\boldsymbol{\zeta} \cdot\mathbf{K} + \boldsymbol{\theta} \cdot\mathbf{J} }.</math>

The converse is also true, but the decomposition of a finite general Lorentz transformation into such factors is nontrivial. In particular,
<math display="block">e^{-\boldsymbol{\zeta} \cdot\mathbf{K} + \boldsymbol{\theta} \cdot\mathbf{J} } \ne e^{-\boldsymbol{\zeta} \cdot\mathbf{K}} e^{\boldsymbol{\theta} \cdot\mathbf{J}},</math>
because the generators do not commute. For a description of how to find the factors of a general Lorentz transformation in terms of a boost and a rotation ''in principle'' (this usually does not yield an intelligible expression in terms of generators {{math|'''J'''}} and {{math|'''K'''}}), see [[Wigner rotation]]. If, on the other hand, ''the decomposition is given'' in terms of the generators, and one wants to find the product in terms of the generators, then the [[Baker–Campbell–Hausdorff formula]] applies.