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advanced_tools:group_theory:lorentz_group [2018/04/06 15:29]
jakobadmin
advanced_tools:group_theory:lorentz_group [2023/05/20 19:32] (current)
edi [Abstract]
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-====== Lorentz ​group ======+<WRAP lag> ​ $SO(3,​1)$</​WRAP>​ 
 +====== Lorentz ​Group ======
  
  
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 <tabbox Intuitive> ​ <tabbox Intuitive> ​
  
-<note tip> +For perfect intuitive introduction to the Lorentz group, see [[https://​www.youtube.com/watch?​v=Rh0pYtQG5wI|this video by minutephyiscs]].
-Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during ​coffee break or at a cocktail party. +
-</note>+
   ​   ​
 <tabbox Concrete> ​ <tabbox Concrete> ​
 +**Definition of the Lorentz transformations**
 +
 +It follows from the postulates of [[models:​special_relativity|special relativity]] that
 +$d s^2 = \eta^{\mu \nu} dx_\mu dx_\nu$ stays exactly the same in all inertial frames of reference:
 +\begin{equation} ds'^2 = dx'​_\mu dx'​_\nu \eta^{\mu\nu} = ds^2 = dx_\mu dx_\nu \eta^{\mu\nu} \, ,​\end{equation}
 +where $\eta^{\mu\nu}$ is the [[advanced_tools:​minkowski_metric|Minkowski metric]]. ​
 +
 +We denote a generic transformation that takes us to another frame with $\Lambda$ and the transformed coordinates $dx_\mu'​$:​
 +
 +\begin{equation} \label{eq:​lorentztrafo1} ​ dx_\mu \rightarrow dx'​_\mu=\Lambda^{ \  \sigma}_{\mu} dx_\sigma. \end{equation}
 +Then we can write the invariance condition from above as:
 +\begin{align}
 +  (ds)^2 &= (ds'​)^2 ​ \notag \\
 +  \rightarrow ​  dx \cdot dx &​\stackrel{!}{=} dx' \cdot dx' ​ \notag \\
 +   ​\rightarrow ​ dx_\mu dx_\nu \eta^{\mu\nu} &​\stackrel{!}{=} dx'​_\mu dx'​_\nu \eta^{\mu\nu} \underbrace{=} \Lambda^{ \ \sigma}_{ \mu} dx_\sigma \Lambda^{ \ \gamma}_{ \nu} dx_\gamma \eta^{\mu\nu} \notag \\
 +  \underbrace{\rightarrow} dx_\mu dx_\nu \eta^{\mu\nu} &​\stackrel{!}{=} \Lambda^{ \ \mu}_{ \sigma} dx_\mu \Lambda^{ \ \nu}_{ \gamma} dx_\nu \eta^{\sigma\gamma} \notag \\
 +  \underbrace{\rightarrow}_{\text{Because the equation holds for  arbitrary } dx_\mu} \eta^{\mu\nu} ​ &​\stackrel{!}{=} ​ \Lambda^{ \ \mu}_{ \sigma} \eta^{\sigma \gamma} ​  ​\Lambda^{ \ \nu}_{ \gamma}
 +\end{align}
 +
 +Or written in matrix notation
 +
 +\begin{equation} \label{eq:​lorentztrafodefequation} \eta = \Lambda^T \eta \Lambda \end{equation}
 +
 +This is the condition that transformations $\Lambda$ which take us from one frame to another allowed frames of reference must fulfill. Such transformations are called Lorentz transformations and the equation can be taken as a definition of Lorentz transformations. Formulated differently,​ the Lorentz transformations are defined as all those transformations that leave the Minkowski metric unchanged. ​
 +
 +----
 +
 +**Explicit form of the Lorentz transformations**
 +
 +__Rotations__
 +First, we note that the rotation matrices of 3-dimensional Euclidean space that only act on space and not on time, fulfil the defining condition. This follows because the spatial part ($\mu=1,​2,​3$) of the Minkowski metric is proportional to the $3 \times 3$ identity matrix. Thus for transformations that only modify space, we get from the condition $\eta = \Lambda^T \eta \Lambda$ that
 +
 +\[-R^T I_{3 \times 3} R =- R^T R \stackrel{!}{=} -  I_{3 \times 3} 
 +\]
 +\[\rightarrow R^T I_{3 \times 3} R =R^T R \stackrel{!}{=} ​  I_{3 \times 3} .
 +\]
 +This is exactly the defining condition of the group $O(3)$. Together with the condition
 +\[ \det(\Lambda) \stackrel{!}{=} 1
 +\]
 +these are the defining conditions of the group $SO(3)$, which describes three-dimensional rotations. We conclude that one type of Lorentz transformation is given by
 +\[ \Lambda_{\mathrm{rot }}= \begin{pmatrix} 1 &  \\ & R_{3 \times 3} \end{pmatrix}
 +\]
 +with the usual rotation matrices ​ $R_{3 \times 3}$:
 +
 +\begin{eqnarray}
 +& & R_x(\phi) = 
 +\begin{pmatrix}
 +1 & 0 & 0 \\ 0 & \cos\phi & \sin\phi \\ 0 & -\sin\phi & \cos\phi
 +\end{pmatrix} \label{eq:​rotx} \\
 +& & R_y(\psi) = 
 +\begin{pmatrix}
 +\cos \psi & 0 & -\sin\psi \\ 0 & 1 & 0 \\ \sin\psi & 0 & \cos\psi
 +\end{pmatrix} \label{eq:​roty} \\
 +& & R_z(\theta) = 
 +\begin{pmatrix}
 +\cos \theta & \sin \theta & 0 \\-\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1
 +\end{pmatrix} \label{eq:​rotz}
 +\end{eqnarray}
 +
 +__Boosts__
 +
 +To investigate all other transformations which transform time //and// space we start, as usual in Lie theory, with an infinitesimal transformation
 +\begin{equation} ​ \Lambda^{\mu}_{\rho} \approx \delta^{\mu}_{\rho}+ \epsilon K^{\mu}_{\rho}. \end{equation}
 +We put this now into the defining condition $\eta = \Lambda^T \eta \Lambda$ and get
 +\[\Lambda^{\mu}_{\rho} \eta_{\mu \nu} \Lambda^{\nu}_{\sigma} ​ \stackrel{!}{=} ​  ​\eta_{\rho \sigma}
 +\]
 +\[ \rightarrow ( \delta^{\mu}_{\rho}+ \epsilon K^{\mu}_{\rho} ) \eta_{\mu \nu} (\delta^{\nu}_{\sigma}+ \epsilon K^{\nu}_{\sigma}) ​ \stackrel{!}{=} ​  ​\eta_{\rho \sigma}  ​
 +\]
 +\[ \rightarrow \eta_{\rho \sigma} + \epsilon K^{\mu}_{\rho}\eta_{\mu \sigma} + \epsilon K^{\nu}_{\sigma} \eta_{\rho \nu} + \underbrace{\epsilon^2 ​  ​K^{\mu}_{\rho}\eta_{\mu \nu}  K^{\nu}_{\sigma}}_{ \approx 0 \text{ because } \epsilon \text{ is infinitesimal }\rightarrow \epsilon^2 \approx 0} = \eta_{\rho \sigma}
 +\]
 +\begin{equation}\rightarrow ​ K^{\mu}_{\rho}\eta_{\mu \sigma} +  K^{\nu}_{\sigma} \eta_{\rho \nu} = 0     ​\end{equation}
 +
 +or in matrix notation
 +
 +\begin{equation} \label{eq:​boost4d} K^T \eta = - \eta K. \end{equation}
 +
 +A transformation that fulfill this equation is called a boost. A boost takes us from one frame to another frame that moves with a different velocity. Explcitly, such transformations can be described by
 +
 +\begin{equation} \Lambda_x =  \begin{pmatrix}
 +  \cosh(\phi)&​i\sinh(\phi) ​ & 0 & 0\\ i\sinh(\phi)&​ \cosh(\phi) &0 &0 \\
 +  0&​0&​1&​0 \\ 0&​0&​0&​1
 + ​\end{pmatrix} ​ \end{equation}
 +
 +\begin{equation} \Lambda_y =  \begin{pmatrix}
 +  \cosh(\phi)&​ 0 & i\sinh(\phi) & 0\\ 0 & 1 &0 &​0 ​  \\
 +  i\sinh(\phi)&​ 0 &  \cosh(\phi) &​0 ​ \\ 0&​0&​0&​1
 + ​\end{pmatrix} ​ \end{equation}
 + 
 +\begin{equation} \label{eq:​boostexplicitz-direction} \Lambda_z =  \begin{pmatrix}
 +  \cosh(\phi)&​0 ​ & 0 & i\sinh(\phi)\\ 0 &1 &0 &0 \\
 +  0&​0&​1&​0 \\ i\sinh(\phi)&​ 0 &0 &​\cosh(\phi)
 + ​\end{pmatrix}. ​ \end{equation}
 +<tabbox Abstract> ​
 +
 +**Representations of the Lorentz group**
  
 At the heart of the representation theory of the Poincare group is the representation theory of the proper orthochronous Lorentz group $SO(1,​3)^{\uparrow}$. We can concentrate on this subset of the Lorentz group, because the Lorentz group can be decomposed as follows: At the heart of the representation theory of the Poincare group is the representation theory of the proper orthochronous Lorentz group $SO(1,​3)^{\uparrow}$. We can concentrate on this subset of the Lorentz group, because the Lorentz group can be decomposed as follows:
Line 103: Line 197:
  </​WRAP>​  </​WRAP>​
  
-  +---- 
-<tabbox Abstract> ​+ 
 +**Graphical Summary** 
 + 
 +The picture below shows the weight diagrams of some important irreducible representations of the (double cover of the) Lorentz group (right) and, for comparison, some irreducible representations of $SU(2)$ (left). For a more detailed explanation of this picture see [[https://​esackinger.wordpress.com/​blog/​lie-groups-and-their-representations/#​lorentz_irreps|Fun with Symmetry]]. 
 + 
 +[{{ :​advanced_tools:​group_theory:​representation_theory:​lorentz_irreps.jpg?​nolink }}] 
 + 
 +The following diagram illustrates the relationship between the groups of rotation $O(3)$ and $O(4)$, in 3D and 4D Euclidean space, respectively,​ and the Lorentz group $O(1,3)$. For a more detailed explanation of this diagram see [[https://​esackinger.wordpress.com/​blog/​lie-groups-and-their-representations/#​rotation_to_lorentz|Fun with Symmetry]]. 
 + 
 +[{{ :​advanced_tools:​group_theory:​representation_theory:​rotation_to_lorentz.jpg?​nolink }}]
  
-<note tip> 
-The motto in this section is: //the higher the level of abstraction,​ the better//. 
-</​note>​ 
  
 <tabbox Why is it interesting?> ​ <tabbox Why is it interesting?> ​
 +
 +The Lorentz group is an important part of the fundamental spacetime symmetry group of the standard model, called the [[advanced_tools:​group_theory:​poincare_group|Poincare group]]. ​
 +
 +It encodes the fact that physics should be the same in all frames of reference and additionally that the speed of light is the same in all such frames. ​
  
 Understanding the representations is crucial for the standard model, because these representations are the tools that we need to describe [[advanced_notions:​elementary_particles|elementary particles]]. ​ Understanding the representations is crucial for the standard model, because these representations are the tools that we need to describe [[advanced_notions:​elementary_particles|elementary particles]]. ​
advanced_tools/group_theory/lorentz_group.1523021347.txt.gz · Last modified: 2018/04/06 13:29 (external edit)