Lorentz Group [The Physics Travel Guide]

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--- advanced_tools:group_theory:lorentz_group [2018/04/06 15:43]
jakobadmin [Concrete]
+++ advanced_tools:group_theory:lorentz_group [2025/03/04 01:00] (current)
edi [Abstract]
@@ Line 9: / Line 9: @@
 <tabbox Concrete>
-It follows from the postulates of [[theories:special_relativity|special relativity]] that
+**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}
+\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 $dx_\mu'$:
@@ Line 29: / Line 32: @@
 \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.
+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}
+& 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&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&1&0 \\ i\sinh(\phi)& 0 &0 &\cosh(\phi)
+ \end{pmatrix}.  \end{equation}
 <tabbox Abstract>
@@ Line 125: / Line 196: @@
 See also, section 5.5 at page 79 in http://www.math.columbia.edu/~woit/QM/qmbook.pdf
  </WRAP>
+----
+**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/appendices/#relativity|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/appendices/#relativity|Fun with Symmetry]].
+[{{ :advanced_tools:group_theory:representation_theory:rotation_to_lorentz.jpg?nolink }}]
 <tabbox Why is it interesting?>

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