advanced_tools:group_theory:lorentz_group
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advanced_tools:group_theory:lorentz_group [2018/04/06 15:56] jakobadmin [Concrete] |
advanced_tools:group_theory:lorentz_group [2025/03/04 01:00] (current) edi [Abstract] |
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| **Definition of the Lorentz transformations** | **Definition of the Lorentz transformations** | ||
| - | It follows from the postulates of [[theories:special_relativity|special relativity]] that | + | 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: | $d s^2 = \eta^{\mu \nu} dx_\mu dx_\nu$ 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} | ||
| Line 51: | Line 51: | ||
| \[ \Lambda_{\mathrm{rot }}= \begin{pmatrix} 1 & \\ & R_{3 \times 3} \end{pmatrix} | \[ \Lambda_{\mathrm{rot }}= \begin{pmatrix} 1 & \\ & R_{3 \times 3} \end{pmatrix} | ||
| \] | \] | ||
| - | with the usual rotation matrices $R_{3 \times 3}$. | + | 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__ | __Boosts__ | ||
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| See also, section 5.5 at page 79 in http://www.math.columbia.edu/~woit/QM/qmbook.pdf | See also, section 5.5 at page 79 in http://www.math.columbia.edu/~woit/QM/qmbook.pdf | ||
| </WRAP> | </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?> | <tabbox Why is it interesting?> | ||
advanced_tools/group_theory/lorentz_group.1523022979.txt.gz · Last modified: 2018/04/06 13:56 (external edit)
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