advanced_tools:group_theory:lorentz_group
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advanced_tools:group_theory:lorentz_group [2018/05/04 09:53] jakobadmin ↷ Links adapted because of a move operation |
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 196: | Line 196: | ||
| 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> | ||
| + | |||
| + | ---- | ||
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| + | **Graphical Summary** | ||
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| + | 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]]. | ||
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| + | [{{ :advanced_tools:group_theory:representation_theory:lorentz_irreps.jpg?nolink }}] | ||
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| + | 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]]. | ||
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| + | [{{ :advanced_tools:group_theory:representation_theory:rotation_to_lorentz.jpg?nolink }}] | ||
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
advanced_tools/group_theory/lorentz_group.1525420416.txt.gz · Last modified: 2018/05/04 07:53 (external edit)
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