advanced_tools:group_theory:su2
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advanced_tools:group_theory:su2 [2018/04/15 16:31] aresmarrero |
advanced_tools:group_theory:su2 [2020/12/05 18:29] edi [Intuitive] |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| + | The Lie group $SU(2)$ describes all possible rotations of a spinorial object, that is, an object that needs to be rotated 720 degrees before returning to its initial state. A good example for such an object is a cube that is attached to a wall by belts: see the animations here [[https://en.wikipedia.org/wiki/Spinor]]. In physics, an important spinorial object is the fermion (e.g., an electron). | ||
| - | <note tip> | + | For small rotations $SU(2)$ is identical to $SO(3)$, that is, both groups have the same Lie algebra. |
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | |
| - | </note> | + | |
| - | | + | |
| <tabbox Concrete> | <tabbox Concrete> | ||
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| Elements of $SU(2)$ can be written as | Elements of $SU(2)$ can be written as | ||
| - | $$ U(x) = e^{i a \vec{r} \vec{\sigma} }= e^{i a \vec{r} \vec{\sigma}} = \cos(a) + i \vec{r} \vec{\sigma} \sin( a )$$ | + | $$ U(x) = e^{i a \vec{r} \vec{\sigma} } = \cos(a) + i \vec{r} \vec{\sigma} \sin( a )$$ |
| - | where $\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$ are the usual Pauli matrices and $ \vec{r} $ is a unit vector. | + | where $\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$ are the usual Pauli matrices and $ \vec{r} $ is a unit vector. This is also known as the version of the the well-known Euler's identity for $2\times2$ matrices. |
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| **Representations** | **Representations** | ||
| - | [{{ :advanced_tools:group_theory:su2reps.png?nolink |Diagram by Eduard Sackinger}}] | + | The diagram below shows the defining 2-dimensional representation of $SU(2)$, which describes spin-$1/2$ particles, in its upper branch. A 3-dimensional representations of the same group, which describes spin-$1$ particles, is shown in the lower branch. For an explanation of this diagram and more representations of $SU(2)$ see [[https://esackinger.wordpress.com/|Fun with Symmetry]]. |
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| + | [{{ :advanced_tools:group_theory:su2_qm_spin.jpg?nolink }}] | ||
| <tabbox Abstract> | <tabbox Abstract> | ||
advanced_tools/group_theory/su2.txt · Last modified: 2023/04/17 03:23 by edi
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