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advanced_tools:group_theory:su2 [2018/04/15 16:31]
aresmarrero
advanced_tools:group_theory:su2 [2023/04/17 03:23] (current)
edi [Concrete]
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 <tabbox Intuitive> ​ <tabbox Intuitive> ​
 +The Lie group $SU(2)$ describes all possible 3D 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 formulasbut 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 $SU(2)$ in its upper branch and a 3-dimensional representations of the same group in the lower branch. An important application of these two representations is the rotation of the quantum state of a spin-1/2 and a spin-1 particle, respectively. For a more detailed explanation of this diagram and more representations of $SU(2)$ see [[https://​esackinger.wordpress.com/​blog/​lie-groups-and-their-representations/#​su2_qm_spin|Fun with Symmetry]]. 
 + 
 +[{{ :​advanced_tools:​group_theory:​su2_qm_spin.jpg?nolink }}]
  
 <tabbox Abstract> ​ <tabbox Abstract> ​
advanced_tools/group_theory/su2.1523802697.txt.gz · Last modified: 2018/04/15 14:31 (external edit)