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advanced_tools:group_theory:su2 [2017/11/03 14:10]
jakobadmin created
advanced_tools:group_theory:su2 [2023/04/17 03:23] (current)
edi [Concrete]
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 ====== SU(2) ====== ====== SU(2) ======
  
-<tabbox Why is it interesting?> ​ 
  
-<​tabbox ​Layman+<​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. +<​tabbox ​Concrete
-</​note>​ +
-  ​ +
-<​tabbox ​Student+
  
-<note tip> +Every $SU(2)$ transformation can be written as $$ g(x) = a_0(x) 1 + i a_i(x) \sigma ,$$ where $\sigma$ are the Pauli matrices. The defining conditions of $SU(2)$ are $g(x)^\dagger g(x)=1$ ​and $det(g(x)=1$and thus we have $$ (a_0)^2 +a_i^2=1 ​$$ which is the defining condition of $S^3$.
-In this section things should ​be explained by analogy and with pictures ​and, if necessarysome formulas. +
-</​note>​ +
-  +
-<tabbox Researcher> ​+
  
-  * [[https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/|A Journey to The Manifold SU(2)]]+Source: page 23 in http://www.iop.vast.ac.vn/theor/conferences/vsop/​18/​files/​QFT-4.pdf
  
 +This is also shown nicely at page 164 in the book Magnetic Monopoles by Shnir.
  
-   +----
-<tabbox Examples> ​+
  
---> Example1#+Elements of $SU(2)$ can be written as
  
-  +$$ U(x) = e^{i a \vec{r} \vec{\sigma} } = \cos(a) + i \vec{r} \vec{\sigma} \sin( a )$$
-<--+
  
---> Example2:#+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. 
 + 
 + 
 +---- 
 + 
 +**Representations** 
 + 
 +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>​  
 + 
 +  * [[https://​www.physicsforums.com/​insights/​journey-manifold-su2mathbbc-part/​|A Journey to The Manifold SU(2)]]
  
-  
-<-- 
  
-<tabbox FAQ> ​ 
   ​   ​
-<​tabbox ​History+<​tabbox ​Why is it interesting?​ 
 +$SU(2)$ is one of the most important groups in modern physics. The group is a crucial ingredient of the [[advanced_tools:​gauge_symmetry|gauge symmetry]] of the [[models:​standard_model|standard model]] of particle physics and, in some sense, explains the structure of weak interactions.  
 + 
 +In addition, the fundamental spacetime symmetry group called Lorentz group, is usually analyzed in terms of its Lie algebra. This Lie algebra can be understood as two copies of the $SU(2)$ Lie algebra. Hence, by understanding the Lie algebra of $SU(2)$, we understand almost everything about the Lorentz group. This analysis is crucial for the understanding what [[basic_notions:​spin|spin]] is, which is one of the most important properties of [[advanced_notions:​elementary_particles|elementary particles]]. ​
  
 </​tabbox>​ </​tabbox>​
  
  
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