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advanced_tools:group_theory:su2

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advanced_tools:group_theory:su2 [2018/04/15 14:30]
aresmarrero [Student]
advanced_tools:group_theory:su2 [2018/04/15 14:31] (current)
aresmarrero
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 ====== SU(2) ====== ====== SU(2) ======
  
-<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 ​Intuitive
-<​tabbox ​Layman+
  
 <note tip> <note tip>
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 </​note>​ </​note>​
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-<​tabbox ​Student+<​tabbox ​Concrete
  
 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 $S^3$. 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 $S^3$.
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 [{{ :​advanced_tools:​group_theory:​su2reps.png?​nolink |Diagram by Eduard Sackinger}}] [{{ :​advanced_tools:​group_theory:​su2reps.png?​nolink |Diagram by Eduard Sackinger}}]
  
-<​tabbox ​Researcher+<​tabbox ​Abstract
  
   * [[https://​www.physicsforums.com/​insights/​journey-manifold-su2mathbbc-part/​|A Journey to The Manifold SU(2)]]   * [[https://​www.physicsforums.com/​insights/​journey-manifold-su2mathbbc-part/​|A Journey to The Manifold SU(2)]]
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-<​tabbox ​Examples+<​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. ​
  
---> Example1# +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]]. ​
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---> Example2:+
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-<tabbox FAQ>  +
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-<tabbox History> ​+
  
 </​tabbox>​ </​tabbox>​
  
  
advanced_tools/group_theory/su2.txt · Last modified: 2018/04/15 14:31 by aresmarrero