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 advanced_tools:group_theory:su2 [2018/04/15 16:30]aresmarrero [Student] advanced_tools:group_theory:su2 [2018/04/15 16:31] (current)aresmarrero Both sides previous revision Previous revision 2018/04/15 16:31 aresmarrero 2018/04/15 16:30 aresmarrero [Student] 2018/03/17 16:03 jakobadmin [Student] 2018/03/17 16:02 jakobadmin [Student] 2018/03/17 16:01 jakobadmin [Student] 2017/12/17 12:02 jakobadmin [Why is it interesting?] 2017/12/17 11:59 jakobadmin [Student] 2017/12/04 09:01 external edit2017/11/03 14:10 jakobadmin created 2018/04/15 16:31 aresmarrero 2018/04/15 16:30 aresmarrero [Student] 2018/03/17 16:03 jakobadmin [Student] 2018/03/17 16:02 jakobadmin [Student] 2018/03/17 16:01 jakobadmin [Student] 2017/12/17 12:02 jakobadmin [Why is it interesting?] 2017/12/17 11:59 jakobadmin [Student] 2017/12/04 09:01 external edit2017/11/03 14:10 jakobadmin created Line 1: Line 1: ====== SU(2) ====== ====== SU(2) ====== -  ​ - $SU(2)$ 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> + Line 11: Line 8: ​ ​ ​ - <​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 $S^3$. Line 34: Line 31: [{{ :​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)]] Line 40: Line 37: ​ ​ - <​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]]. ​ - + - + - <-- + - + - --> Example2:# + - + - + - <-- + - + - + - + -  ​ +