advanced_tools:group_theory:su2
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advanced_tools:group_theory:su2 [2018/04/15 16:30] aresmarrero [Student] |
advanced_tools:group_theory:su2 [2020/09/07 04:19] 14.161.7.200 [Concrete] |
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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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| 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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| [{{ :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]]. |
| - | + | ||
| - | + | ||
| - | <-- | + | |
| - | + | ||
| - | --> Example2:# | + | |
| - | + | ||
| - | + | ||
| - | <-- | + | |
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| - | <tabbox FAQ> | + | |
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| - | <tabbox History> | + | |
| </tabbox> | </tabbox> | ||
advanced_tools/group_theory/su2.txt · Last modified: 2023/04/17 03:23 by edi
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