advanced_tools:group_theory:su2
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advanced_tools:group_theory:su2 [2018/03/17 16:01] jakobadmin [Student] |
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?> | ||
| - | $SU(2)$ 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> | + | 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 formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | <tabbox Concrete> |
| - | </note> | + | |
| - | | + | |
| - | <tabbox Student> | + | |
| 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 | ||
| - | $$ U(x) = e^{i f(x) \vec{r} \vec{\sigma} },$$ | + | $$ U(x) = e^{i a \vec{r} \vec{\sigma} } = \cos(a) + i \vec{r} \vec{\sigma} \sin( a )$$ |
| - | where $\vec{\sigma}=(\sigma_1,\sigma_2,\sigma_3)$ are the usual Pauli matrices and $ \vec{r} $ is a unit vector. The condition $U(x) \to 1$ for $|x| \to \infty $ therefore means $f(x) \to 2\pi n$ for $|x| \to \infty $, where $n$ is an arbitrary integer, because we can write the matrix exponential as | + | 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. |
| - | $$e^{i f(x) \vec{r} \vec{\sigma}} = \cos(f(x)) + i \vec{r} \vec{\sigma} \sin( f(x) ) .$$ | ||
| - | <tabbox Researcher> | ||
| - | * [[https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/|A Journey to The Manifold SU(2)]] | + | ---- |
| + | **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]]. | |
| - | <tabbox Examples> | + | |
| - | --> Example1# | + | [{{ :advanced_tools:group_theory:su2_qm_spin.jpg?nolink }}] |
| - | + | <tabbox Abstract> | |
| - | <-- | + | |
| - | --> Example2:# | + | * [[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> | ||
advanced_tools/group_theory/su2.1521298891.txt.gz · Last modified: 2018/03/17 15:01 (external edit)
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