advanced_tools:group_theory:su2
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advanced_tools:group_theory:su2 [2020/12/05 18:35] edi [Concrete] |
advanced_tools:group_theory:su2 [2023/04/17 03:23] (current) edi [Concrete] |
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| - | The Lie group $SU(2)$ describes all possible 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). | + | 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). |
| For small rotations $SU(2)$ is identical to $SO(3)$, that is, both groups have the same Lie algebra. | For small rotations $SU(2)$ is identical to $SO(3)$, that is, both groups have the same Lie algebra. | ||
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| **Representations** | **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/|Fun with Symmetry]]. | + | 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 }}] | [{{ :advanced_tools:group_theory:su2_qm_spin.jpg?nolink }}] | ||
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