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 advanced_tools:group_theory:su2 [2020/12/05 18:29]edi [Intuitive] advanced_tools:group_theory:su2 [2020/12/26 22:52]edi [Intuitive] Both sides previous revision Previous revision 2020/12/26 22:52 edi [Intuitive] 2020/12/05 18:35 edi [Concrete] 2020/12/05 18:29 edi [Intuitive] 2020/11/28 18:40 edi [Concrete] 2020/09/07 04:19 [Concrete] 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 Next revision Previous revision 2020/12/26 22:52 edi [Intuitive] 2020/12/05 18:35 edi [Concrete] 2020/12/05 18:29 edi [Intuitive] 2020/11/28 18:40 edi [Concrete] 2020/09/07 04:19 [Concrete] 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 3: Line 3:  ​  ​ - 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. Line 27: Line 27: **Representations** **Representations** - The diagram below shows the defining 2-dimensional representation of $SU(2)$, which describes spin-$1/2$ particles, ​in its upper branch. A 3-dimensional representations of the same group, which describes spin-$1$ particles, is shown in the lower branch. ​ For an 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/​|Fun with Symmetry]]. [{{ :​advanced_tools:​group_theory:​su2_qm_spin.jpg?​nolink }}] [{{ :​advanced_tools:​group_theory:​su2_qm_spin.jpg?​nolink }}]