advanced_tools:group_theory:quotient_group
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advanced_tools:group_theory:quotient_group [2017/12/17 13:01] jakobadmin [Examples] |
advanced_tools:group_theory:quotient_group [2018/05/15 06:58] jakobadmin ↷ Links adapted because of a move operation |
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | Quotient groups are crucial to understand, for example, [[advanced_notions:symmetry_breaking|symmetry breaking]]. When a [[advanced_tools:group_theory|group]] $G$ breaks to a [[advanced_tools:group_theory:subgroup|subgroup]] $H$ the resulting [[advanced_notions:symmetry_breaking:goldstones_theorem|Goldstone bosons]] live in the quotient space: $G/H$. | + | Quotient groups are crucial to understand, for example, [[advanced_notions:symmetry_breaking|symmetry breaking]]. When a [[advanced_tools:group_theory|group]] $G$ breaks to a [[advanced_tools:group_theory:subgroup|subgroup]] $H$ the resulting [[theorems:goldstones_theorem|Goldstone bosons]] live in the quotient space: $G/H$. |
| Moreover, quotient groups are a powerful way to understand geometry. Instead of a long list of axioms one can study geometry by treating the corresponding space as a homogeneous space (= coset space) and then study invariants of transformation groups of this homogeneous space. | Moreover, quotient groups are a powerful way to understand geometry. Instead of a long list of axioms one can study geometry by treating the corresponding space as a homogeneous space (= coset space) and then study invariants of transformation groups of this homogeneous space. | ||
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| **Definition:** | **Definition:** | ||
| - | For a group $G$ and a normal subgroup of it $N$, we call | + | For a group $G$ and a [[advanced_tools:group_theory:subgroup|normal subgroup]] of it $N$, we call |
| $$ G/N=\{gN:g\in G\} $$ | $$ G/N=\{gN:g\in G\} $$ | ||
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| --> Goldstone Bosons# | --> Goldstone Bosons# | ||
| - | The famous [[advanced_notions:symmetry_breaking:goldstones_theorem|Goldstone bosons]] that appear through the process of spontaneous [[advanced_notions:symmetry_breaking|symmetry breaking]] of a group $G$ to some subgroup $H$ live in the coset space $G/H$. | + | The famous [[theorems:goldstones_theorem|Goldstone bosons]] that appear through the process of spontaneous [[advanced_notions:symmetry_breaking|symmetry breaking]] of a group $G$ to some subgroup $H$ live in the coset space $G/H$. |
| This can be understood by considering an explicit example: | This can be understood by considering an explicit example: | ||
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| In this sense, when we mod out $SO(2)$ rotations from $SO(3)$, we can identify elements of the resulting $SO(3)/SO(2)$ with elements of $S^2$. Without the $SO(2)$ rotations, we have a one-to-one correspondence between the remaining rotations ( = elements of $SO(3)/SO(2)$) and the two-sphere $S^2$. To every point on $S^2$ there is a unique element of $SO(3)/SO(2)$, namely the rotation that rotates, for example, the north pole into this point. | In this sense, when we mod out $SO(2)$ rotations from $SO(3)$, we can identify elements of the resulting $SO(3)/SO(2)$ with elements of $S^2$. Without the $SO(2)$ rotations, we have a one-to-one correspondence between the remaining rotations ( = elements of $SO(3)/SO(2)$) and the two-sphere $S^2$. To every point on $S^2$ there is a unique element of $SO(3)/SO(2)$, namely the rotation that rotates, for example, the north pole into this point. | ||
| - | Take note that $S^2$ is not a Lie group, because $SO(2)$ is not a [[group_theory:notions:subgroups#normal_subgroups|normal subgroup]] of $SO(3)$. | + | Take note that $S^2$ is not a Lie group, because $SO(2)$ is not a [[advanced_tools:group_theory:subgroup|normal subgroup]] of $SO(3)$. |
| In general, the quotient space $SO(n)/SO(n-1)$ is $S^{n-1}$ (= the $n-1$-sphere). | In general, the quotient space $SO(n)/SO(n-1)$ is $S^{n-1}$ (= the $n-1$-sphere). | ||
advanced_tools/group_theory/quotient_group.txt · Last modified: 2023/07/29 01:41 by 38.114.114.173
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