advanced_tools:group_theory:quotient_group
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
advanced_tools:group_theory:quotient_group [2017/12/17 13:01] jakobadmin |
advanced_tools:group_theory:quotient_group [2023/07/29 01:41] 38.114.114.173 [Student] |
||
|---|---|---|---|
| Line 3: | Line 3: | ||
| <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. | ||
| Line 19: | Line 19: | ||
| <tabbox Student> | <tabbox Student> | ||
| - | A helpful (but slightly wrong) way to think about $G/N$ is that is consist of all elements in $G%$ that are not elements of $N$. A subgroup $N$ is defined through some special condition that its members must fulfil. Formulated differently: elements of $N$ are elements of $G$ that have some special property. | + | A helpful (but slightly wrong) way to think about $G/N$ is that it consist of all elements in $G%$ that are not elements of $N$. A subgroup $N$ is defined through some special condition that its members must fulfil. Formulated differently: elements of $N$ are elements of $G$ that have some special property. |
| For example, the subgroup $n\mathbb{Z}$ of $\mathbb{Z}$ consist of all integers that are a multiple of $n$. (More explicitly: members of $3\mathbb{Z}\subset\mathbb{Z}$ are all integers that are divisible by $3$). Or another example: the subgroup $SO(N)$ of $O(N)$ consists of all $N \times N$ matrices with determinant equal to $1$. Now, $G/N$ consist of all elements that do not have this extra property. //This isn't really correct.// | For example, the subgroup $n\mathbb{Z}$ of $\mathbb{Z}$ consist of all integers that are a multiple of $n$. (More explicitly: members of $3\mathbb{Z}\subset\mathbb{Z}$ are all integers that are divisible by $3$). Or another example: the subgroup $SO(N)$ of $O(N)$ consists of all $N \times N$ matrices with determinant equal to $1$. Now, $G/N$ consist of all elements that do not have this extra property. //This isn't really correct.// | ||
| Line 64: | Line 64: | ||
| --> 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: | ||
advanced_tools/group_theory/quotient_group.txt · Last modified: 2023/07/29 01:41 by 38.114.114.173
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
