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 [Examples] |
advanced_tools:group_theory:quotient_group [2023/07/29 01:41] (current) 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 40: | Line 40: | ||
**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 |
$$ G/N=\{gN:g\in G\} $$ | $$ G/N=\{gN:g\in G\} $$ | ||
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: | ||
Line 136: | Line 136: | ||
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). |