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advanced_tools:group_theory:quotient_group [2017/12/17 12:59] jakobadmin [Student] |
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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<tabbox Examples> | <tabbox Examples> | ||
- | --> Example1# | + | --> Goldstone Bosons# |
- | + | 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: | ||
+ | |||
+ | For an $SO(N)$ gauge theory with a fundamental scalar representation $\phi$ the scalar potential reads | ||
+ | |||
+ | $$ V(\phi) = \frac{\lambda}{4}(\phi^2-v^2). $$ | ||
+ | |||
+ | The fundamental representation acts on $\mathbb{R}^N$. Thus, we can divide $\mathbb{R}^N$ into **orbits** by using the group action. Each orbit consists of all the points that can be transformed into each other through $SO(N)$ transformations. These orbits are are $N-1$ dimensional spheres (except for the one-dimensional orbit that consist of the origin that isn't moved around at all by $SO(N)$ transformations). | ||
+ | |||
+ | We denote the radius of such a sphere by $\rho$ and with $\varphi_\alpha$, where $\alpha=1, \ldots, N$ the coordinates on the sphere (= angles). | ||
+ | |||
+ | Now, we assume the scalar representation develops a non-zero vacuum expectation value, i.e. breaks the $SO(N)$ symmetry. Scalars in the fundamental representation always break $SO(N)$ to $SO(N-1)$, because any VEV can be brought into the following form $<\phi> = (0,\ldots, v)$ through $SO(N)$ transformations. | ||
+ | |||
+ | The radius $\rho$ is then the Higgs field while the $\varphi_\alpha$ are the Goldstone bosons. This way, we can see that the Goldstone bosons are simply gauge degrees of freedom, while the Higgs field is not. (We can move around on a sphere $S^{N-1}$ through $SO(N)$ transformations). | ||
+ | |||
+ | In this sense, the Goldstone bosons live on the sphere $S^{N-1}$, which is exactly the coset space | ||
+ | |||
+ | $$ SO(N)/SO(N-1) = S^{N-1} . $$ | ||
+ | |||
+ | Source: https://arxiv.org/pdf/0910.5167v1.pdf | ||
+ | |||
+ | See also: http://physics.stackexchange.com/questions/108722/in-what-sense-do-goldstone-bosons-live-in-the-coset | ||
<-- | <-- | ||
- | --> Example2:# | + | --> $\mathbb{Z}/3\mathbb{Z}$# |
+ | |||
+ | $\mathbb{Z}$ is the set of all integers | ||
+ | |||
+ | $$ \mathbb{Z} = \{ \ldots, -3,-2,-1,0,1,2,3,\ldots \} $$ | ||
+ | |||
+ | and forms a group under addition. | ||
+ | |||
+ | We can divide this set into subsets, for example, using as a criterion their remainder after a division by $3$. This way $\mathbb{Z}$ gets divided into three distinct subsets: | ||
+ | |||
+ | $$ 3\mathbb{Z} = \{ \ldots, ,-6,-3,0,3,6,\ldots \} =[0], $$ | ||
+ | $$ 3\mathbb{Z}+1 = \{ \ldots, ,-5,-2,1,4,7,\ldots \}=[1], $$ | ||
+ | $$ 3\mathbb{Z}+2 = \{ \ldots, ,-4,-1,2,5,8,\ldots \}=[2]. $$ | ||
+ | |||
+ | The three sets $[0],[1],[2]$ form a group under addition. For example, $[1]+[2]=[0]$. This means, we can take any element of $[1]$ and add to it an element of $[2]$ and this always yields an element of $[1]$. | ||
+ | |||
+ | The set $\{[0],[1],[2] \}$ is called the quotient group | ||
+ | $$ \mathbb{Z}/3\mathbb{Z} = \{[0],[1],[2] \} $$ | ||
+ | |||
+ | The subgroup $3 \mathbb{Z}$ of $\mathbb{Z}$ becomes the identity element $[0]$ of the quotient group. The sets $[0]$, $[1]$ and $[3]$ are the cosets of $3 \mathbb{Z}$ in $\mathbb{Z}$ and $[0]$ is called the trivial coset. | ||
+ | |||
+ | <WRAP tip>This means that $ \mathbb{Z}/3\mathbb{Z}$ are all integers that aren't multiples of $3$, i.e. all integers that do not have the defining property of the set $3 \mathbb{Z}$. The cosets are defined as subsets of the integer number which are group together how much they do not satisfy the defining property of the set $3 \mathbb{Z}$. For example, the integers in the coset $[1]$ fail to be elements of $3 \mathbb{Z}$ by $1$ and $[2]$ fail to be elements of $3 \mathbb{Z}$ by $2$. </WRAP> | ||
+ | |||
+ | Source: https://www.quora.com/Whats-an-intuitive-explanation-of-quotient-group | ||
+ | <-- | ||
+ | |||
+ | --> $SO(3)/SO(n-1)$ # | ||
+ | |||
+ | $SO(n)$ is the set of all rotations in $\mathbb{R}^n$. Such rotations can be understood by investigating how points on a sphere get moved around. For example, $SO(3)$ moves points on the two-sphere $S^2$ into each other. In contrast, $SO(2)$ described rotations in $\mathbb{R}^2$ and therefore acts on the unit circle $S^1$. | ||
+ | |||
+ | Now, we are interested in the coset $g SO(2)$ for some fixed $g \in SO(3)$. If we say $SO(2)$ described rotations around some arbitrary, but fixed axis like the $z$-axis, we can say roughly that $SO(3)/SO(2)$ contains all three dimensional rotations that do not rotate around the $z$-axis. | ||
+ | |||
+ | For concreteness, we pick an arbitrary base point, say the north pole $\hat{e}_z= (0,0,1)$ on the two-sphere $S^2$. This point can be rotated into any other point $\hat u$ on $S^2$. However, it is not possible to identify points on $S^2$ with elements of $SO(3)$. The reason is that there are several rotations that rotate the north pole $\hat{e}_z$ into $\hat u$. Say an element $g_1$ of $SO(3)$ rotates $\hat{e}_z$ into $\hat u$: | ||
+ | |||
+ | $$ g_1 \hat{e}_z = \hat u . $$ | ||
+ | |||
+ | We can then immediately write down another rotation that rotates $\hat{e}_z$ into $\hat u$ by combining $g_1$ with a rotation $h$ around the $z$-axis: | ||
+ | $$ g_2 := g_1 h$$ | ||
+ | $$ g_2 \hat{e}_z = g_1 h \hat{e}_z = \hat u ,$$ | ||
+ | because the north pole is unchanged under rotations around the $z$-axis: $ h\hat{e}_z = \hat{e}_z$. | ||
+ | |||
+ | Rotations in a plane, like, for example, rotations in the $x-y$ plane (= rotations around the $z$-axis) are described by $SO(2)$. | ||
+ | |||
+ | 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 [[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). | ||
+ | |||
+ | $$S^{n} = \{ x \in \mathbb{R}^n : |x|=1\}$$ | ||
+ | |||
+ | Source: Bott, R. and Tu, L. W., Differential Forms in Algebraic Topology, Springer (1982) page 195 and [[https://books.google.de/books?id=5Dy1hlKvmCYC&lpg=PA590&ots=BQWJTqWzCH&dq=SO(3)%2FSO(2)%20equivalence%20classes&hl=de&pg=PA590#v=onepage&q&f=false|page 590 in Zees' Einstein Gravity in a Nutshell]]. | ||
+ | |||
+ | |||
+ | |||
+ | <-- | ||
+ | |||
+ | --> $GL(\mathbb{R},n)/SL(\mathbb{R},n)$ # | ||
+ | |||
+ | The group $GL(\mathbb{R},n)$ consist of all invertible $n \times n$ matrices and is called the general linear group. The subgroup $SL(\mathbb{R},n) \subset GL(\mathbb{R},n)$ consist of all invertible $n \times n$ matrices with determinant $1$. Now, a (slightly wrong but) helpful view is that $GL(\mathbb{R},n)/SL(\mathbb{R},n)$ is the set of all invertible $n \times n$ matrices with determinant not equal to $1$. | ||
+ | |||
+ | Each coset $gSL(\mathbb{R},n)$ for $g\in GL(\mathbb{R},n)$ consist of all invertible $n \times n$ matrices with a given value for the determinant. For example, all matrices with determinant $2$ live in the same coset and all matrices with determinant $\sqrt{3}$ live in another coset. (Two matrices $A$ and $B$ live in the same coset if $\det(AB^{-1})=1$, which is only true for $\det(A)= \det(B)$ because $\det(AB^{-1})=\det(A)\det(B^{-1})= \det(A)\det(B)^{-1}$). | ||
+ | |||
+ | This can be understood as follows: | ||
+ | |||
+ | We pick an element $g$ of $GL(\mathbb{R},n)$ and the corresponding coset $gSL(\mathbb{R},n)$ is the set of all matrices that we get when we act on $g$ with elements of $SL(\mathbb{R},n)$. Because $SL(\mathbb{R},n)$ is the set of invertible $n\times n$ matrices with determinant $1$ and $\det(AB) = \det(A)\det(B)$ holds it follows that all elements of a given coset have the same determinant. With a slight abuse of notation: | ||
+ | |||
+ | $$ \det(gSL(\mathbb{R},n))= \det(g) \det(SL(\mathbb{R},n)) = \det(g) 1 = \det(g)$$ | ||
+ | |||
+ | Again, as in the $\mathbb{Z}/3\mathbb{Z}$ example, the cosets are elements that are grouped together depending on how far off they are from the defining condition of the subgroup that gets modded out. The cosets $gSL(\mathbb{R},n)$ are sets that are grouped depending on how far away their determinant is from $1$. Moreover, again the subgroup that gets modded out, in this case $SL(\mathbb{R},n)$ becomes the identity element of the quotient group. | ||
+ | |||
+ | Each coset is an equivalence class labelled through the value of the determinant. For the elements of $GL(\mathbb{R},n)$ the determinant can be any real number and therefore we have a one-to-one correspodence between the set of all cosets and $\mathbb{R}^n$: | ||
+ | |||
+ | $$ GL(\mathbb{R},n)/SL(\mathbb{R},n) \cong \mathbb{R}^n$$. | ||
+ | |||
+ | Source: http://www.math3ma.com/mathema/2016/12/15/a-quotient-of-the-general-linear-group-intuitively | ||
+ | |||
<-- | <-- | ||