This is a study note of Group Theory and Quantum Mechanics. It is not complete and maybe never will be. I write this just because I have written it neatly on papers and it looks good. So I decided to turn it into a more convenient and explicit electronic version. It’s converted from picture using OCR, so there may be some typo I didn’t notice. Don’t try to ask me questions. I don’t know.
2 Abstract Group Theory
2.1 Definition of Groups
A group is a set of elements that satisfy group multiplication, therefore must satisfy:
- Closed. The product of two elements is in the set. ie $A B=C$
- Associative. $A(B C)=(A B) C$
- Identity. Unitary element must in set. $E A=A E=A$
- Inverse. There is an inverse of each element. $A^{-1} A=A A^{-1}=E$
The number of elements $h$ is called order.
If group multiplication is commutative, i.e. $A B=B A$ the group is called Abelian Group.
2.2 Rearrangement Theorem
In the multiplication table, each element appears in each column or row once and only once.
2.3 Subgroups ard Cosets
A subgroup is a subset of a larger group, and itself is a group.
Let $\mathscr{S}=E, S_2, S_3, \cdots, S_{g}$ be a subgroup of a larger group $\mathscr{G}$ of order $h$.
Then we call the set of $g$ elements $EX, S_2X, S_3X, \cdots, S_{g}X$ a right coset $\mathscr{S}X$, if $X$ not in $\mathscr{S}$. Similary, we can define left coset $X \mathscr{S}$
$\rightarrow$ cosets cannot be subgroups, since they cannot have identity element $E$.
$\rightarrow$ A coset $\mathscr{S}X$ contains no common elements with subgroup $\mathscr{S}$
$\rightarrow$ Two right (or left) cosets of subgroup $\mathscr{S}$ in $\mathscr{G}$ either are identical or have no common elements.
The order $g$ of a subgroup $\mathscr{S}$ must be an integer divisor of the order $h$ of entire group $\mathscr{G}$, i.e. $h/g=l,$ where $l$ is called the index of the subgroup $\mathscr{S}$ in $\mathscr{G}$
Proof:
Each of the $h$ elements of $\mathscr{G}$ must appear either in $\mathscr{S}$ or in $\mathscr{S}X$, for some $X$. Thus each element must appear in one of the sets $\mathscr{S},\mathscr{S}X_2,\mathscr{S}X_3,\cdots, \mathscr{S}X_l$. And we know there are no common elements within these $l$ sets. Hense, the $l$ sets averagely divide $\mathscr{G}$, i.e. $h=l \times g$
Ps. $\mathscr{S}$ is letter $S$, the LaTeX code is
$\mathscr{S}$. Similary, $\mathscr{G},\mathscr{K},\mathscr{R}$ are $G,K,R$ respectively.
2.4 Conjugate Elements and Class Structure
An element $B$ is said to be conjugate to $A$ if
\[B=XAX^{-1}\]where $X$ is some mumber of the group.
$\rightarrow$ If $B$ is conjugate to $A$, then $A$ is conjugate to $B$. Since $A=X^{-1} B X$
$\rightarrow$ If $B$ and $C$ are both conjugates to $A$, then $B$ and $C$ are conjugate to each other.
The collection of all mutually conjugate elements is called a class.
$\rightarrow$ In Abelian groups, each element is in a class by itself. Since $X A X^{-1}=A X^{-1} X=A$, i.e. element $A$ can only be conjugated to itself.
2.5 Normal Divisors and Factor Groups
If a subgroup $\mathscr{S}$ of a larger group $\mathscr{G}$ consists entirely of complete classes, it is called an invariant subgroup or normal divisors, ie if $A$ in $\mathscr{G}$, then all $X A X^{-1}$ in $\mathscr{G}$.
Now introduce the notion of complex.
$\mathscr{K}=\left(K_{1}, K_{2}, \cdots, K_{n}\right)$ is a colllection of group elements disregarding order.
$\mathscr{K} X=\left(K_1 X, K_2 X, \cdots, K_{n} X\right)$
$\mathscr{K}\mathscr{R}=\left(K_1 R_{1}, K_{1} R_{2},\cdots, K_{n} R_{m}\right)$
Elements are considered to be included only once, regardless of how often they are generated.
Then we can state our argument using complexes.
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$\mathscr{S}\mathscr{S}=\mathscr{S}$ closure of groups.
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$X^{-1}\mathscr{S}X=\mathscr{S}$ for invariant subgroup $\mathscr{S}$ and all $X$ in $\mathscr{G}$
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$\mathscr{S}X=X\mathscr{S}$ left and right cosets of an invariant subgroup are identical.
In Sec.02-3 we prove that there are a finite number $(l-1)$ of distinct cosets for any subgroup $\mathscr{S}$. The collection of $\mathscr{S}$ ard the $(l-1)$ distinct cosets can themselves be regarded as elements of a new group.
This group $\mathscr{S},\mathscr{S}X_2,\mathscr{S}X_3,\cdots, \mathscr{S}X_l$ is called the factor group of $\mathscr{G}$ with respect to the normal divisor $\mathscr{S}$. In this factor group, $\mathscr{S}$ forms the unit element, since
\[\mathscr{S}\mathscr{K}_i = \mathscr{S} (\mathscr{S} {K}_i) = (\mathscr{S}\mathscr{S}) {K}_i = \mathscr{S} K_i =\mathscr{K}_i\]2.6 Isomorphy and homomorphy
Two groups are said to be isomorphy if there exist a one-to-one correspondence between the elements $A, B,\cdots$ of one group and elements $A^{\prime}, B^{\prime},\cdots$ of the other group, such that $A B=C$ implies $A^{\prime} B^{\prime}=C^{\prime},$ and vice versa.
$\rightarrow$ Tuo groups are isomorphy when they have the same multiplication table.
Two groups are said to be homomorphy if there exist a correspondence between two groups that $A \leftrightarrow A_{1}^{\prime} A_{2}^{\prime}, \cdots, \quad B \leftrightarrow B_{1}^{\prime}, B_{2}^{\prime}, \ldots, \quad C \leftrightarrow C_{1}^{\prime}, C_{2}^{\prime}, \ldots \quad$ such that $A B=C$ implies $A_i^\prime B_j^\prime $will be a number of the set $C_{k}^{\prime}$
In general, a homomorphism is a many-to-one correspondence. It specializes to isomorphism if the correspondence is one-to-one.
2.7 Class Multiplication
! In this section, $\mathscr{R} = \mathscr{K} $ implies that each element appear as often in $\mathscr{R}$ as in $\mathscr{K}$.
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3 Theory of Group Representation
3.1 Representation of an abstract group
A rep of an abstract group is in general any group composed of concrete mathematical entities which is homomorphic to the abstract group. However, we are interested in square matrices, i.e we associate each group element $A \leftrightarrow \hat{\Gamma}(A)$ with a matrix such that
\[\hat{\Gamma}(A) \hat{\Gamma}(B)=\hat{\Gamma}(A B)\]$\rightarrow$ Clearly, $\hat{\Gamma}(E)=\hat{E}$ or $\hat{1}$
$\rightarrow$ number of rows or columns of the matrix is called the dimensionality of the representation.
If each matrix is different, then the two groups are isomorphic (rather then only homomorphic), and the representation is said to be true or faithful.
If several elements correspond to a single matrix, then all elements corresponding to the unit matrix $\hat{E}$ form an invariant subgroup of the full group. And elements corresponding to other matrices rather than $\hat{E}$ form the distinct cosets of that invariant group. And these matrices form a true representation of the factor group of this invariant subgroup.
If a similarity transformation leaves matrix equations unchanged, i.e.
\[\begin{aligned} \hat{\Gamma}(A) &=\hat{S}^{-1} \Gamma(A) \hat{S}\\ \downarrow \\ \hat{\Gamma} (A) \hat{\Gamma} (B) &= \left[\hat{S}^{-1}\Gamma (A) \hat{S} \right] \left[\hat{S}^{-1}\Gamma (B) \hat{S} \right] =\hat{S}^{-1} \hat{\Gamma} (A)\hat{\Gamma} (B)\hat{S} = \hat{\Gamma} (AB) \end{aligned}\]then the transformed matrics $\hat{\Gamma}^{\prime}$ form a representation, as long as $\Gamma$ do, ie they are equivalent.
Reference: Tinkham, M. Group Theory and Quantum Mechanics.
Author: Mike Lyou
Link: https://blog.mikelyou.com/2020/03/22/group-theory/
Liscense: This work is licensed under a CC BY-NC 4.0 International License.
