Theory of groups for physics applications

Week 1

Introduction

The origins of Group theory is in Premutations (the algebra obeyed) and Geometry (rotations) corresponding to Discrete and Continuous groups respectively.
Geometric rotations, in general do not commute.
Continuous groups is essentially Trignomentry.
In quantum mechanics, symmetry group substitutes for the geometry of shape and size.

Algebraic preliminaries

Sets and maps

Mathematicians usually classify maps as:

Surjective/onto: Range is completely covered.
Injective/into: One to One, but need not exhaust range.
Bijective: Both one-to-one and covers the whole range.

Algebraic Structure: There exists a binary relation with properties:

commutatitive: $a \circ b$ = $b \circ a$
associative: $a \circ (b \circ c) = (a \circ b) \circ c$
- When not associative, Jacobi identity is used as an alternative.
identity element: $a \circ e = a = e \circ a$
inverse: $a \circ a^{-1} = e = a^{-1} \circ a$

Homomorphism: Maps which preserves algebraic structure.

Isomorphism: Maps which are bijective and preserves the algrbraic structure i.e., a map M: $S_1 \rightarrow S_2$ with algrbraic structures $a \circ_1 b \in S_1$ and $a’ \circ_2 b’ \in S_2$ has the property - $(a \circ_1 b) \rightarrow (a’ \circ_2 b’)$ if $a \rightarrow a’$ and $b \rightarrow b’$.

Groups

Group: A set which satisfies:

Closure: $\forall a, b \in S\ \exists\ a \circ b \in S$
Associative: $a \circ (b \circ c) = (a \circ b) \circ c$
Identity: $\exists\ e \in S$ such that $a \circ e = a = e \circ a$
Inverse: $\forall a \in S\ \exists\ a^{-1} \in S$ such that $a \circ a^{-1} = e = a^{-1} \circ a$

Examples of groups: set of matrices with determinant $\ne$ 0; set of all posible rotational configurations of a rigid object.

Abelian group: A group which satisfies an additional property - Commutativiy: $a \circ b$ = $b \circ a$. Example: Rotations in 2D plane - SO(2).

Linear vector space

A set $V$ with + (addition) and $\bullet$ (scalar multiplication) and an auxiliary set of scalars (s) which satisfies:

Abelian group under +
Under $\bullet$:
- $a \bullet v \in V$ when $a \in s$
- $a \bullet (v_1 + v_2) = a \bullet v_1 + a \bullet v_2$: multiplication is distributive
- Additionally, scalars have their own abelian structure with + so that $a_1+a_2 \in s$ etc.
- $(a_1+a_2) \bullet v = a_1 \bullet v + a_2 \bullet v$

Permutations

Permutation group is the group of all possible ways of rearranging n objects. The “possible ways” are elements of the group.

Any discrete group is a sub-group of some permutation group.

Can be represented as matrices.

Equivalence realtion

For a set $s$ a “relation” $R$ is a conditional about a,b etc. $\in s$ such that:

$a R a$ : a is always related to a: Reflexivity
$a R b \Rightarrow b R a$: Symmetry
$a R b$ and $b R c \Rightarrow a R c$: Tansitivity

Any relation $R$ with above properties is called an Equivalence relation. Example: “parallel” relation of st. lines in a plane is an equivalence relation ; “perpendicular” is not an equivalence relation (does not satisfy first req.)

Theorem: An equivalence relation divides a set into disjoint subsets whose union makes up the whole set $s$.

Proof: Let $s_1, s_2, …$ be some subsets. Let $s_1$ be such that all elements in it are related by R. Similarly, consider $s_2$. Now, $s_1 \cap s_2$ because $a \in s_1$ and $b \in s_2$ and hypothesis $a R b$ forces $s_1$ and $s_2$ to be same subsets due to transitivity and symmetry.

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