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∘b = b∘a
• associative: a∘(b∘c)=(a∘b)∘c
  • When not associative, Jacobi identity is used as an alternative.
• identity element: a∘e=a=e∘a
• inverse: a∘a−1=e=a−1∘a

Homomorphism: Maps which preserves algebraic structure.

Isomorphism: Maps which are bijective and preserves the algrbraic structure i.e., a map M: S1→S2 with algrbraic structures a∘1b∈S1 and a′∘2b′∈S2 has the property - (a∘1b)→(a′∘2b′) if a→a′ and b→b′.

Groups

Group: A set which satisfies:

1. Closure: ∀a,b∈S ∃ a∘b∈S
2. Associative: a∘(b∘c)=(a∘b)∘c
3. Identity: ∃ e∈S such that a∘e=a=e∘a
4. Inverse: ∀a∈S ∃ a−1∈S such that a∘a−1=e=a−1∘a

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

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

Linear vector space

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

• Abelian group under +
• Under ∙:
  • a∙v∈V when a∈s
  • a∙(v1+v2)=a∙v1+a∙v2: multiplication is distributive
  • Additionally, scalars have their own abelian structure with + so that a1+a2∈s etc.
  • (a1+a2)∙v=a1∙v+a2∙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. ∈s such that:

• aRa : a is always related to a: Reflexivity
• aRb⇒bRa: Symmetry
• aRb and bRc⇒aRc: 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 s1 be such that all elements in it are related by R. Similarly, consider s2. Now, s1∩s2 because a∈s1 and b∈s2 and hypothesis aRb forces s1 and s2 to be same subsets due to transitivity and symmetry.

Week 2

Lagrange’s Theorem:

If H is a subgroup

Reference books:

• Morton Hammermesh
• Sadri Hassani Ch. 23 and 24
• Brian C. Hall
• Ramadevi’s draft book for applications