Theory of groups for physics applications

Notes for the [[NPTEL lecture][https://onlinecourses.nptel.ac.in/noc18_ph11/course]] titled the same.

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:

  1. Closure: \(\forall a, b \in S\ \exists\ a \circ b \in S\)
  2. Associative: \(a \circ (b \circ c) = (a \circ b) \circ c\)
  3. Identity: \(\exists\ e \in S\) such that \(a \circ e = a = e \circ a\)
  4. 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.

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