Unit 5 Notes - DSC 401 Physics Semester 7

Group theory is a fundamental branch of mathematics that studies algebraic structures known as groups. Groups are sets equipped with a binary operation that combines any two elements to form a third element while satisfying certain axioms. This field is crucial in various areas of physics, chemistry, and computer science for understanding symmetries.

Abstract Groups

An abstract group is a set G combined with a binary operation * (such as addition or multiplication) that satisfies four fundamental axioms:

Closure: For all elements a, b in G, the result of a * b is also in G.
Associativity: For all elements a, b, c in G, (a * b) * c = a * (b * c).
Identity Element: There exists an element e in G such that for every element a in G, a * e = e * a = a.
Inverse Element: For every element a in G, there exists an element a^-1 in G such that a * a^-1 = a^-1 * a = e.

Subgroups

A subset H of a group G is a subgroup of G if H itself forms a group under the same operation as G.

To check if a subset H is a subgroup of G, one can use the two-step subgroup test:

H is non-empty.
For any a, b ∈ H, a * b^-1 ∈ H.

Alternatively, the one-step subgroup test states that a non-empty subset H is a subgroup if for every a, b ∈ H, a * b^-1 ∈ H. For finite groups, if a non-empty subset H is closed under the operation, it is a subgroup.

Classes (Conjugacy Classes)

Two elements a and b in a group G are said to be conjugate if there exists an element g ∈ G such that b = g * a * g^-1. The set of all elements conjugate to an element a forms a conjugacy class of a.

Properties of conjugacy classes:

Conjugacy is an equivalence relation (reflexive, symmetric, transitive).
The conjugacy classes partition the group G.
The identity element forms a conjugacy class by itself: {e}.
In an Abelian group, each element forms its own conjugacy class.

Cosets

Given a group G and a subgroup H, for any element g ∈ G, the set gH = {g * h | h ∈ H} is called a left coset of H in G. Similarly, Hg = {h * g | h ∈ H} is a right coset of H in G.

Important properties of cosets:

Any two left (or right) cosets are either identical or disjoint.
All cosets of H have the same number of elements as H.
The union of all distinct left (or right) cosets of H forms the entire group G.
Lagrange's Theorem: If G is a finite group and H is a subgroup of G, then the order of H (number of elements in H) divides the order of G (number of elements in G). That is, |G| = |H| * [G:H], where [G:H] is the index of H in G, representing the number of distinct cosets.

Factor Groups (Quotient Groups)

A factor group (or quotient group), denoted G/N, can be formed when N is a normal subgroup of G. The elements of G/N are the cosets of N in G, and the group operation is defined by (aN) * (bN) = (ab)N.

For G/N to be a group, the operation must be well-defined, which is guaranteed if and only if N is a normal subgroup.

Normal Subgroups

A subgroup N of a group G is called a normal subgroup (or self-conjugate subgroup) if for all g ∈ G and all n ∈ N, g * n * g^-1 ∈ N. This is equivalent to saying that gN = Ng for all g ∈ G (i.e., left and right cosets are identical).

Key characteristics of normal subgroups:

Every subgroup of an Abelian group is normal.
If [G:N] = 2 (i.e., N has only two distinct cosets), then N is a normal subgroup.
Normal subgroups are precisely the kernels of homomorphisms.

Direct Product of Groups

The direct product of two groups G₁ and G₂, denoted G₁ × G₂, is the set of all ordered pairs (g₁, g₂) where g₁ ∈ G₁ and g₂ ∈ G₂, with the group operation defined component-wise: (g₁, g₂) * (h₁, h₂) = (g₁ * h₁, g₂ * h₂).

This is the external direct product. There is also an internal direct product, which describes how a group G can be expressed as a product of two of its normal subgroups.

Homomorphism & Isomorphism

Homomorphism

A map φ: G → H between two groups G and H is a homomorphism if it preserves the group operation, i.e., for all a, b ∈ G, φ(a * b) = φ(a) * φ(b).

Important concepts related to homomorphisms:

Kernel of a homomorphism: The set of elements in G that map to the identity element in H. Denoted Ker(φ) = {g ∈ G | φ(g) = e_H}. The kernel is always a normal subgroup of G.
Image of a homomorphism: The set of all elements in H that are images of elements in G. Denoted Im(φ) = {h ∈ H | h = φ(g) for some g ∈ G}. The image is always a subgroup of H.
First Isomorphism Theorem: If φ: G → H is a group homomorphism, then G/Ker(φ) is isomorphic to Im(φ).

Isomorphism

An isomorphism is a bijective (both injective and surjective) homomorphism. If an isomorphism exists between two groups G and H, then G and H are said to be isomorphic, denoted G ≅ H.

Isomorphic groups are essentially the same from an algebraic perspective; they have identical structural properties. An automorphism is an isomorphism from a group to itself.

Representations

A representation of a group G on a vector space V is a homomorphism ρ: G → GL(V), where GL(V) is the group of invertible linear transformations from V to itself. This allows abstract group elements to be represented by concrete matrices, facilitating their study through linear algebra.

Reducible and Irreducible Representations

A representation (ρ, V) is reducible if there exists a proper, non-trivial subspace W of V such that for all g ∈ G, ρ(g)W ⊆ W (i.e., W is an invariant subspace under all transformations ρ(g)). If no such invariant subspace exists, the representation is irreducible.

Irreducible representations are the building blocks of all representations. Every finite-dimensional representation of a finite group can be decomposed into a direct sum of irreducible representations.

Unitary Representations

A representation (ρ, V) is unitary if the vector space V is a Hilbert space (a complete inner product space), and for every g ∈ G, ρ(g) is a unitary operator (i.e., ρ(g)^†ρ(g) = I, where † denotes the conjugate transpose).

Unitary representations are particularly important in quantum mechanics because they preserve inner products and thus probabilities. For finite groups, every representation over a complex vector space is equivalent to a unitary representation.

Schur's Lemma and Orthogonality Theorems

Schur's Lemma:

First Schur's Lemma: Let (ρ₁, V₁) and (ρ₂, V₂) be two irreducible representations of a group G. If T: V₁ → V₂ is a linear map such that Tρ₁(g) = ρ₂(g)T for all g ∈ G (i.e., T is an intertwining operator), then either T is the zero map or T is an isomorphism.

Second Schur's Lemma: Let (ρ, V) be an irreducible representation of a group G over a complex vector space V. If T: V → V is an intertwining operator (i.e., Tρ(g) = ρ(g)T for all g ∈ G), then T must be a scalar multiple of the identity operator (T = cI for some c ∈ ℂ).

Orthogonality Theorems: These theorems are crucial for analyzing representations and characters. For irreducible unitary representations (ρ⁽ⁱ⁾, V_i) and (ρ^(j), V_j) of a finite group G, and their matrix elements ρ⁽ⁱ⁾(g)_mn and ρ^(j)(g)_pq:

Great Orthogonality Theorem:
∑_g∈G ρ⁽ⁱ⁾(g)_mn [ρ^(j)(g)_pq]* = ( |G| / d_i ) δ_ij δ_mp δ_nq

where |G| is the order of the group, d_i is the dimension of the i-th irreducible representation, and δ denotes the Kronecker delta. This theorem implies that matrix elements of distinct irreducible representations (or different elements within the same representation) are orthogonal.

Characters of Representation

The character of a representation (ρ, V) of a group G is a function χ: G → ℂ defined by χ(g) = Tr(ρ(g)), where Tr denotes the trace of the matrix ρ(g).

Properties of characters:

The character is a class function: χ(g) = χ(hgh^-1) for all g, h ∈ G. This means the character is constant on conjugacy classes.
χ(e) = d, the dimension of the representation.
χ(g^-1) = [χ(g)]*.
Characters of irreducible representations (irreducible characters) form an orthonormal basis for the space of class functions.

Character Orthogonality Theorem: For irreducible characters χ⁽ⁱ⁾ and χ^(j) of a finite group G:
(1 / |G|) ∑_g∈G χ⁽ⁱ⁾(g) [χ^(j)(g)]* = δ_ij

This theorem is used to determine if a representation is irreducible and to decompose reducible representations. Character tables summarize the characters of irreducible representations for a group.

Direct Product of Representations

Given two representations (ρ₁, V₁) and (ρ₂, V₂) of a group G, their direct product (or tensor product) representation (ρ₁ ⊗ ρ₂, V₁ ⊗ V₂) is defined on the tensor product space V₁ ⊗ V₂. The representation operators are given by:

(ρ₁ ⊗ ρ₂)(g) = ρ₁(g) ⊗ ρ₂(g)

The character of the direct product representation is the product of the individual characters: χ_1⊗2(g) = χ₁(g) χ₂(g). Direct products are crucial for understanding how symmetries combine, for example, in coupling angular momenta in quantum mechanics.

Introduction to Continuous Groups

Continuous groups are groups whose elements are parameterized by continuous variables. They are topological groups, meaning they have a topological structure where the group operations are continuous. They are fundamental in physics for describing continuous symmetries.

Lie Groups

A Lie group is a group that is also a differentiable manifold, such that the group operations (multiplication and inversion) are smooth (infinitely differentiable) maps. They are named after Sophus Lie.

Examples of Lie groups include:

General Linear Group GL(n, ℝ) or GL(n, ℂ): The group of all invertible n × n matrices with real or complex entries.
Orthogonal Group O(n): The group of n × n orthogonal matrices (R^TR = I).
Special Orthogonal Group SO(n): The subgroup of O(n) with determinant +1.
Unitary Group U(n): The group of n × n unitary matrices (U^†U = I).
Special Unitary Group SU(n): The subgroup of U(n) with determinant +1.

Lie groups are analyzed using their associated Lie algebras, which are vector spaces equipped with a Lie bracket (a bilinear, antisymmetric, and Jacobi identity-satisfying operation). The Lie algebra captures the local structure of the Lie group.

Rotation and Unitary Groups

Rotation Groups SO(n): These groups describe rotations in n-dimensional Euclidean space. SO(n) consists of all n × n orthogonal matrices with determinant +1. For example, SO(3) describes rotations in 3D space, which are crucial in classical mechanics and quantum mechanics for angular momentum.

Unitary Groups U(n) and Special Unitary Groups SU(n): U(n) consists of n × n unitary matrices (U^†U = I). SU(n) is the subgroup of U(n) where the determinant of the matrices is +1. These groups are fundamental in quantum mechanics, especially SU(2) for spin and SU(3) for strong interactions (color symmetry).

Representation of SO(3), SU(2)

Representation of SO(3)

The rotation group SO(3) is non-Abelian and has a continuous set of elements. Its irreducible representations are indexed by a non-negative integer j (or half-integer in the context of its covering group SU(2)), representing the total angular momentum quantum number. The dimension of such a representation is 2j + 1.

These representations are often denoted as D^j.
For integer j, they correspond to the representations of SO(3). For example, j=0 is the scalar representation, j=1 is the vector representation (rotations of 3D vectors), j=2 is the quadrupole representation.
The Lie algebra of SO(3) is isomorphic to 𝔰𝔬(3), which is spanned by three generators corresponding to rotations around the x, y, and z axes. These generators are identified with the angular momentum operators J_x, J_y, J_z in quantum mechanics.

Representation of SU(2)

SU(2) is the group of 2 × 2 unitary matrices with determinant +1. It is the universal covering group of SO(3), meaning there is a homomorphism from SU(2) onto SO(3) that is 2-to-1 (two elements in SU(2) map to one element in SO(3)). This relationship is vital for understanding spin in quantum mechanics.

The irreducible representations of SU(2) are also indexed by j, which can be a non-negative integer or half-integer (j = 0, 1/2, 1, 3/2, ...).
The dimension of the representation D^j is 2j + 1.
The representations with half-integer j (e.g., j=1/2 for spin-1/2 particles) are called spinor representations and are representations of SU(2) but not of SO(3) directly (they pick up a sign change under a 2π rotation in SO(3)).
The Lie algebra of SU(2) is isomorphic to 𝔰𝔲(2), which is generated by the Pauli matrices (up to a factor). These generators are also identified with angular momentum operators.

The study of SU(2) representations is directly applicable to the quantum mechanical theory of angular momentum and spin, where the total angular momentum (or spin) of a system corresponds to the label j of an irreducible representation.

Unit 5: Comprehensive Study Guide to Group Theory