Group Theory: An Intuitive Approach

The following sections are included:

• Preface

• Table of Contents

• List of Tables

• List of Figures

The following sections are included:

• CHANGE IN THE WORLD AROUND US

• FOUNDATIONS
  • The need for observers
  • Observations must be related

• WHAT ARE TRANSFORMATIONS?
  • Transformations of real objects
  • Transformations and invariance
  • Transformations that almost are symmetries
  • What transformations should we consider?
  • Why we must examine the meaning of transformations
  • The centrality of transformations
  • The example of orbital angular momentum
  • If symmetry starts to break down

• WHAT WE THINK PHYSICAL TRANSFORMATIONS ARE LIKE
  • The group axioms
  • Why groups of transformations are important
  • Pseudo-counterexamples

• GROUPS
  • Uniqueness
  • Some related concepts
  • Transformation groups and symmetry groups
  • The value of group theory

• GROUPS AND QUANTUM-MECHANICAL STATES
  • Labeling the states of the hydrogen atom
  • The reason for the labels
  • The procedure for labeling states
  • Why is angular momentum connected with group theory?

The following sections are included:

• GROUPS IN THE REAL WORLD

• GROUPS OF PHYSICAL OBJECTS
  • The group table
  • Two-element groups
    • The group of a pair of hands
    • The group of a held football
  • The group of a triangle
  • The water group
  • The pyramidal group
  • The groups of the square
  • The double-rectangle group
  • The group of the cube
  • Groups of given order

• CONTINUOUS GROUPS
  • Rotation (orthogonal) groups
  • The SU(2) group
  • The isospin group
  • Unitary groups
  • The Lorentz group
  • The translation group
  • Time translations
  • The Poincaré group
  • Inversions
  • Infinite discrete groups

• GROUPS OF MATHEMATICAL OBJECTS
  • Abelian groups
  • Cyclic groups
  • Dihedral groups
  • Symmetric groups
    • The ordering convention
    • Alternating groups
  • The group of the roots of unity
  • The groups of numbers
  • Matrix groups
  • The group of a polynomial
  • Functions forming groups
  • Vector Spaces

• THE GROUPS OF PEOPLE AND OTHER FAMILIAR OBJECTS

• THE PHYSICS OF GROUP TABLES
  • How the dimension of space affects the group
  • The connection between groups and objects

The following sections are included:

• EXAMPLES AND RELATIONS

• THE LEAST COMPLICATED GROUPS
  • Cyclic groups
  • Abelian groups
  • Groups of order 4
  • Groups of order 6
  • Groups of order 8
  • The tetrahedral group
  • The cubic and icosahedral groups
  • Dihedral groups
  • Groups and their tables

• GENERATORS AND PRESENTATIONS

• RELATIONS AMONG SETS SUCH AS GROUPS — MORPHISM
  • Equivalence relations
  • Isomorphisms
  • Homomorphisms
  • Endomorphisms
  • Automorphisms
    • Inner and outer automorphisms
    • The automorphism group of a group
    • Automorphisms and isomorphisms
  • Enantiomorphisms
  • Isometries

• COMBINATIONS OF SETS AND GROUPS
  • Sums of sets
  • Products of sets and groups
  • Direct products

• DEVELOPING ALGORITHMS TO FIND GROUPS

The following sections are included:

• USES OF SUBSETS

• OPERATIONS ON OPERATORS: SIMILARITY TRANSFORMATIONS

• SUBGROUPS
  • Conjugate subgroups
  • Intersections and unions
  • Cyclic groups and cyclic subgroups
  • Subgroups of symmetric groups
    • Groups having, and being, symmetric subgroups
    • All finite groups are subgroups of symmetric groups
    • The group of the square as a subgroup of a symmetric group
    • Implications of Cayley's Theorem

• CENTRALIZERS AND NORMALIZERS

• CLASSES
  • Why classes exist and are important
  • How geometry gives the classes of the square group
  • Classes are given by similarity transformations
  • Classes of rotation groups
  • Classes of groups and subgroups
  • Class multiplication

• COSETS
  • The group, and the (co)sets obtained from a subgroup
  • The index of a subgroup in a group
    • The order of subgroups and the order of groups: Lagrange's Theorem
    • Groups and subgroups are limited

• INVARIANT SUBGROUPS
  • Examples of normal subgroups
  • The reasons for normal subgroups

• MULTIPLYING AND DIVIDING COSETS AND GROUPS
  • Products of cosets
  • Products of groups
  • The (factor) group of the cosets of a normal subgroup
    • Examples of factor groups
    • Properties of factor groups
    • Factor groups and homomorphisms

• SIMPLE GROUPS
  • Semisimple groups
  • The importance of simple groups

The following sections are included:

• THE PHYSICALLY MOST RELEVANT PART OF GROUP THEORY

• HOW REPRESENTATIONS ARE DISGUISED
  • Systems with cylindrical symmetry
    • The condition symmetry places on the solutions
    • How rotations determine the solutions
    • The only possible statefunctions give all representations
  • Disguising the three-dimensional rotation group
    • The basis states for spherical symmetric systems
    • What does finding a representation mean?
  • Removing the disguise from quantum mechanics

• WHAT ARE REPRESENTATIONS?
  • A mathematical view of representations
  • Why groups have representations
  • Realizations

• TYPES OF REPRESENTATIONS
  • Must a representation be faithful to its group?
  • Equivalent representations are much the same
  • Degeneracy and equivalence
  • Reducible representations
  • Decomposability and reducibility

• EXAMPLES OF REPRESENTATIONS
  • The two-dimensional rotation group, SO(2)
  • Representations of a finite group, D₃
  • Representations of symmetric groups
    • Finding the sums forming the states
    • The definition of, and requirements on, a product
  • The group of an equilateral triangle

The following sections are included:

• THE REGULAR REPRESENTATION
  • The relationship between group elements and basis vectors
  • Basis vectors of any, and of the regular, representation
  • There is always a symmetric representation

• NUMBER OF APPEARANCES WITHIN THE REGULAR REPRESENTATION
  • Why the number of appearances equals the dimension
  • Representation dimensions and group order

• THE MATRICES OF THE REGULAR REPRESENTATION

• EQUIVALENCE CLASSES
  • Classes and equivalence classes
  • Labeling representations and states

• EXAMPLES OF THE REGULAR REPRESENTATION AND ITS REDUCTION

• THEOREM OF THE REGULAR REPRESENTATION

The following sections are included:

• REPRESENTATIONS AND FUNCTIONS ON THEM

• UNITARY REPRESENTATIONS
  • Representations of finite groups can be taken unitary
    • All representations are unitary for all products
    • Every representation is equivalent to a unitary one
  • Unitary representations are decomposable
  • Decomposability of finite group representations
  • Why finite group representations can be taken unitary

• SCHUR'S LEMMAS
  • Schur's first lemma
  • Schur's second lemma
  • Why Schur's lemmas?

• CHARACTERS
  • Traces are characters and determine the representations
  • Characters are class functions

• ORTHOGONALITY THEOREMS
  • Orthogonality of the representation matrices
  • The reason for the orthogonality of the matrices
  • Orthogonality of the characters
    • The character orthogonality condition for classes
    • The necessary and sufficient condition for equivalence of representations

• APPLICATIONS OF CHARACTERS
  • Reduction of representations using characters
  • How to tell if a representation is completely reduced
  • Equality of number of classes and representations
  • Characters of the regular representation
    • Number of elements and representation dimension
    • How simplicity of a group affects its character table
  • Uniqueness of the reduction of a representation

• COMPUTATION OF CHARACTERS
  • Abelian groups and their characters
    • Characters of cyclic and dihedral groups
    • Description of the characters of Abelian groups
  • Characters of symmetric groups
  • Characters for the group of the square
  • Characters of the tetrahedral group
  • Examples of character tables

The following sections are included:

• SYMMETRIC GROUPS

• PROPERTIES OF SYMMETRIC GROUPS
  • Cycle structure, subgroups and simplicity
  • Realizations of the symmetric group
  • Transpositions
  • Representations and the regular representation
  • Categories of representations

• LABELS
  • Young frames
  • Young tableaux
  • Labeling states
  • Conjugate frames
  • Ordering frames and tableaux
    • Ordering frames
    • Ordering tableaux
  • How states are related to their labels
  • The problems with these states

• CRITERIA FOR CORRECTNESS OF THE LABELS
  • Number of frames and number of equivalence classes
  • Frames give different classes
  • Tableaux correctly label representations and states
    • Independence of tableaux
    • Why the standard tableaux label the states
    • What about nonstandard multinomials?
    • What does relabeling do?

• STATES OF THE ORTHOGONAL CATEGORY
  • Basis state operators
    • Basis state operators can be taken as states
    • States and standard tableaux
    • Why are these the basis state operators?
  • Expansion of a monomial in terms of states

• THE ORTHOGONAL-REPRESENTATION MATRICES
  • Matrices of the neighboring transpositions
    • The axial distance
    • The nonzero elements for the transposition
    • The matrices for the semi-normal representations
  • Derivation of the transposition matrix elements
    • Finding the matrices for S_n by induction
    • Computing the axial distance
    • The reasons for the form of the matrices
  • Equivalent representations of S₃

• SYMMETRY OF STATES AND MATRICES
  • Symmetry properties of the states
  • Symmetry properties of representation matrices

The following sections are included:

• OBJECTS FOR THE SYMMETRIC GROUP

• REPRESENTATION DIMENSIONS
  • The formula for the dimension
  • Proof of the dimension formula
  • The dimensions and the regular representation

• REPRESENTATIONS OF ALTERNATING GROUPS

• PHYSICAL REALIZATIONS OF TWO-DIMENSIONAL REPRESENTATIONS

• PRODUCTS OF REPRESENTATIONS OF SYMMETRIC GROUPS
  • The outer product
    • The frames in the outer products
    • Frames are independent of choice of subsets
    • The states in the expansion
  • Inner products

• AUTOMORPHISMS OF SYMMETRIC GROUPS

The following sections are included:

• LIE GROUPS

• THE ROTATION GROUPS
  • What are the rotation groups?
  • Mathematical definitions of rotations

• THE TWO-DIMENSIONAL ROTATION GROUP
  • Defining SO(2)
  • SO(2) is infinitely connected

• DESCRIPTION OF SO(3)
  • Parametrizations of the groups
    • The number of parameters
    • Parametrization of SO(3)
  • Why rotations do not commute
    • Noncommutivity and the bounded parameter range
    • Commutators of rotations
  • The classes of the rotation groups

• THE UNITARY, UNIMODULAR GROUP SU(2)
  • Parametrization of SU(2)
  • The effect of SU(2) on complex vectors

• THE SU(2), SO(3) HOMOMORPHISM
  • The relationship of complex vectors to real vectors
  • The rotation group is doubly connected
    • Why the rotation group is not infinitely connected
    • Connectivity of groups and subgroups
  • Why only two groups?

• ANGULAR MOMENTUM OPERATORS AND THEIR ALGEBRA
  • The transformation matrices when angles are small
    • The SO(2) group for small angle
    • The SO(3) and SU(2) groups when parameters are small
  • Lie Algebras of Lie groups
  • A differential realization of the rotation algebra
  • What the commutation relations require
  • The physical significance of the rotation Lie algebra
  • Why Lie algebras are important

• CONSERVATION LAWS AND SPACE
  • Conservation and commutation
  • Why momentum must be conserved
  • Homogeneity, isotropy, conservation

The following sections are included:

• REPRESENTATIONS OF GROUPS AND OF THEIR ALGEBRAS
  • Representations of linear groups
  • Representations of SO(3) and SU(2)
  • Representations of algebras

• REPRESENTATIONS OF THE ALGEBRA OF SO(3) AND SU(2)
  • The reality conditions
  • Labeling the representations and states
    • The diagonal generators
    • The eigenvalues of the diagonal generators
    • The values of the labels
    • Values of the labels differ for the two groups

• THE REPRESENTATIONS OF SO(2)

• THE REPRESENTATIONS OF SO(3)
  • An equation invariant under rotations
    • Solution by separation of variables
    • Group theoretical reasons for solution by separation
  • The representation basis states
    • The P equation
    • Reducing the representations
  • Spherical harmonics
  • Spherical harmonics and rotations
  • Properties of spherical harmonics
  • The addition formula for spherical harmonics
  • The number of parameters in spherical harmonics
  • Irreducibility of the rotation group representations

• THE REPRESENTATIONS OF SU(2)
  • Basis states of the SU(2) defining representation
  • The basis vectors of arbitrary SU(2) representations
  • Computing the SU(2) matrix elements
  • The unitarity of the representations
  • Completeness
  • Connectivity
  • Irreducibility
  • How SO(3) and SU(2) representations are related
  • The characters of the representations

• WHAT HAVE WE LEARNED FROM THESE GROUPS?

The following sections are included:

• PROPERTIES AND APPLICATIONS OF THE REPRESENTATIONS

• PRODUCTS OF GROUPS AND REPRESENTATIONS
  • The vector addition model
  • The representations in the decomposition of products
    • The range
    • Finding the linear combinations
    • Computing the Clebsch-Gordan coefficients
  • Unitarity and reciprocity

• TENSOR OPERATORS AND THE WIGNER-ECKART THEOREM
  • Tensor operators
  • The Wigner-Eckart theorem
  • Why does the Wigner-Eckart theorem hold?

• THE GROUP O(3) AND PARITY
  • The meaning of parity
  • The physical distinction between O(n) and SO(n)
  • The physical effect of parity
  • The group surfaces of O(3) and SO(3)

• SYSTEMS WITH BROKEN SPHERICAL SYMMETRY
  • The Stark and Zeeman effects
  • The Stark effect
    • The eigenstates
    • The hydrogen atom
    • The role of group theory in studying the Stark effect
  • The Zeeman effect
    • Spin-orbit coupling dominant
    • Calculation of the splitting
    • The quadratic Zeeman effect
  • Perturbation theory of the Zeeman effect
  • For how long can we use the rotation group?

The following sections are included:

• ALGEBRAS ON THEIR OWN

• THE DEFINITION OF A LIE ALGEBRA
  • The nonassociativity of Lie algebras
  • The associative algebra of a Lie algebra
  • Combining the algebra symbols
  • Scalars are also needed to define algebras
  • Compactness
  • Isomorphic algebras
  • The adjoint representation of a Lie algebra
  • Subalgebras
  • Ideals

• TYPES OF LIE ALGEBRAS
  • Solvable algebras, and nilpotent ones
    • Definition of solvable algebras
    • Definition of nilpotent algebras
    • Abelian algebras
  • Semisimple algebras
    • Simple algebras
    • What shall we call simple algebras?

• EXAMPLES OF LIE ALGEBRAS
  • Simple, semisimple and non-semisimple algebras
    • The direct sum of simple algebras
    • The direct sum of a simple and an Abelian algebra
  • Inhomogeneous algebras

• THE CLASSICAL LIE ALGEBRAS
  • The general linear and the unitary algebras
  • The orthogonal algebras
  • The symplectic algebras

• THE CONDITION THAT AN ALGEBRA BE SEMISIMPLE
  • The metric
  • The metric gives the condition for semisimplicity

• THE BEST WAY OF WRITING SIMPLE LIE ALGEBRAS
  • The Cartan subalgebra
  • The canonical form of the commutation relations
    • The symmetric form of the generators
    • What has to be proved?
  • The simple roots

• PROOF OF THE CANONICAL COMMUTATION RELATIONS
  • The step generators as eigenvectors
  • The Jacobi identities and the roots
  • Conditions on the roots
  • The values of the N's
  • Diagonalization of the metric
  • Why semisimple algebras have this canonical form
  • The Cartan matrix
  • The simplest simple algebras

The following sections are included:

• WHAT IS A REPRESENTATION OF AN ALGEBRA?
  • Why these are algebra representations
  • Completeness of the representation
  • The defining representation
  • The adjoint representation

• HOW TO GIVE A REPRESENTATION
  • The enveloping algebra
  • Invariants — Casimir operators
  • Semisimplicity, conservation laws and invariants

• WEIGHTS
  • Ordering weights
  • Simple and nonsimple weights
  • Extending properties of roots to weights
  • Weights as vectors in root space
    • Scalar products
    • Restrictions on products and implications
  • The action of step generators
  • Expressing weights in terms of simple roots

• USING WEIGHTS TO LABEL REPRESENTATIONS
  • Simplicity of the highest weight
  • Representations are labeled by their highest weights
  • Are there labeling operators?

• DECOMPOSABILITY OF THE REPRESENTATIONS
  • Complete reducibility for a group and for its algebra
  • All representations of semisimple algebras are decomposable
    • Why the representations are decomposable
    • Complete reducibility and unitarity
    • The physics of complete reducibility
  • When, and why, are representations finite dimensional?

The following sections are included:

• Appendix A SU(3) and States of Some of its Representations
  • Matrices of the Defining Representation
  • Representations

• References

• Index