It was with some trepidation that I decided to write yet another book on
·group theory and physics. There are, after all, quite a large number
already. However, in the first place some of the best of these were,
inexplicably, out of print, and secondly I thought I perceived a niche for
a book which, while by no means skimping the physical applications,
exposes the student to the power and elegance of abstract mathematics.
There is, indeed, a fascinating interplay between mathematics and
physics. In some cases the need to understand and formulate a physical
problem provides a stimulus for the development of the relevant
mathematics, as for example in Newton's development of the calculus as
a tool for calculating planetary orbits. In others a mathematical formalism
already developed turns out to be tailor-made for physics. Thus the
theory of vector spaces is exactly what one needs for the general
formulation of quantum mechanics, and the representation theory of
groups is precisely the mathematical framework needed when one
considers the action of symmetry transformations on quantum systems.
With this in mind, I have developed the mathematical theory in a
more formal fashion than is strictly necessary for physical applications.
Thus, for example, I have dealt with representations in the coordinatefree
language of linear transformations acting on vector spaces, rather
than simply as matrices. It is my experience that students, far from
being put off by this, are fascinated by what may well be their first
exposure to the rigorous axiomatic approach, and ultimately obtain a
deeper understanding of the subject. However, since the book is aimed
primarily at physicists, I have been careful to include many examples
and illustrations of the various mathematical structures and theorems. I
have also included for the reader's convenience a short glossary of the
mathematical symbols used.
Author(s): H. F. Jones
Edition: 2
Publisher: Taylor & Francis Group
Year: 1998
Language: English
Pages: 344
Tags: Groups; representations; physics; finite groups; Lie groups; Lie algebras; quantum mechanics; particle physics; gauge theories; quantum field theory
Contents
1 Introduction
1.1 Symmetry in physics; groups and reps
1.2 Definition of a group; some simple examples
Definition
Examples
Counterexamples
1.3 Some simple point groups
(i) The cyclic groups C_n
(ii) The dihedral group D_n
Group D_3
Group D_4
1.4 The permutation group S_n
Group S_2
Group S_3
Group S_4
Cayley's theorem
2 General properties of groups and mappings
2.1 Conjugacy and conjugacy classes
Conjugacy
Definition
Conjugacy classes
Examples
(1) C_n
(2) D_n
(3) S_n
2.2 Subgroups
Definition
Cosets
Lagrange's theorem
Example
2.3 Normal subgroups
Example: D_3
Quotient group
Definition
Example: H in D_3
Counterexample: H in D_3
Direct product
Definition
Example
Counterexample: D_3
2.4 Homomoprhisms
Image
Group homomorphism
Kernel
Isomorphism theorem
Example: D_3
3 Group representations
3.1 A simple example; formal definition
Example: C_3
Definition
3.2 Induced transformation of the quantum mechanical wavefunction
The inverse explained
Examples
3.3 Equivalence of reps; characters; reducibility
Definition equivalence
Definition character
Reducibility
3.4 Groups acting on vector spaces
Axioms of a vector space
Examples
1. Ordinary 3-vectors
2. Function space
Definition linearly independent
Definition basis
Definition n-dim vector space
Linear transformations
Definition
Similarity
G-modue
Reducibility
3.5 Scalar product; unitary repsl Maschke's theorem
Scalar product
Orthonormal basis
Unitary transformations
Complete reducibility
Maschke's theorem
4 Properties of irreducible representations
4.1 Schur's lemmas
(i) The first lemma
(ii) The second lemma
4.2 The fundamental orthogonality theorem
Restriction on the number of irreps
4.3 Orthogonality of characters
Decomposition of reducible reps
The regular rep
4.4 Construction of the character table
Character table of C_3
Character table of D_3
4.5 Direct products of reps and their decomposition
5 Physical applications
5.1 Macroscopic properties of crystals
(i) Ferromagnetism and ferroelectricity
(ii) Conductivity tensor
5.2 Molecular vibrations (H20)
(i) Small oscillations and normal modes
(ii) Example: The water molecule
(a) Decomposition of D^(3N)
(b) Elimination of translation and rotations
(c) Displacement patterns of vibrational modes
(d) Calculation of normal mode frequencies
5.3 Raising of degeneracy
(i) Degeneracy
(ii) Breaking of degeneracy
Example: crystalline splitting of atmoic levels
6 Continuous groups SO(N)
6.1 SO(2)
The infinitesimal generator J_z
6.2 SO(3) (SU(2))
Infinitesimal generators
Commutation relations
Irreducible representations of SO(3)
Characters
Orthogonality
The Clebsch-Gordan series
6.3 Clebsch-Gordan coefficients
Tensor operators and the Wigner-Eckart theorem
Definition
Exammples
Dipole selection rules
Lande g-factor in l-s coupling
7 Further applications
7.1 Energy levels of atoms in Hartree-Fock scheme
Example: Carbon
7.2 `Accidental' degeneracy of the H atom and SO(4)
7.3 The partial wave expansion; unitarity
(i) Scattering theory
(ii) Partial waves
7.4 Isotopic spin; pi N scattering
8 The SU(N) groups and particle physics
8.1 The relation between SU(2) and SO(3)
8.2 SU(2)
(i) Quarks; isospin as SU(2)
(ii) SU(2) tensors
8.3 SU(3)
(i) Strangeness
(ii) SU(3) tensors
8.4 SU(N); Young tableaux
(i) Young tableaux
(ii) Rules for dimensionality
(iii) Rules for CG series
9 General treatment of simple Lie groups
9.1 The adjoint rep and the Killing form
9.2 The Cartan basis of a Lie algebra
9.3 Properties of the roots and root vectors
9.4 Quantiation of the roots
9.5 Simple roots—Dynkin diagrams
SU(n)
SO(2r)
SO(2r+1)
Sp(2r)
The exceptional groups
9.6 Reps and weights
10 Representations of the Poincare group
10.1 Lorentz transformations
10.2 4-vector notation
10.3 The Lorentz group SO(3,1)
Boosts + rotations + inversions
Generators of SO(3,1)
10.4 The Poincare group
The inhomogeneous Lorentz group (Lorentz + spacetime transl.)
Massive reps; helicity
Massless reps
10.5 Angular momentum states
(i) Single-particle states; potential scattering
(ii) Two-particle states; relativistic scattering
11 Gauge groups
11.1 The electromagnetic potentials; gauge transformations
11.2 Interaction with non-relativistic electrons
11.3 Relativistic formulation of electromagnetism
11.4 Relativistic equation of motion for the electron
11.5 Quantum fields and their interactions
Scalar field \phi
Dirac field \psi
Maxwell field A_\mu
11.6 Gauge field theories
(i) U(1)—QED
(ii) SU(2)
(iii) SU(3)—QCD
A Dirac notation in QM
A1 State vectors
A2 Orthonormal basis
A3 Linear operators
A4 Hermitian operators
Reality of eigenvalues
Orthogonality of eigenvectors
Completness of eigenvectors
Rep of H in terms of eigenvectors
Defining a function f(H)
The exponential function
A matrix operator
A5 Unitary operators
A6 Continuous eigenvalues
A7 Interpretative formalism; wavefunctions
Axioms
Wavefunctions
B Eigenstates of angular momentum in QM
Commutation relations
Addition of angular momenta
C Group-invariant measure for SO(3)
D Calculation of roots for SO(n) and Sp(2r)
SO(2r)
SO(2r+1)
Sp(2r)
E Covariant normalization and relativistic scattering
F Lagrangian mechanics
Lagrange's equations
Hamilton's principle
Hamilton's equations
Lagrangian for Lorentz force
Lagrangian field theory
Glossary of mathematical symbols
Bibliography
Problem solutions
1
2
3
4
5
6
7
8
9
10
11
Index
