group theory provides the natural mathematical language to
formulate symmetry principles and to derive their consequences in
mathematics and in physics. the "special functions" of mathematical
physics, which pervade mathematical analysis,classical physics, and
quantum mechanics, invariably originate from underlying symmetries
of the problem although the traditional presentation of such topics
may not expressly emphasize this universal feature. modern
developments in all branches of physics are putting more and more
emphasis on the role of symmetries of the underlying physical
systems. thus the use of group theory has become increasingly
important in recent years. however, the incorporation of group
theory into the undergraduate or graduate physics curriculum of
most universities has not kept up with this development. at best,
this subject is offered as a special topic course, catering to a
restricted class of students. symptomatic of this unfortunate gap
is the lack of suitable textbooks on general group-theoretical
methods in physics for all serious students of experimental and
theoretical physics at the beginning graduate and advanced
undergraduate level. this book is written to meet precisely this
need.

there already exist, of course, many books on group theory and
its applications in physics. foremost among these are the old
classics by weyl, wigner, and van der waerden. for applications to
atomic and molecular physics, and to crystal lattices in solid
state and chemical physics, there are many elementary textbooks
emphasizing point groups, space groups, and the rotation group.
reflecting the important role played by group theory in modern
elementary particle theory, many current books expound on the
theory of lie groups and lie algebras with emphasis suitable for
high energy theoretical physics. finally, there are several useful
general texts on group theory featuring comprehensiveness and
mathematical rigor written for the more mathematically oriented
audience. experience indicates, however, that for most students, it
is difficult to find a suitable modern introductory text which is
both general and readily understandable.

preface

chapter 1 introduction

1.1 particle on a one-dimensional lattice

1.2 representations of the discrete translation operators

1.3 physical consequences of translational symmetry

1.4 the representation functions and fourier analysis

1.5 symmetry groups of physics

chapter 2 basic group theory

2.1 basic definitions and simple examples

2.2 further examples, subgroups

2.3 the rearrangement lemma and the symmetric (permutation)
group

2.4 classes and invariant subgroups

2.5 cosets and factor (quotient) groups

2.6 homomorphisms

2.7 direct products

problems

chapter 3 group representations

3.1 representations

3.2 irreducible, inequivalent representations

3.3 unitary representations

3.4 schur's lemmas

3.5 orthonormality and completeness relations of irreducible
representation matrices

3.6 orthonormality and completeness relations of irreducible
characters

3.7 the regular representation

3.8 direct product representations, clebsch-gordan
coefficients

problems

chapter 4 general properties of irreducible vectors and
operators

4.1 irreducible basis vectors

4.2 the reduction of vectors--projection operators for irreducible
components

4.3 irreducible operators and the wigner-eckart theorem

problems

chapter 5 representations of the symmetric groups

5.1 one-dimensional representations

5.2 partitions and young diagrams

5.3 symmetrizers and anti-symmetrizers of young tableaux

5.4 irreducible representations of sn

5.5 symmetry classes of tensors

problems

chapter 6 one-dimensional continuous groups

6.1 the rotation group so(2)

6.2 the generator of so(2)

6.3 irreducible representations of so(2)

6.4 invariant integration measure, orthonormality and completeness
relations

6.5 multi-valued representations

6.6 continuous translational group in one dimension

6.7 conjugate basis vectors

problems

chapter 7 rotations in three-dimensional space--the group
so(3)

7.1 description of the group so(3)

7.1.1 the angle-and-axis parameterization

7.1.2 the euler angles

7.2 one parameter subgroups, generators, and the lie algebra

7.3 irreducible representations of the so(3) lie algebra

7.4 properties of the rotational matrices dj(a, fl, 7)

7.5 application to particle in a central potential

7.5.1 characterization of states

7.5.2 asymptotic plane wave states

7.5.3 partial wave decomposition

7.5.4 summary

7.6 transformation properties of wave functions and
operators

7.7 direct product representations and their reduction

7.8 irreducible tensors and the wigner-eckart theorem

problems

chapter 8 the group su(2) and more about so(3)

8.1 the relationship between so(3) and su(2)

8.2 invariant integration

8.3 Orthonormality and completeness relations of dj

8.4 projection operators and their physical applications

8.4.1 single particle state with spill

8.4.2 two particle states with spin

8.4.3 partial wave expansion for two particle scattering with
spin

8.5 differential equations satisfied by the dj-functions

8.6 group theoretical interpretation of spherical harmonics

8.6.1 transformation under rotation

8.6.2 addition theorem

8.6.3 decomposition of products of yim with the same
arguments

8.6.4 recursion formulas

8.6.5 symmetry in m

8.6.6 Orthonormality and completeness

8.6.7 summary remarks

8.7 multipole radiation of the electromagnetic field

problems

chapter 9 euclidean groups in two- and three-dimensional
space

9.1 the euclidean group in two-dimensional space e2

9.2 unitary irreducible representations of e2--the
angular-momentum basis

9.3 the induced representation method and the plane-wave
basis

9.4 differential equations, recursion formulas,and addition
theorem of the bessel function

9.5 group contraction--so(3) and e2

9.6 the euclidean group in three dimensions: e3

9.7 unitary irreducible representations of e3 by the induced
representation method

9.8 angular momentum basis and the spherical bessel function

problems

chapter 10 the lorentz and poincarie groups, and space-time
symmetries

10.1 the lorentz and poincare groups

10.1.1 homogeneous lorentz transformations

10.1.2 the proper lorentz group

10.1.3 decomposition of lorentz transformations

10.1.4 relation of the proper lorentz group to sl(2)

10.1.5 four-dimensional translations and the poincare group

10.2 generators and the lie algeebra

10.3 irreducible representations of the proper lorentz group

10.3.1 equivalence of the lie algebra to su(2) x su(2)

10.3.2 finite dimensional representations

10.3.3 unitary representations

10.4 unitary irreducible representations of the poincare
group

10.4.1 null vector case (pu= 0)

10.4.2 time-like vector case (c1＞3 0)

10.4.3 the second casimir operator

10.4.4 light-like case (c1 = 0)

10.4.5 space-like case (c1＜0)

10.4.6 covariant normalization of basis states and integration
measure

10.5 relation between representations of the lorentz and poincare
groups--relativistic wave functions, fields, and wave
equations

10.5.1 wave functions and field operators

10.5.2 relativistic wave equations and the plane wave
expansion

10.5.3 the lorentz-poincare connection

10.5.4 "deriving" relativistic wave equations

problems

chapter 11 space inversion invariance

11.1 space inversion in two-dimensional euclidean space

11.1.1 the group 0(2)

11.1.2 irreducible representations of 0(2)

11.1.3 the extended euclidean group e2 and its irreducible
representations

11.2 space inversion in three-dimensional euclidean space

11.2.1 the group 0(3) and its irreducible representations

11.2.2 the extended euclidean group e3 and its irreducible
representations

11.3 space inversion in four-dimensional minkowski space

11.3.1 the complete lorentz group and its irreducible
representations

11.3.2 the extended poincare group and its irreducible
representations

11.4 general physical consequences of space inversion

11.4.1 eigenstates of angular momentum and parity

11.4.2 scattering amplitudes and electromagnetic multipole
transitions

problems

chapter 12 time reversal invariance

12.1 preliminary discussion

12.2 time reversal invariance in classical physics

12.3 problems with linear realization of timereversal
transformation

12.4 the anti-unitary time reversal operator

12.5 irreducible representations of the full poincare group in the
time-like case

12.6 irreducible representations in the light-like case (c1 = c2 =
0)

12.7 physical consequences of time reversal invariance

12.7.1 time reversal and angular momentum eigenstates

12.7.2 time-reversal symmetry of transition amplitudes

12.7.3 time reversal invariance and perturbation amplitudes

problems

chapter 13 finite-dimensional representations of the classical
groups

13.1 gl(m): fundamental representations and the associated vector
spaces

13.2 tensors in v x v, contraction, and gl(m)
transformations

13.3 irreducible representations of gl(m) on thespace of general
tensors

13.4 irreducible representations of other classical linear
groups

13.4.1 unitary groups u(m) and u(m+, m_)

13.4.2 special linear groups sl(m) and special unitary groups
su(m+, m_)

13.4.3 the real orthogonal group o(m+,m_; r) and the special real
orthogonal group so(m +, m_; r)

13.5 concluding remarks

problems

appendix i notations and symbols

i.1 summation convention

i.2 vectors and vector indices

i.3 matrix indices

appendix ii summary of linear vector spaces

ii.1 linear vector space

ii.2 linear transformations (operators) on vector spaces

ii.3 matrix representation of linear operators

ii.4 dual space, adjoint operators

ii.5 inner (scalar) product and inner product space

ii.6 linear transformations (operators) on inner product
spaces

appendix iii group algebra and the reduction of regular
representation

iii. 1 group algebra

1ii.2 left ideals, projection operators

iii.3 idempotents

iii.4 complete reduction of the regular representation

appendix iv supplements to the theory of symmetric groups sn

appendix v clebsch-gordan coefficients and spherical
harmonics

appendix vi rotational and lorentz spinors

appendix vii unitary representations of the proper lorentz
group

appendix viii anti-linear operators

references and bibliography

index