群论是现代数学的一个分支，即抽象代数学(近世代数学)的一部分。群论也是用来描写物理世界对称性的一个非常有用的数学工具。

《物理学中的群论》为物理学中涉及的群论知识的简明教程，适合理工科各相关专业学生使用。全书介绍了群论的基本概念，讨论群的表示等内容。本书由吴基东编。

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 3group 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 4general 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 8the group su（2） and more about so（3）

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

8.2 invariant integration

8.30rthonormality 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

8.6.60rthonormality and completeness

8.6.7 summary remarks

8.7 multipole radiation of the electromagnetic field problems

chapter 9euclidean 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.i 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 remarksproblems appendix inotations 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 illgroup 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 ivsupplements to the theory of symmetric groups sn

appependix vclebsch-gordan coefficients and spherical harmonics

appendix virotational and lorentz spinors

appendix viiunitary representations of the proper lorentz group

appendix viii anti-linear operators

references and bibliography

index