# The Application of Group Theory in Physics

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The Application of Group Theory in Physics is a 17-chapter text based on a course of lectures concerning the principles, concepts, and application of group theory in physics, given at the Gorki University in Kharkov. This text presents first the parts of the theory of representations of finite and continuous groups that are most important in application. Considerable chapters cover the groups of theory of interest in theoretical physics and demonstrate the principles according to which the abstract concepts and the theorems of representation theory are applied in theoretical physics. The remaining chapters provide representations of the rotation group and the Lorentz group. The closing part of this work contains tables of the detailed description of the 230 space groups and for the characters of certain groups. This book is intended primarily for physicists specializing in theoretical physics

## Table of Contents

Preface

Chapter I Elements of the Theory of Groups

1. Groups

2. Subgroups

Isomorphism and Homomorphism of Groups

Chapter II Some Specific Groups

4. The Permutation Group

5. The Rotation Group

6. The Full Orthogonal Group

7. The Euclidean Group

8. The Point Groups

9. The Point Groups of the First Kind

10. The Point Groups of the Second Kind

11. The Translation Group

12. Syngonies

13. The Symmetry of Crystals

Chapter III The Theory of Group Representations

14. Representation of a Group

15. Equivalent Representations

16. The Averaging Functional

17. Reducible Representations

18. Irreducible Representations and Orthogonality Properties

19. The Completeness Theorem

20. The Theory of Characters

Chapter IV Operations with Group Representations

21. The Product of Representations

22. Conjugate Representation

23. Real Representations

24. The Direct Product

25. Symmetrized Multiple Products of Representations

26. Decomposition of a Reducible Representation into Irreducible Representations.

Chapter V Representations of Certain Groups

27. Representations of the Permutation Group Sn

28. The Irreducible Representations of Point Groups

29. Representations of Translation Groups

30. Representations of Space Groups

Chapter VI Small Oscillations of Symmetrical Systems

31. Normal Coordinates and Eigen-Frequencies

32. Symmetrical Coordinates

33. The Lagrangian in Symmetrical Coordinates

34. The Oscillatory Representation

35. An Example: The Molecule CHC13(117)

Chapter VII Second Order Phase Transitions

36. Formulation of the problem

37. Active Representations

38. An Example

Chapter VIII Crystals

39. Sound in Crystals

40. Electron Levels in a Crystal

41. Tensors in Crystals

Chapter IX Infinite Groups

42. Specific Properties of Infinite Groups

43. Elements of the Theory of Lie Groups

44. Infinitesimal Representation of a Lie Group

Chapter X Representations of the Rotation Groups in Two and Three Dimensions and of the Full Orthogonal Group

45. The Irreducible Representations of the Two-dimensional Rotation Group Z

46. Classification of the Irreducible Representations of Three-Dimensional Rotation Group

47. The Matrix Elements of the Irreducible Representations

48. Properties of the Irreducible Representations of the Rotation Group

49. The Product of Representations of the Rotation Group

50. Spinor Algebra

51. Tensor Algebra

52. Representations of the Full Orthogonal Group

53. Double-Valued Representations of Point Groups

Chapter XI Clebsch-Gordon and Racah Coefficients

54. Evaluation of the Clebsch-Gordon Coefficients

55. Properties of the Clebsch-Gordon Coefficients

56. Racah Coeificients

Chapter XII The Schrödinger Equation

57. Conservation Laws

58. Classification of States

Chapter XIII Equations Invariant Under the Euclidean Group of Motions in Space

59. Spherical Harmonics with Spin

60. Equations Invariant Under the Group of Euclidean Motions in Space

61. An Example

Chapter XIV Absorption and Raman Scattering of Light

62. Quantum Mechanical Introduction

63. Selection Rules for the Absorption of Light by Atoms and Molecules

64. Raman Scattering of Light by Atoms and Molecules

Chapter XV Representations of the Lorentz Group

65. The Lorentz Group

66. Infinitesimal Operators of the Lorentz Group

67. Classification of the Irreducible Representations of the Lorentz Group

68. Product of Irreducible Representations of the Lorentz Group

69. Complex-Conjugate Representations

70. Spinor Algebra

71. Tensor Algebra

72. Representations of the Full Lorentz Group

Chapter XVI Relativistically Invariant Equations

73. The Wave Function

74. Relativistically Invariant Equations

75. The Lagrangiang

76. Conservation Laws

77. Spin

78. The Relativistically Invariant Operation of Time Inversion and the Pauli Theorem

79. The Dirac Equation

Chapter XVII Nuclear Reactions

80. The Scattering Matrix

81. Angular Distribution of the Products of a Nuclear Reaction

82. Angular Distribution of the Products of a Nuclear Reaction (Continued)

Appendices

I. Characters of Irreducible Representations of the Permutation Groups S4,S5,S6,S7.

II. Characters of Irreducible Representations of Point Groups.

III. Two-Valued Representations of Point Groups.

IV. Space Groups.

V. Racah Coefficients.

Bibliography

Subject Index

## Product details

- No. of pages: 392
- Language: English
- Copyright: © Pergamon 1960
- Published: January 1, 1960
- Imprint: Pergamon
- eBook ISBN: 9781483225982

## About the Author

### G.Ya. Lyubarskii

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