Jean Hladik

Spinors in Physics

Translated by J. Michael Cole

Springer

Contents

Preface v

I Spinors in Three-Dimensional Space 1

1 Two-Component Spinor Geometry 3 1.1 Definition of a Spinor 4

1.1.1 Stereographic Projection 4 1.1.2 Vectors Associated with a Spinor 5 1.1.3 The Definition of a Spinor 7

1.2 Geometrical Properties 9 1.2.1 Plane Symmetries 9 1.2.2 Rotations 11 1.2.3 The Olinde-Rodrigues Parameters 14 1.2.4 Rotations Defined in Terms of the Euler Angles . . . 15

1.3 Infinitesimal Properties of Rotations 17 1.3.1 The Infinitesimal Rotation Matrix 17 1.3.2 The Pauli Matrices 18 1.3.3 Properties of the Pauli Matrices 20

1.4 Algebraic Properties of Spinors 22 1.4.1 Operations on Spinors 22 1.4.2 Properties of Operations on Spinors 23 1.4.3 The Basis of the Vector Space of Spinors 24 1.4.4 Hermitian Vector Spaces 25 1.4.5 Properties of the Hermitian Product 26 1.4.6 The Use of an Antisymmetric Metrie Tensor 28

1.5 Solved Problems 29

2 Spinors and SU(2) Group Representations 35 2.1 Lie Groups 35

2.1.1 Examples of Continuous Groups 35

viii Contents

2.1.2 Analytic Definition of Continuous Groups 37 2.1.3 Linear Representations 39 2.1.4 Infinitesimal Generators 41 2.1.5 Infinitesimal Matrices 43 2.1.6 Exponential Mapping 46 2.1.7 The Nomenclature of Continuous Linear Groups . . 47

2.2 Unimodular Unitary Groups 48 2.2.1 The Unitary Group U(2) 48 2.2.2 The Unitary Unimodular Group SU(2) 50 2.2.3 Three-Dimensional Representations 52 2.2.4 Representations of the Groups SU(2) 53 2.2.5 Irreducible Representations of SU(2) 55

2.3 Solved Problems 57

3 Spinor Representation of SO (3) 67 3.1 The Rotation Group SO(3) 67

3.1.1 Rotations About a Point 67 3.1.2 The Infinitesimal Matrices of the Group 68 3.1.3 Rotations About a Given Axis 70 3.1.4 The Exponential Matrix of a Rotation About

a Given Axis 72 3.2 Irreducible Representations of SO(3) 73

3.2.1 The Structure Equations 73 3.2.2 The Infinitesimal Matrices of the Representations

of the Group Sö(3) 75 3.2.3 Eigenvectors and Eigenvalues of the Infinitesimal

Matrices of the Representations 76 3.2.4 Irreducible Representations 78 3.2.5 The Infinitesimal Matrices of an Irreducible

Representation in the Canonical Basis 79 3.2.6 The Characters of the Rotation Matrices

of a Representation , 81 3.3 Spherical Harmonics 82

3.3.1 The Infinitesimal Operators in Spherical Coordinates 82

3.3.2 Spherical Harmonics 83 3.4 Spinor Representations 85

3.4.1 The Two-Dimensional Irreducible Representation . . 85 3.4.2 The Three-Dimensional Irreducible Representation . 87 3.4.3 (2j + l)-Dimensional Irreducible Representations . . 91

3.5 Solved Problems 93

4 Pauli Spinors 99 4.1 Spin and Spinors 99 4.2 The Linearized Schrödinger Equations 100

Contents ix

4.2.1 The Free Particle 100 4.2.2 Particle in an Electromagnetic Field 104 4.2.3 The Spinors in Pauli's Equation 105

4.3 Spinor and Vector Fields 108 4.3.1 The Transformation of a Vector Field by a Rotation 108 4.3.2 The Rotation of a Spinor Field 110

4.4 Solved Problems 112

II Spinors in Four-Dimensional Space 119

5 The Lorentz Group 121 5.1 The Generalized Lorentz Group 121

5.1.1 Rotations and Reflections 121 5.1.2 Orthochronous and Anti-Orthochronous

Transformations 123 5.1.3 Sheets of the Generalized Lorentz Group 123

5.2 The Four-Dimensional Rotation Group 125 5.2.1 Four-Dimensional Orthogonal Transformations . . . 125 5.2.2 Matrix Representations of the Group SO(4) 126 5.2.3 Infinitesimal Matrices 128 5.2.4 Irreducible Representations 130

5.3 Solved Problems 131

6 Representations of the Lorentz Groups 135 6.1 Irreducible Representations 135

6.1.1 Relations Between the Groups SO(3,1)T and SO(4) 135 6.1.2 Infinitesimal Matrices 137 6.1.3 Irreducible Representations 139

6.2 The Group SL(2,C) 140 6.2.1 Two-Component Spinors 140 6.2.2 Higher-Order Spinors 141 6.2.3 Representations of the Groups SL(2, C) 142 6.2.4 Irreducible Representations 144

6.3 Spinor Representations of the Lorentz Group 145 6.3.1 Four-Dimensional Irreducible Representations . . . . 145 6.3.2 Two-Dimensional Representations 147 6.3.3 The Direct Product of Irreducible Representations . 149

6.4 Solved Problems 150

7 Dirac Spinors 157 7.1 The Dirac Equation 157

7.1.1 The Classical Relativistic Wave Equation 157 7.1.2 The Dirac Equation for a Free Particle 158 7.1.3 A Particle in an Electromagnetic Field 159

x Contents

7.2 Relativistic Invariance of the Dirac Equation 160 7.2.1 The Relativistic Invariance Condition 160 7.2.2 The Type of Representation for the Wave Function . 161 7.2.3 The Link Between a Spinor and a Four-Vector . . . 162 7.2.4 Dirac's Equation in the Spinor Representation . . . 164 7.2.5 The Symmetrie Form of the Dirac Equation 165

7.3 Solved Problems 166

8 Clifford and Lie Algebras 171 8.1 Lie Algebras 171

8.1.1 The Definition of an Algebra 171 8.1.2 Lie Algebras 172 8.1.3 Isomorphic Lie Algebras 173

8.2 Representations of Lie Algebras 174 8.2.1 Definition 174 8.2.2 Representations of a Lie Group and of Its

Lie Algebra 176 8.2.3 Connected Groups 178 8.2.4 Reducible and Irreducible Representations 178

8.3 Clifford Algebras 179 8.3.1 Definition 179 8.3.2 Examples of Clifford Algebras 180 8.3.3 Clifford and Lie Algebras 184 8.3.4 Spinor Groups 186

8.4 Solved Problems , 189

Appendix: Groups and Their Representations 197 A.l The Definition of a Group 197

A.l.l Examples of Groups 197 A.l.2 The Axioms Defining a Group 200 A.l.3 Elementary Properties of Groups 201

A.2 Linear Operators 203 A.2.1 The Operator Representing an Element of a Group . 203 A.2.2 The Operators Acting on the Vectors of

Geometrie Space 204 A.2.3 The Operators Acting on Wave Functions 205 A.2.4 Operators Representing a Group 207

A.3 Matrix Representations 207 A.3.1 The Rotation Matrix Acting on the Vectors

of a Three-Dimensional Space 207 A.3.2 The Matrix of an Operator Acting on Functions . . 208 A.3.3 The Matrices Representing the Elements of a Group 209

A.4 Matrix Representations 210 A.4.1 The Definition of a Matrix Representation 210

Contents xi

A.4.2 The Fundamental Property of the Matrices of a Representation 210

A.4.3 Representation by Regulär Matrices 211 A.4.4 Equivalent Representations 212

A.5 Reducible and Irreducible Representations 214 A.5.1 The Direct Sum of Two Vector Spaces 214 A.5.2 The Direct Sum of Two Representations 214 A.5.3 Irreducible Representations 215

A.6 The Direct Product of Representations 216 A.6.1 The Direct Product of Two Matrices 216 A.6.2 Properties of Tensor Products of Matrices 217 A.6.3 The Direct Product of Two Representations 219

References 221

Index 223

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