Theory of spinors : an introduction

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Preface......Page 8
Contents......Page 10
1.1.1 Group and Subgroup......Page 16
1.1.3 Isomorphism and Homomorphism......Page 18
1.2.1 The Euler Angles......Page 19
1.3.1 Homomorphism between the Groups SO(3) and SU(2)......Page 21
1.4.1 Invariant Integral over the Group SO(3)......Page 23
1.4.2 Invariant Integral over the Group SU(2)......Page 24
1.6 References for Further Reading......Page 25
2.1.1 Linear Operators......Page 26
2.1.2 Finite-Dimensional Representations......Page 27
2.2.1 Weyl's Method......Page 28
2.2.2 Infinitesimal Generators......Page 29
2.2.4 Canonical Basis......Page 31
2.2.5 Unitary Matrices Corresponding to Rotations......Page 32
2.3 Matrix Elements of Representations......Page 33
2.3.1 The Spinor Representation of the Group SU(2)......Page 34
2.3.2 Matrix Elements of Representations......Page 35
2.3.3 Properties of (u)......Page 36
2.4 Differential Operators of Rotations......Page 37
2.4.1 Representation of SO(3) in Space of Functions......Page 38
2.4.2 The Differential Operators......Page 39
2.4.3 Angular Momentum Operators......Page 41
2.5.1 Banach Space......Page 42
2.5.2 Hilbert Space......Page 44
2.5.4 General Definition of a Representation......Page 45
2.5.6 Unitary Representations......Page 46
2.6 References for Further Reading......Page 47
3.1.1 Postulates of Special Relativity......Page 50
3.1.2 The Galilean Transformation......Page 52
3.1.4 Derivation of the Lorentz Transformation......Page 53
3.1.5 The Cosmological Transformation......Page 56
3.2 The Lorentz Group......Page 59
3.2.2 Subgroups of the Lorentz Group......Page 61
3.3.1 Infinitesimal Lorentz Matrices......Page 62
3.3.2 Infinitesimal Operators......Page 64
3.3.3 Determination of the Representation by its Infinitesimal Operators......Page 66
3.3.4 Conclusions......Page 67
3.3.5 Unitarity Conditions......Page 68
3.4.1 The Group SL(2,C)......Page 69
3.4.2 Homomorphism of the Group SL(2,C) on the Lorentz Group L......Page 71
3.4.3 Kernel of Homomorphism......Page 73
3.4.4 Subgroups of the Group SL(2,C)......Page 74
3.4.5 Connection with Lobachevskian Motions......Page 75
3.5 Problems......Page 76
3.6 References for Further Reading......Page 77
4.1.1 The Space of Polynomials......Page 80
4.1.2 Realization of the Spinor Representation......Page 81
4.1.3 Two-Component Spinors......Page 83
4.1.4 Examples......Page 86
4.2.1 One-Parameter Subgroups......Page 88
4.2.2 Infinitesimal Operators......Page 89
4.2.3 Matrix Elements of the Spinor Operator D (g)......Page 90
4.3.1 Principal Series of Representations......Page 92
4.3.2 Infinite-Dimensional Spinors......Page 94
4.5 References for Further Reading......Page 96
5.1 Maxwell's Theory......Page 98
5.1.1 Maxwell's Equations in Curved Spacetime......Page 101
5.2.1 Correspondence between Spinors and Tensors......Page 103
5.2.3 Properties of the o Matrices......Page 104
5.2.4 The Metric 9AB'CD' and the Minkowskian Metric 77?......Page 105
5.2.5 Hermitian Spinors......Page 106
5.3.1 Spinor Affine Connections......Page 107
5.3.2 Spin Covariant Derivative......Page 108
5.4.1 The Electromagnetic Potential Spinor......Page 110
5.4.3 Decomposition of the Electromagnetic Spinor......Page 111
5.4.4 Intrinsic Spin Structure......Page 112
5.4.5 Pauli, Dirac and Maxwell Equations......Page 113
5.5 Problems......Page 115
5.6 References for Further Reading......Page 122
6.1 Elements of General Relativity......Page 124
6.1.1 Riemannian Geometry......Page 125
6.1.2 Principle of Equivalence......Page 133
6.1.3 Principle of General Covariance......Page 134
6.1.4 Gravitational Field Equations......Page 135
6.1.5 The Schwarzschild Solution......Page 138
6.1.6 Experimental Tests of General Relativity......Page 142
6.1.7 Equations of Motion......Page 148
6.1.8 Decomposition of the Riemann Tensor......Page 157
6.2 The Curvature Spinor......Page 158
6.2.1 Spinorial Ricci Identity......Page 159
6.2.2 Symmetry of the Curvature Spinor......Page 160
6.3 Relation to the Riemann Tensor......Page 161
6.3.1 Bianchi Identities......Page 162
6.4.1 Decomposition of the Riemann Tensor......Page 163
6.4.2 The Gravitational Spinor......Page 165
6.4.3 The Ricci Spinor......Page 167
6.4.4 The Weyl Spinor......Page 168
6.5 Problems......Page 172
6.6 References for Further Reading......Page 178
7.1.1 Gauge Invariance......Page 182
7.1.3 Conservation of Isotopic Spin and Invariance......Page 183
7.1.5 Isotopic Gauge Transformation......Page 184
7.1.6 Field Equations......Page 186
7.2.1 The Yang-Mills Spinor......Page 187
7.2.2 Energy-Momentum Spinor......Page 189
7.2.4 Spinor Indices......Page 191
7.3.1 Four-Index Tensor......Page 192
7.3.2 Spinor Formulation......Page 194
7.3.3 Comparison with the Gravitational Field......Page 195
7.3.4 Ricci and Einstein Spinors......Page 197
7.4 References for Further Reading......Page 198
8.1 Euclidean Spacetime......Page 200
8.1.1 The Euclidean Dirac Equation......Page 201
8.1.2 Algebra of the Matrices s_\nv......Page 203
8.2.1 0(4) Two-Component Spinors......Page 205
8.2.2 Self-Dual and Anti-Self-Dual Fields......Page 208
8.4 References for Further Reading......Page 209
Index......Page 212
Back Cover......Page 228