9789810242619

Theory of Spinors : An Introduction

by ;
  • ISBN13:

    9789810242619

  • ISBN10:

    9810242611

  • Format: Hardcover
  • Copyright: 2000-04-01
  • Publisher: WORLD SCIENTIFIC PUB CO INC
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Summary

This textbook expounds the relationship between spinors and representations of groups. Authors present the origin of spinors from representation theory and apply the theory of spinors to general relativity theory. Part of text is devoted to curved space-time applications. For advanced undergraduate and graduate students in physics and mathematics and as a resource for researchers.

Table of Contents

Preface vii
Introduction to Group Theory
1(10)
Review of Group Theory
1(3)
Group and Subgroup
1(2)
Normal Subgroup and Factor Group
3(1)
Isomorphism and Homomorphism
3(1)
The Pure Rotation Group SO(3)
4(2)
The Euler Angles
4(2)
The Special Unitary Group SU(2)
6(2)
Homomorphism between the Groups SO(3) and SU(2)
6(2)
Invariant Integrals over Groups
8(2)
Invariant Integral over the Group SO(3)
8(1)
Invariant Integral over the Group SU(2)
9(1)
Problems
10(1)
References for Further Reading
10(1)
Representation Theory
11(24)
Some Basic Concepts
11(2)
Linear Operators
11(1)
Finite-Dimensional Representations
12(1)
Unitary Representations
13(1)
Representations of SO(3) and SU(2)
13(5)
Weyl's Method
13(1)
Infinitesimal Generators
14(2)
Basic Infinitisimal Operators
16(1)
Canonical Basis
16(1)
Unitary Matrices Corresponding to Rotations
17(1)
Matrix Elements of Representations
18(4)
The Spinor Representation of the Group SU(2)
19(1)
Matrix Elements of Representations
20(1)
Properties of Djmn (u)
21(1)
Differential Operators of Rotations
22(5)
Representation of SO(3) in Space of Functions
23(1)
The Differential Operators
24(2)
Angular Momentum Operators
26(1)
Infinite-Dimensional Representations
27(5)
Banach Space
27(2)
Hilbert Space
29(1)
Operators in a Banach Space
30(1)
General Definition of a Representation
30(1)
Continuous Representations
31(1)
Unitary Representations
31(1)
References for Further Reading
32(3)
The Lorentz and SL (2,C) Groups
35(30)
Elements of Special Relativity
35(9)
Postulates of Special Relativity
35(2)
The Galilean Transformation
37(1)
The Lorentz Transformation
38(1)
Derivation of the Lorentz Transformation
38(3)
The Cosmological Transformation
41(3)
The Lorentz Group
44(3)
Orthochronous Lorentz Transformation
46(1)
Subgroups of the Lorentz Group
46(1)
The Infinitesimal Approach
47(7)
Infinitesimal Lorentz Matrices
47(2)
Infinitesimal Operators
49(2)
Determination of the Representation by its Infinitesimal Operators
51(1)
Conclusions
52(1)
Unitarity Conditions
53(1)
The Group SL(2, C) and the Lorentz Group
54(7)
The Group SL(2,C)
54(2)
Homomorphism of the Group SL(2,C) on the Lorentz Group L
56(2)
Kernel of Homomorphism
58(1)
Subgroups of the Group SL(2,C)
59(1)
Connection with Lobachevskian Motions
60(1)
Problems
61(1)
References for Further Reading
62(3)
Two-Component Spinors
65(18)
Spinor Representation of SL(2,C)
65(8)
The Space of Polynomials
65(1)
Realization of the Spinor Representation
66(2)
Two-Component Spinors
68(3)
Examples
71(2)
Operators of the Spinor Representation
73(4)
One-Parameter Subgroups
73(1)
Infinitesimal Operators
74(1)
Matrix Elements of the Spinor Operator D (g)
75(2)
Further Properties of Spinor Representations
77(1)
Infinite-Dimensional Spinors
77(4)
Principal Series of Representations
77(2)
Infinite-Dimensional Spinors
79(2)
Problems
81(1)
References for Further Reading
81(2)
Maxwell, Dirac and Pauli Spinors
83(26)
Maxwell's Theory
83(5)
Maxwell's Equations in Curved Spacetime
86(2)
Spinors in Curved Spacetime
88(4)
Correspondence between Spinors and Tensors
88(1)
Raising and Lowering Spinor Indices
89(1)
Properties of the σ Matrices
89(1)
The Metric gAB'CD' and the Minkowskian Metric ημν
90(1)
Hermitian Spinors
91(1)
Covariant Derivative of a Spinor
92(3)
Spinor Affine Connections
92(1)
Spin Covariant Derivative
93(2)
A Useful Formula
95(1)
The Electromagnetic Field Spinors
95(5)
The Electromagnetic Potential Spinor
95(1)
The Electromagnetic Field Spinor
96(1)
Decomposition of the Electromagnetic Spinor
96(1)
Intrinsic Spin Structure
97(1)
Pauli, Dirac and Maxwell Equations
98(2)
Problems
100(7)
References for Further Reading
107(2)
The Gravitational Field Spinors
109(58)
Elements of General Relativity
109(34)
Riemannian Geometry
110(8)
Principle of Equivalence
118(1)
Principle of General Covariance
119(1)
Gravitational Field Equations
120(3)
The Schwarzschild Solution
123(4)
Experimental Tests of General Relativity
127(6)
Equations of Motion
133(9)
Decomposition of the Riemann Tensor
142(1)
The Curvature Spinor
143(3)
Spinorial Ricci Identity
144(1)
Symmetry of the Curvature Spinor
145(1)
Relation to the Riemann Tensor
146(2)
Bianchi Identities
147(1)
The Gravitational Field Spinors
148(9)
Decomposition of the Riemann Tensor
148(2)
The Gravitational Spinor
150(2)
The Ricci Spinor
152(1)
The Weyl Spinor
153(4)
The Bianchi Identities
157(1)
Problems
157(6)
References for Further Reading
163(4)
The Gauge Field Spinors
167(18)
The Yang-Mills Theory
167(5)
Gauge Invariance
167(1)
Isotopic Spin
168(1)
Conservation of Isotopic Spin and Invariance
168(1)
Isotopic Spin and Gauge Fields
169(1)
Isotopic Gauge Transformation
169(2)
Field Equations
171(1)
Gauge Potential and Field Strength
172(5)
The Yang-Mills Spinor
172(2)
Energy-Momentum Spinor
174(2)
SU(2) Spinors
176(1)
Spinor Indices
176(1)
The Geometry of Gauge Fields
177(6)
Four-Index Tensor
177(2)
Spinor Formulation
179(1)
Comparison with the Gravitational Field
180(2)
Ricci and Einstein Spinors
182(1)
References for Further Reading
183(2)
The Euclidean Gauge Field Spinors
185(12)
Euclidean Spacetime
185(5)
The Euclidean Dirac Equation
186(2)
Algebra of the Matrices sμ
188(2)
The Euclidean Gauge Field Spinors
190(4)
O(4) Two-Component Spinors
190(3)
Self-Dual and Anti-Self-Dual Fields
193(1)
Problems
194(1)
References for Further Reading
194(3)
Index 197

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