3-D Spinors, Spin-Weighted Functions and Their Applications

by Del Castillo, Gerardo F. Torres; Torres Del Castillo, G. F.

• ISBN13: 9780817632496
• ISBN10: 0817632492
• Format: Hardcover
• Copyright: 2003-08-01
• Publisher: Birkhauser

Summary

This systematic and self-contained treatment of the theory of three-dimensional spinors and their applications fills an important gap in the literature. Without using the customary Clifford algebras frequently studied in connection with the representations of orthogonal groups, spinors are developed in this work for three-dimensional spaces in a language analogous to the spinor formalism used in relativistic spacetime. Unique features of this work: * Systematic, coherent exposition throughout * Introductory treatment of spinors, requiring no previous knowledge of spinors or advanced knowledge of Lie groups * Three chapters devoted to the definition, properties and applications of spin-weighted functions, with all background given. * Detailed treatment of spin-weighted spherical harmonics, properties and many applications, with examples from electrodynamics, quantum mechanics, and relativity * Wide range of topics, including the algebraic classification of spinors, conformal rescalings, connections with torsion and Cartan's structural equations in spinor form, spin weight, spin-weighted operators and the geometrical meaning of the Ricci rotation coefficients * Bibliography and index This work will serve graduate students and researchers in mathematics and mathematical and theoretical physics; it is suitable as a course or seminar text, as a reference text, and may also be used for self-study.

Table of Contents

Preface	vii
Rotations and Spinors	1	(36)
Representation of the rotations	2	(5)
Spinors	7	(12)
Elementary applications	19	(5)
Spinors in spaces with indefinite metric	24	(13)
Spin-Weighted Spherical Harmonics	37	(22)
Spherical harmonics	37	(5)
Spin weight	42	(9)
Wigner functions	51	(8)
Spin-Weighted Spherical Harmonics. Applications	59	(52)
Solution of the vector Helmholtz equation	59	(12)
The source-free electromagnetic field	71	(8)
The equation for elastic waves in an isotropic medium	79	(5)
The Weyl neutrino equation	84	(4)
The Dirac equation	88	(9)
The spin-2 Helmholtz equation	97	(4)
Linearized Einstein theory	101	(6)
Magnetic monopole	107	(4)
Spin-Weighted Cylindrical Harmonics	111	(40)
Definitions and basic properties	111	(4)
Representation of the Euclidean group of the plane	115	(4)
Applications	119	(20)
Solution of the vector Helmholtz equation	120	(6)
Elastic waves in an isotropic elastic medium	126	(4)
Solution of the equations of equilibrium for an isotropic elastic medium	130	(2)
Solution of the Dirac equation	132	(4)
Solution of the spin-2 Helmholtz equation	136	(3)
Parabolic and elliptic coordinates	139	(7)
Spin-weighted parabolic harmonics	143	(1)
Spin-weighted elliptic harmonics	144	(2)
Applications	146	(5)
Solution of the vector Helmholtz equation	146	(1)
Divergenceless vector fields	147	(1)
Solution of the Dirac equation	147	(4)
Spinor Algebra	151	(26)
The spinor equivalent of a tensor	151	(4)
The orthogonal and spin groups	155	(5)
Positive definite metric	157	(1)
Indefinite metric	158	(2)
Algebraic classification	160	(14)
The triad defined by a spinor	174	(3)
Spinor Analysis	177	(38)
Covariant differentiation	177	(4)
Curvature	181	(4)
Spin weight and priming operation	185	(12)
Positive definite metric	185	(9)
Indefinite metric	194	(3)
Metric connections with torsion	197	(3)
Congruences of curves	200	(4)
Applications	204	(11)
Applications to General Relativity	215	(20)
Spacelike hypersurfaces	216	(8)
Timelike hypersurfaces	224	(1)
Stationary space-times	225	(10)
Appendix: Spinors in the Four-Dimensional Space-Time	235	(4)
References	239	(4)
Index	243