E-Book, Englisch, Band 59, 177 Seiten
Torres del Castillo Spinors in Four-Dimensional Spaces
2010
ISBN: 978-0-8176-4984-5
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark
E-Book, Englisch, Band 59, 177 Seiten
Reihe: Progress in Mathematical Physics
ISBN: 978-0-8176-4984-5
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark
Without using the customary Clifford algebras frequently studied in connection with the representations of orthogonal groups, this book gives an elementary introduction to the two-component spinor formalism for four-dimensional spaces with any signature. Some of the useful applications of four-dimensional spinors, such as Yang-Mills theory, are derived in detail using illustrative examples. Spinors in Four-Dimensional Spaces is aimed at graduate students and researchers in mathematical and theoretical physics interested in the applications of the two-component spinor formalism in any four-dimensional vector space or Riemannian manifold with a definite or indefinite metric tensor. This systematic and self-contained book is suitable as a seminar text, a reference book, and a self-study guide.
Autoren/Hrsg.
Weitere Infos & Material
1;Preface;6
2;Contents
;8
3;1 Spinor Algebra;10
3.1;1.1 Orthogonal Groups;11
3.2;1.2 Null Tetrads and the Spinor Equivalent of a Tensor;13
3.3;1.3 Spinorial Representation of the Orthogonal Transformations;26
3.3.1;1.3.1 Euclidean Signature;30
3.3.2;1.3.2 Lorentzian Signature;38
3.3.3;1.3.3 Ultrahyperbolic Signature;48
3.4;1.4 Reflections;54
3.5;1.5 Clifford Algebra. Dirac Spinors;58
3.6;1.6 Inner Products. Mate of a Spinor;66
3.7;1.7 Principal Spinors. Algebraic Classification;68
3.8;Exercises;73
4;2 Connection and Curvature;76
4.1;2.1 Covariant Differentiation;76
4.2;2.2 Curvature;86
4.2.1;2.2.1 Curvature Spinors;87
4.2.2;2.2.2 Algebraic Classification of the Conformal Curvature;100
4.3;2.3 Conformal Rescalings;102
4.4;2.4 Killing Vectors. Lie Derivative of Spinors;104
4.5;Exercises;108
5;3 Applications to General Relativity;112
5.1;3.1 Maxwell's Equations;113
5.2;3.2 Dirac's Equation;124
5.3;3.3 Einstein's Equations;131
5.3.1;3.3.1 The Goldberg–Sachs Theorem;132
5.3.2;3.3.2 Space-Times with Symmetries. Ernst Potentials;140
5.4;3.4 Killing Spinors;146
5.5;Exercises;153
6;4 Further Applications;158
6.1;4.1 Self-Dual Yang–Mills Fields;158
6.2;4.2 H and HH Spaces;161
6.3;4.3 Killing Bispinors. The Dirac Operator;171
6.4;Exercises;174
7;Appendix
A Bases Induced by Coordinate Systems;176
8;References;180
9;Index;184
