Spinors in Four-Dimensional Spaces (eBook)

Gerardo F. Torres Del Castillo (Autor)

eBook Download: PDF

2010 | 2010
VIII, 177 Seiten
Birkhäuser Boston (Verlag)
978-0-8176-4984-5 (ISBN)

Infos

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.

Inhalt

Preface 6
Contents 
8 
1 Spinor Algebra 10
1.1 Orthogonal Groups 11
1.2 Null Tetrads and the Spinor Equivalent of a Tensor 13
1.3 Spinorial Representation of the Orthogonal Transformations 26
1.3.1 Euclidean Signature 30
1.3.2 Lorentzian Signature 38
1.3.3 Ultrahyperbolic Signature 48
1.4 Reflections 54
1.5 Clifford Algebra. Dirac Spinors 58
1.6 Inner Products. Mate of a Spinor 66
1.7 Principal Spinors. Algebraic Classification 68
Exercises 73
2 Connection and Curvature 76
2.1 Covariant Differentiation 76
2.2 Curvature 86
2.2.1 Curvature Spinors 87
2.2.2 Algebraic Classification of the Conformal Curvature 100
2.3 Conformal Rescalings 102
2.4 Killing Vectors. Lie Derivative of Spinors 104
Exercises 108
3 Applications to General Relativity 112
3.1 Maxwell's Equations 113
3.2 Dirac's Equation 124
3.3 Einstein's Equations 131
3.3.1 The Goldberg–Sachs Theorem 132
3.3.2 Space-Times with Symmetries. Ernst Potentials 140
3.4 Killing Spinors 146
Exercises 153
4 Further Applications 158
4.1 Self-Dual Yang–Mills Fields 158
4.2 H and HH Spaces 161
4.3 Killing Bispinors. The Dirac Operator 171
Exercises 174
Appendix 
176 
References 180
Index 184

Produkt-Details

• Erscheint lt. Verlag: 23.7.2010
• Reihe/Serie: Progress in Mathematical Physics
• Reihe/Serie: Progress in Mathematical Physics
• Zusatzinfo: VIII, 177 p.
• Verlagsort: Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Relativitätstheorie
• Themenwelt: Technik
• Schlagworte: Algebra • Conformal Curvature • Curvature Spinors • Dirac Spinors • Einstein’s Equations • Killing Bispinors • manifold • Potential • Riemannian Geometry • Self-Dual Yang–Mills Fields • Spinor Algebra • theoretical physics • Vector Space
• ISBN-10: 0-8176-4984-0 / 0817649840
• ISBN-13: 978-0-8176-4984-5 / 9780817649845