Spinors in Four-Dimensional Spaces

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.

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.

Key topics and features:
Uniform treatment of the spinor formalism for four-dimensional spaces of any signature, not only the usual signature (+ + + ) employed in relativity
Examples taken from Riemannian geometry and special or general relativity are
discussed in detail, emphasizing the usefulness of the two-component spinor formalism
Exercises in each chapter
The relationship of Clifford algebras and Dirac four-component spinors is established
Applications of the two-component formalism, focusing mainly on general relativity, are
presented in the context of actual computations

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.

Reviews from the author's previous book, 3-D Spinors, Spin-Weighted Functions and their Applications:

In summary the book gathers much of what can be done with 3-D spinors in an easy-to-read, self-contained form designed for applications that will supplement many available spinor treatments. The book should be appealing to graduate students and researchers in relativity and mathematical physics.
Mathematical Reviews

The present book provides an easy-to-read and unconventional presentation of the spinor formalism for three-dimensional spaces with a definite or indefinite metric...Following a nice and descriptive introduction the final chapter contains some applications of the formalism to general relativity.
Monatshefte für Mathematik

Systematic, coherent exposition throughout Introductory treatment of spinors, requiring no previous knowledge of spinors or advanced knowledge of Lie groups Includes a detailed bibliography and index Includes supplementary material: sn.pub/extras

Inhalt
1 Spinor Algebra.-1.1 Orthogonal Groups.-1.2 Null Tetrads and the Spinor Equivalent of a Tensor.-1.3 Spinorial Representation of the Orthogonal Transformations.-1.3.1 Euclidean Signature.-1.3.2 Lorentzian Signature.-1.3.3 Ultrahyperbolic Signature.-1.4 Reflections.-1.5 Clifford Algebra. Dirac Spinors.-1.6 Inner Products. Mate of a Spinor.-1.7 Principal Spinors. Algebraic Classification.-Exercises.-2 Connection and Curvature.-2.1 Covariant Differentiation .- 2.2 Curvature.-2.2.1 Curvature Spinors.-2.2.2 Algebraic Classification of the Conformal Curvature.-2.3 Conformal Rescalings.-2.4 Killing Vectors. Lie Derivative of Spinors.-Exercises.- 3 Applications to General Relativity.-3.1 Maxwell's Equations.-3.2 Dirac's Equation .-3.3 Einstein's Equations.-3.3.1 The GoldbergSachs Theorem.-3.3.2 Space-Times with Symmetries. Ernst Potentials.-3.4 Killing Spinors.-Exercises.-4 Further Applications.-4.1 Self-Dual YangMills Fields.-4.2 H and H H Spaces.-4.3 Killing Bispinors. The Dirac Operator.-Exercises.-A Bases Induced by Coordinate Systems.-References.