Without using the customary Clifford algebras frequently studied in connection with therepresentations 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 YangMills theory, are derived in detail using illustrative examples.Spinors in Four-Dimensional Spaces is aimed at graduate students and researchers inmathematical 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.

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 Maxwells Equations.-3.2 Diracs Equation .-3.3 Einsteins 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.