Spinors in Four-Dimensional Spaces

• 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 Goldberg–Sachs Theorem.-3.3.2 Space-Times with Symmetries. Ernst Potentials.-3.4 Killing Spinors.-Exercises.-4 Further Applications.-4.1 Self-Dual Yang–Mills Fields.-4.2 H and H H Spaces.-4.3 Killing Bispinors. The Dirac Operator.-Exercises.-A Bases Induced by Coordinate Systems.-References.

From the reviews:“This is a very welcome addition to the bookshelves of those for whom it is mainly intended: graduate students and more experienced researchers using spinors in relativity or quantum field theory. It is a detailed and careful explanation, beginning with the important algebraic steps, and pushing carefully onward to the use of spinors to make powerful results in these fields of physics much easier to obtain. … This is definitely a book I will be very happy to have on my shelves … .” (J. D. Finley, III, Mathematical Reviews, Issue 2011 g)“This book contains an elementary and essentially self-contained exposition of the two-component spinor formalism for a four-dimensional space-time having an arbitrary signature without appealing to the Clifford algebra methodology. … Each chapter includes numerous examples, and exercises, which make it ideal for the use as a textbook, a reference, or for independent study. It is a notable and welcome addition to the existing literature.” (Joseph D. Zund, Zentralblatt MATH, Vol. 1206, 2011)