Spinors in Four-Dimensional Spaces

ISBN-13: 9780817649838 / Angielski / Twarda / 2010 / 177 str.

Gerardo F. Torres Castillo; Gerardo F. Torres Del Castillo; Gerardo F. Torres del Castillo

Opis

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.

Recenzje

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)

Spis treści

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.

Blurb

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