The spinor calculus employed in general relativity is a very useful tool; many expressions and computations are considerably simplified if one makes use of spinors instead of tensors. Some advantages of the spinor formalism applied in the four-dimensional space-time of general relativity come from the fact that each spinor index takes two values only, which simplifies the algebraic manipulations. Spinors for spaces of any dimension can be defined in connection with rep resentations of orthogonal groups and in the case of spaces of dimension three, the spinor indices also take two values only, which allows us to apply some of the results found in the two-component spinor formalism of four-dimensional space-time. The spinor formalism for three-dimensional spaces has been partially developed, mainly for spaces with a definite metric, also in connection with gen eral relativity (e.g., in space-plus-time decompositions of space-time), defining the spinors of three-dimensional space from those corresponding to four-dimensional space-time, but the spinor formalism for three-dimensional spaces considered on their own is not widely known or employed. One of the aims of this book is to give an account of the spinor formalism for three-dimensional spaces, with definite or indefinite metric, and its applications in physics and differential geometry. Another is to give an elementary treatment of the spin-weighted functions and their various applications in mathematical physics.
This systematic and self-contained treatment of the theory of three-dimensional spinors and their applications fills an important gap in the literature. Without using the customary Clifford algebras frequently studied in connection with the representations of orthogonal groups, spinors are developed in this work for three-dimensional spaces in a language analogous to the spinor formalism used in relativistic spacetime.
Unique features of this work:
* Systematic, coherent exposition throughout
* Introductory treatment of spinors, requiring no previous knowledge of spinors or advanced knowledge of Lie groups
* Three chapters devoted to the definition, properties and applications of spin-weighted functions, with all background given.
* Detailed treatment of spin-weighted spherical harmonics, properties and many applications, with examples from electrodynamics, quantum mechanics, and relativity
* Wide range of topics, including the algebraic classification of spinors, conformal rescalings, connections with torsion and Cartan's structural equations in spinor form, spin weight, spin-weighted operators and the geometrical meaning of the Ricci rotation coefficients
* Bibliography and index
This work will serve graduate students and researchers in mathematics and mathematical and theoretical physics; it is suitable as a course or seminar text, as a reference text, and may also be used for self-study.
