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
CONTENT
1 Rotations and Spinors -- 1.1 Representation of the rotations -- 1.2 Spinors -- 1.3 Elementary applications -- 1.4 Spinors in spaces with indefinite metric -- 2 Spin-Weighted Spherical Harmonics -- 2.1 Spherical harmonics -- 2.2 Spin weight -- 2.3 Wigner functions -- 3 Spin-Weighted Spherical Harmonics. Applications -- 3.1 Solution of the vector Helmholtz equation -- 3.2 The source-free electromagnetic field -- 3.3 The equation for elastic waves in an isotropic medium -- 3.4 The Weyl neutrino equation -- 3.5 The Dirac equation -- 3.6 The spin-2 Helmholtz equation -- 3.7 Linearized Einstein theory -- 3.8 Magnetic monopole -- 4 Spin-Weighted Cylindrical Harmonics -- 4.1 Definitions and basic properties -- 4.2 Representation of the Euclidean group of the plane -- 4.3 Applications -- 4.4 Parabolic and elliptic coordinates -- 4.5 Applications -- 5 Spinor Algebra -- 5.1 The spinor equivalent of a tensor -- 5.2 The orthogonal and spin groups -- 5.3 Algebraic classification -- 5.4 The triad defined by a spinor -- 6 Spinor Analysis -- 6.1 Covariant differentiation -- 6.2 Curvature -- 6.3 Spin weight and priming operation -- 6.4 Metric connections with torsion -- 6.5 Congruences of curves -- 6.6 Applications -- 7 Applications to General Relativity -- 7.1 Spacelike hypersurfaces -- 7.2 Timelike hypersurfaces -- 7.3 Stationary space-times -- Appendix: Spinors in the Four-Dimensional Space-Time -- References
Physics
Topological groups
Lie groups
Applied mathematics
Engineering mathematics
Gravitation
Physics
Physics general
Applications of Mathematics
Topological Groups Lie Groups
Mathematical Methods in Physics
Classical and Quantum Gravitation Relativity Theory
