Clifford Algebras and Spinors

ISBN-13: 9780521005517 / Angielski / Miękka / 2001 / 352 str.

Pertti Lounesto; J. W. S. Cassels; N. J. Hitchin

Opis

This second edition of a popular and unique introduction to Clifford algebras and spinors has three new chapters. The beginning chapters cover the basics: vectors, complex numbers and quaternions are introduced with an eye on Clifford algebras. The next chapters, which will also interest physicists, include treatments of the quantum mechanics of the electron, electromagnetism and special relativity. A new classification of spinors is introduced, based on bilinear covariants of physical observables. This reveals a new class of spinors, residing among the Weyl, Majorana and Dirac spinors. Scalar products of spinors are categorized by involutory anti-automorphisms of Clifford algebras. This leads to the chessboard of automorphism groups of scalar products of spinors. On the algebraic side, Brauer/Wall groups and Witt rings are discussed, and on the analytic, Cauchy's integral formula is generalized to higher dimensions.

Recenzje

'This book cannot be underestimated in its current influence.' B. Fauser, Zentralblatt für Mathematik

Spis treści

Preface; Mathematical notation; 1. Vectors and linear spaces; 2. Complex numbers; 3. Bivectors and the exterior algebra; 4. Pauli spin matrices and spinors; 5. Quaternions; 6. The fourth dimension; 7. The cross product; 8. Electromagnetism; 9. Lorentz transformations; 10. The Dirac equation; 11. Fierz identities and boomerangs; 12. Flags, poles and dipoles; 13. Tilt to the opposite metric; 14. Definitions of the Clifford algebra; 15. Witt rings and Brauer groups; 16. Matrix representations and periodicity of 8; 17. Spin groups and spinor spaces; 18. Scalar products of spinors and the chessboard; 19. Möbius transformations and Vahlen matrices; 20. Hypercomplex analysis; 21. Binary index sets and Walsh functions; 22. Chevalley's construction and characteristic 2; 23. Octonions and triality; A history of Clifford algebras; Selected reading; Index.