Clifford Algebras and SpinorsCambridge University Press, 2001-05-03 - 338 psl. 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. |
Turinys
1 Vectors and linear spaces | 1 |
2 Complex numbers | 18 |
3 Bivectors and the exterior algebra | 33 |
4 Pauli spin matrices and spinors | 50 |
5 Quaternions | 67 |
6 The fourth dimension | 80 |
7 The cross product | 92 |
8 Electromagnetism | 100 |
15 Witt rings and Brauer groups | 195 |
16 Matrix representations and periodicity of 8 | 205 |
17 Spin groups and spinor spaces | 219 |
18 Scalar products of spinors and the chessboard | 231 |
19 Mobius transformations and Vahlen matrices | 244 |
20 Hypercomplex analysis | 255 |
21 Binary index sets and Walsh functions | 279 |
22 Chevalleys construction and characteristic 2 | 288 |
9 Lorentz transformations | 118 |
10 The Dirac equation | 135 |
11 Fierz identities and boomerangs | 152 |
12 Flags poles and dipoles | 162 |
13 Tilt to the opposite metric | 174 |
14 Definitions of the Clifford algebra | 188 |
23 Octonions and triality | 300 |
A history of Clifford algebras | 320 |
Selected reading | 331 |
| 335 | |
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3-dimensional 3-vector angle anticommute antisymmetric associative algebra automorphism group B₁ basis e1 bilinear covariants bivector Cl(Q Cl₂ Clifford algebra Clifford algebra Cl3 Clifford product Clp,q complex conjugation complex numbers components Compute cross product denoted differential dimension Dirac spinor division algebra e₁ electromagnetic Euclidean space exterior algebra exterior product field F function geometric grade involution Hestenes idempotent inverse isomorphic Lie algebra Lorentz transformation Lounesto Math matrix algebra Maxwell equations minimal left ideal Minkowski Möbius transformations monogenic multivector non-zero octonion oriented orthogonal orthonormal basis plane quadratic form quadratic space quaternion real algebra real linear space real numbers representation Riesz rotation satisfies scalar product space R³ space-time spin group spinor operator spinor space square subalgebra subspace symmetric tensor theory triality U₁ unit vector Vahlen vector field Weyl მყ