An introduction to Clifford algebras and spinors /
Clifford algebras and spinors
Jayme Vaz, Jr., Roldão da Rocha, Jr.
- First edition.
- xiv, 242 pages ; 26 cm
Includes bibliographical references (pages 231-237) and index.
Preliminaries -- Vectors and Covectors -- The Tensor Product -- Tensor Algebra -- Exercises -- Exterior Algebra and Grassmann Algebra -- Permutations and the Alternator -- p-Vectors and p-Covectors -- The Exterior Product -- The Exterior Algebra (V) -- The Exterior Algebra as the Quotient of the Tensor Algebra -- The Contraction, or Interior Product -- Orientation, and Quasi-Hodge Isomorphisms -- The Regressive Product -- The Grassmann Algebra -- The Hodge Isomorphism -- Additional Readings -- Exercises -- Clifford, or Geometric, Algebra -- Definition of a Clifford Algebra -- Universal Clifford Algebra as a Quotient of the Tensor Algebra -- Some General Considerations -- Prom the Grassmann Algebra to the Clifford Algebra -- Grassmann Algebra versus Clifford Algebra -- Notation -- Additional Readings -- Exercises -- Classification and Representation of the Clifford Algebras -- Theorems on the Structure of Clifford Algebras -- The Classification of Clifford Algebras -- Idempotents and Representations -- Clifford Algebra Representations -- Additional Readings -- Exercises -- Clifford Algebras, and Associated Groups -- Orthogonal Transformations and the Cartan -- Dieudonne Theorem -- The Clifford -- Lipschitz Group -- The Pin Group and the Spin Group -- Conformal Transformations in Clifford Algebras -- Additional Readings -- Exercises -- Spinors -- The Babel of Spinors -- Algebraic Spinors -- Classical Spinors -- Spinor Operators -- A Comparison of the Different Definitions of Spinors -- The Inner Product in the Space of Algebraic Spinors -- The Triality Principle in the Clifford Algebraic Context -- Pure Spinors -- Dual Rotations, and the Penrose Flagpole -- Weyl Spinors in Cl3,0 -- Weyl Spinors in the Clifford Algebra Cl0,3 H H -- Spinor Transformations -- Spacetime Vectors as Paravectors of Cl3,0 from Weyl Spinors -- Paravectors of Cl4,1 in Cl3,0 via the Periodicity Theorem -- Twistors as Geometric Multivectors -- Spinor Classification According to Bilinear Covariants -- Additional Readings -- Exercises -- A The Standard Two-Component Spinor Formalism -- Weyl Spinors -- Contravariant Undotted Spinors -- Covariant Undotted Spinors -- Contravariant Dotted Spinors -- Covariant Dotted Spinors -- Null Flags and Flagpoles -- The Supersymmetry Algebra -- List of Symbols. Machine generated contents note: 1. 1.1. 1.2. 1.3. 1.4. 2. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. 2.10. 2.11. 2.12. 3. 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 4. 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 5. 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 6. 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.10. 6.11. 6.12. 6.13. 6.14. 6.15. 6.16. 6.17. 6.18. Appendix A.1. A.2. A.3. A.4. A.5. A.6. A.7. Appendix B