An Introduction to Clifford algebras and spinors : Jayme Vaz, Jr. (IMECC, Universidade Estadual de Campinas, Campinas, SP, Brazil), Roldão da Rocha, Jr.(CMCC - Universidade Federal do ABC, Santo André, SP, Brazil)

## Abstract This book is unique in the literature on spinors and Clifford algebras in that it is accessible to both students and researchers while maintaining a formal approach to these subjects. Besides thoroughly introducing several aspects of Clifford algebras, it provides the geometrical aspects underlying the Clifford algebras, as well as their applications, particularly in physics. Previous books on spinors and Clifford algebras have either required the reader to have some prior expertise in these subjects, and thus were difficult to access, or did not provide a deep approach. In contrast, although this book is mathematically complete and precise, it demands little in the way of prerequisites—indeed, a course in linear algebra is the sole prerequisite. This book shows how spinors and Clifford algebras have fuelled interest in the no man’s land between physics and mathematics, an interest resulting from the growing awareness of the importance of algebraic and geometric properties in many physical phenomena. There is much common ground between Clifford algebras, including the geometry arising from those algebras, the classical groups, and the so-called spinors and their three definitions, including pure spinors and twistors, with their main point of contact being the representations of Clifford algebras and the periodicity theorems. Clifford algebras constitute a highly intuitive formalism and have an intimate relationship with quantum field theory; thus, this book will be useful for physicists as well as for mathematicians. This Work Is Unique Compared To The Existing Literature. It Is Very Didactical And Accessible To Both Students And Researchers, Without Neglecting The Formal Character And The Deep Algebraic Completeness Of The Topic Along With Its Physical Applications. Machine Generated Contents Note: 1. Preliminaries -- 1.1. Vectors And Covectors -- 1.2. The Tensor Product -- 1.3. Tensor Algebra -- 1.4. Exercises -- 2. Exterior Algebra And Grassmann Algebra -- 2.1. Permutations And The Alternator -- 2.2. P-vectors And P-covectors -- 2.3. The Exterior Product -- 2.4. The Exterior Algebra (v) -- 2.5. The Exterior Algebra As The Quotient Of The Tensor Algebra -- 2.6. The Contraction, Or Interior Product -- 2.7. Orientation, And Quasi-hodge Isomorphisms -- 2.8. The Regressive Product -- 2.9. The Grassmann Algebra -- 2.10. The Hodge Isomorphism -- 2.11. Additional Readings -- 2.12. Exercises -- 3. Clifford, Or Geometric, Algebra -- 3.1. Definition Of A Clifford Algebra -- 3.2. Universal Clifford Algebra As A Quotient Of The Tensor Algebra -- 3.3. Some General Considerations -- 3.4. Prom The Grassmann Algebra To The Clifford Algebra -- 3.5. Grassmann Algebra Versus Clifford Algebra -- 3.6. Notation -- 3.7. Additional Readings -- 3.8. Exercises -- 4. Classification And Representation Of The Clifford Algebras -- 4.1. Theorems On The Structure Of Clifford Algebras -- 4.2. The Classification Of Clifford Algebras -- 4.3. Idempotents And Representations -- 4.4. Clifford Algebra Representations -- 4.5. Additional Readings -- 4.6. Exercises -- 5. Clifford Algebras, And Associated Groups -- 5.1. Orthogonal Transformations And The Cartan -- Dieudonne Theorem -- 5.2. The Clifford -- Lipschitz Group -- 5.3. The Pin Group And The Spin Group -- 5.4. Conformal Transformations In Clifford Algebras -- 5.5. Additional Readings -- 5.6. Exercises -- 6. Spinors -- 6.1. The Babel Of Spinors -- 6.2. Algebraic Spinors -- 6.3. Classical Spinors -- 6.4. Spinor Operators -- 6.5. A Comparison Of The Different Definitions Of Spinors -- 6.6. The Inner Product In The Space Of Algebraic Spinors -- 6.7. The Triality Principle In The Clifford Algebraic Context -- 6.8. Pure Spinors -- 6.9. Dual Rotations, And The Penrose Flagpole -- 6.10. Weyl Spinors In Cl3,0 -- 6.11. Weyl Spinors In The Clifford Algebra Cl0,3 H H -- 6.12. Spinor Transformations -- 6.13. Spacetime Vectors As Paravectors Of Cl3,0 From Weyl Spinors -- 6.14. Paravectors Of Cl4,1 In Cl3,0 Via The Periodicity Theorem -- 6.15. Twistors As Geometric Multivectors -- 6.16. Spinor Classification According To Bilinear Covariants -- 6.17. Additional Readings -- 6.18. Exercises -- Appendix A The Standard Two-component Spinor Formalism -- A.1. Weyl Spinors -- A.2. Contravariant Undotted Spinors -- A.3. Covariant Undotted Spinors -- A.4. Contravariant Dotted Spinors -- A.5. Covariant Dotted Spinors -- A.6. Null Flags And Flagpoles -- A.7. The Supersymmetry Algebra -- Appendix B List Of Symbols. Jayme Vaz, Jr., (imecc, Universidade Estadual De Campinas, Sp, Brazil), Roldão Da Rocha, Jr. (cmcc-universidade Federal Do Abc, Santo André, Sp, Brazil). Includes Bibliographical References (pages 231-237) And Index. This text explores how Clifford algebras and spinors have been sparking a collaboration and bridging a gap between Physics and Mathematics. This collaboration has been the consequence of a growing awareness of the importance of algebraic and geometric properties in many physical phenomena, and of the discovery of common ground through various touch points: relating Clifford algebras and the arising geometry to so-called spinors, and to their three definitions (both from the mathematical and physical viewpoint). The main point of contact are the representations of Clifford algebras and the periodicity theorems. Clifford algebras also constitute a highly intuitive formalism, having an intimate relationship to quantum field theory. The text strives to seamlessly combine these various viewpoints and is devoted to a wider audience of both physicists and mathematicians. Among the existing approaches to Clifford algebras and spinors this book is unique in that it provides a didactical presentation of the topic and is accessible to both students and researchers. It emphasizes the formal character and the deep algebraic and geometric completeness, and merges them with the physical applications. The style is clear and precise, but not pedantic. The sole pre-requisites is a course in Linear Algebra which most students of Physics, Mathematics or Engineering will have covered as part of their undergraduate studies. This book provides a unique pedagogical introduction to clifford algebras, with a focus on spinors. It bridges the gap between mathematics and physics, merging both applications and the formal approach. It provides detailed worked examples throughout to help understand the ideas presented. "This work is unique compared to the existing literature. It is very didactical and accessible to both students and researchers, without neglecting the formal character and the deep algebraic completeness of the topic along with its physical applications."--Résumé de l'éditeur This book is unique compared to the existing literature. It is very didactical and accessible to both students and researchers, without neglecting the formal character and the deep algebraic completeness of the topic along with its physical applications.