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

Format:

Book

Author/Creator:

Vaz Jr., Jayme, author.

Rocha Jr., Roldão da, 1976- author.

Contributor:

Jessie A. Rodman Fund.

Language:

English

Subjects (All):

Clifford algebras.

Spinor analysis.

Physical Description:

xiv, 242 pages ; 26 cm

Edition:

First edition.

Place of Publication:

Oxford : Oxford University Press, 2016.

Summary:

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 points 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 emphasises 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-requisite is a course in linear algebra which most students of physics, mathematics, or engineering will have covered as part of their undergraduate studies. Book jacket.

Contents:

1 Preliminaries 1

1.1 Vectors and Covectors 1

1.2 The Tensor Product 10

1.3 Tensor Algebra 16

1.4 Exercises 19

2 Exterior Algebra and Grassmann Algebra 21

2.1 Permutations and the Alternator 21

2.2 p-Vectors and p-Covectors 22

2.3 The Exterior Product 24

2.4 The Exterior Algebra Δ(V) 29

2.5 The Exterior Algebra as the Quotient of the Tensor Algebra 33

2.6 The Contraction, or Interior Product 36

2.7 Orientation, and Quasi-Hodge Isomorphisms 41

2.8 The Regressive Product 47

2.9 The Grassmann Algebra 48

2.10 The Hodge Isomorphism 50

2.11 Additional Readings 54

2.12 Exercises 54

3 Clifford, or Geometric, Algebra 57

3.1 Definition of a Clifford Algebra 57

3.2 Universal Clifford Algebra as a Quotient of the Tensor Algebra 60

3.3 Some General Considerations 66

3.4 From the Grassmann Algebra to the Clifford Algebra 73

3.5 Grassmann Algebra versus Clifford Algebra 78

3.6 Notation 83

3.7 Additional Readings 84

3.8 Exercises 85

4 Classification and Representation of the Clifford Algebras 87

4.1 Theorems on the Structure of Clifford Algebras 87

4.2 The Classification of Clifford Algebras 93

4.3 Idempotents and Representations 102

4.4 Clifford Algebra Representations 108

4.5 Additional Readings 119

4.6 Exercises 120

5 Clifford Algebras, and Associated Groups 121

5.1 Orthogonal Transformations and the Cartan-Dieudonné Theorem 121

5.2 The Clifford-Lipschitz Group 126

5.3 The Pin Group and the Spin Group 131

5.4 Conformal Transformations in Clifford Algebras 138

5.5 Additional Readings 143

5.6 Exercises 144

6 Spinors 145

6.1 The Babel of Spinors 145

6.2 Algebraic Spinors 148

6.3 Classical Spinors 149

6.4 Spinor Operators 152

6.5 A Comparison of the Different Definitions of Spinors 158

6.6 The Inner Product in the Space of Algebraic Spinors 165

6.7 The Triality Principle in the Clifford Algebraic Context 170

6.8 Pure Spinors 179

6.9 Dual Rotations, and the Penrose Flagpole 186

6.10 Weyl Spinors in Cl3_3,0 190

6.11 Weyl Spinors in the Clifford Algebra Cl_3,0 ≅ H ⊕ H 193

6.12 Spinor Transformations 194

6.13 Spacetime Vectors as Paravectors of Cl_3,0 from Weyl Spinors 196

6.14 Paravectors of Cl_4,1 in Cl_3,0 via the Periodicity Theorem 199

6.15 Twistors as Geometric Multivectors 200

6.16 Spinor Classification According to Bilinear Covariants 203

6.17 Additional Readings 205

6.18 Exercises 206.

Notes:

Includes bibliographical references (pages 231-237) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Jessie A. Rodman Fund.

ISBN:

9780198782926

0198782926

OCLC:

933337412

Publisher Number:

99968476280