Lectures on Clifford Geometric Algebras and

This text, written by established mathematicians and physicists, provides a systematic, unified exposition of Clifford (geometric) algebras. Beginning with an introductory chapter, the book covers the mathematical structure of Clifford algebras and the basic concepts of Clifford analysis, and then provides a detailed examination of the many applications of Clifford algebras to differential geometry, physics, computer vision and robotics. No prior knowledge of the subject is assumed. The booka??s breadth will appeal to graduate students and researchers in mathematics, physics, and engineering. Contents: P. Lounesto, Introduction to Clifford Algebras; I. Porteous, Mathematical Structure of Clifford Algebras; J. Ryan, Clifford Analysis; W. Baylis, Applications of Clifford Algebras in Physics; J. Selig, Clifford Algebras in Engineering; T. Branson, Clifford Bundles and Clifford Algebras; R. Ablamowicz and G. Sobczyk, Appendix: Software for Clifford (Geometric) Algebras

Preface

Rafal Abl amowicz and Garret Sobczyk

xiii

Lecture 1: Introduction to Clifford Algebras

Pertti Lounestot

1

(30)

1.1 Introduction

1

(1)

1.2 Clifford algebra of the Euclidean plane

2

1.3 Quaternions

1

(10)

1.4 Clifford algebra of the Euclidean space R

11

(3)

1.5 The electorn spin in a magnetic field

14

(4)

1.6 From column spinors to spinor operators

18

(5)

1.7 In 4D: Clifford algebra Cl4 of R4

23

(1)

1.8 Clifford algebra of Minkowski sparetime

24

(1)

1.9 The exterior algebra and contractions

25

(1)

1.10 The trassmann-Cayley algebra and shuffle products

26

(1)

1.11 Alternative definitions of the Clifford algebra

26

(3)

1.12 References

29

(2)

Lecture 2: Mathematical Structure of Clifford Algebras

Ian Porteous

31

(22)

2.1 Clifford algebras

31

(9)

2.2 Conjugation

40

(11)

2.3 References

51

(2)

Lecture 3: Clifford Analysis

John Ryan

53

(38)

3.1 Introduction

53

(1)

3.2 Foundations of Clifford analysis

54

(8)

3.3 Other types of Clifford holomorphic functions

62

(3)

3.4 The equation Dkf = 0

65

(6)

3.5 Conformal groups and Clifford analysis

71

(3)

3.6 Conformally flat spin manifolds

74

(1)

3.7 Boundary behavior and Hardy spaces

75

(3)

3.8 More on Clifford analysis on the sphere

78

(4)

3.9 The Fourier transform and Clifford analysis

82

(4)

3.10 Complex Clifford analysis

86

(1)

3.11 References

87

(4)

Lecture 4: Applications of Clifford Algebras in Physics

William E. Baylis

91

(44)

4.1 Introduction

91

(1)

4.2 Three Clifford algebras

92

(10)

4.3 Paravectors and relativity

102

(11)

4.4 Eigenspinors

113

(3)

4.5 Maxwell's equation

116

(14)

4.6 Quantum theory

130

(2)

4.7 Conclusions

132

(1)

4.8 References

132

(3)

Lecture 5: Clifford Algebras in Engineering

J.M. Selig

135

(16)

5.1 Introduction

135

(2)

5.2 Quaternions

137

(4)

5.3 Biquaternions

141

(3)

5.4 Points, lines, and planes

144

(6)

5.5 Computer vision example

150

(1)

5.6 Robot kinematics

151

(3)

5.7 Concluding remarks

154

(1)

5.8 References

155

Lecture 6: Clifford Bundles and Clifford Algebras

Thomas Branson

151

(38)

6.1 Spin geometry

157

(7)

6.2 Conformal structure

164

(12)

6.3 Tractor constructions

176

(10)

6.4 References

186

(3)

Appendix

Rafal Ablamowicz and Garret Sobczyk

189

(22)

7.1 Software for Clifford algebras

189

(17)

7.2 References

206

(5)

Index

211

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Lectures on Clifford Geometric Algebras and Applications

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