Lectures on Clifford (Geometric) Algebras and Applications

Advances in technology over the last 25 years have created a situation in which workers in diverse areas of computerscience and engineering have found it neces­ sary to increase their knowledge of related fields in order to make further progress. Clifford (geometric) algebra offers a unified algebraic framework for the direct expression of the geometric ideas underlying the great mathematical theories of linear and multilinear algebra, projective and affine geometries, and differential geometry. Indeed, for many people working in this area, geometric algebra is the natural extension of the real number system to include the concept of direction. The familiar complex numbers of the plane and the quaternions of four dimen­ sions are examples of lower-dimensional geometric algebras. During "The 6th International Conference on Clifford Algebras and their Ap­ plications in Mathematical Physics" held May 20--25, 2002, at Tennessee Tech­ nological University in Cookeville, Tennessee, a Lecture Series on Clifford Ge­ ometric Algebras was presented. Its goal was to to provide beginning graduate students in mathematics and physics and other newcomers to the field with no prior knowledge of Clifford algebras with a bird's eye view of Clifford geometric algebras and their applications. The lectures were given by some of the field's most recognized experts. The enthusiastic response of the more than 80 partici­ pants in the Lecture Series, many of whom were graduate students or postdocs, encouraged us to publish the expanded lectures as chapters in book form.

Table of Contents:
Lecture 1: Introduction to Clifford Algebras.- 1.1 Introduction.- 1.2 Clifford algebra of the Euclidean plane.- 1.3 Quaternions.- 1.4 Clifford algebra of the Euclidean space ?3.- 1.5 The electorn spin in a magnetic field.- 1.6 From column spinors to spinor operators.- 1.7 In 4D: Clifford algebra C?4 of ?4.- 1.8 Clifford algebra of Minkowski spacetime.- 1.9 The exterior algebra and contractions.- 1.10 The Grassmann-Cayley algebra and shuffle products.- 1.11 Alternative definitions of the Clifford algebra.- 1.12 References.- Lecture 2: Mathematical Structure of Clifford Algebras.- 2.1 Clifford algebras.- 2.2 Conjugation.- 2.3 References.- Lecture 3: Clifford Analysis.- 3.1 Introduction.- 3.2 Foundations of Clifford analysis.- 3.3 Other types of Clifford holomorphic functions.- 3.4 The equation Dkf=0.- 3.5 Conformal groups and Clifford analysis.- 3.6 Conformally flat spin manifolds.- 3.7 Boundary behavior and Hardy spaces.- 3.8 More on Clifford analysis on the sphere.- 3.9 The Fourier transform and Clifford analysis.- 3.10 Complex Clifford analysis.- 3.11 References.- Lecture 4: Applications of Clifford Algebras in Physics.- 4.1 Introduction.- 4.2 Three Clifford algebras.- 4.3 Paravectors and relativity.- 4.4 Eigenspinors.- 4.5 Maxwell’s equation.- 4.6 Quantum theory.- 4.7 Conclusions.- 4.8 References.- Lecture 5: Clifford Algebras in Engineering.- 5.1 Introduction.- 5.2 Quaternions.- 5.3 Biquaternions.- 5.4 Points, lines, and planes.- 5.5 Computer vision example.- 5.6 Robot kinematics.- 5.7 Concluding remarks.- 5.8 References.- Lecture 6: Clifford Bundles and Clifford Algebras.- 6.1 Spin geometry.- 6.2 Conformal structure.- 6.3 Tractor constructions.- 6.4 References.- 211.