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Lecture 1: Introduction to Clifford Algebras Pertti Lounesto 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 electron spin in a magnetic field • 1.6 From column spinors to spinor operators • 1.7 In 4D: Clifford algebra Cℓ 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 Ian Porteous 2.1 Clifford algebras • 2.2 Conjugation • 2.3 References Lecture 3: Clifford Analysis John Ryan 3.1 Introduction • 3.2 Foundations of Clifford analysis • 3.3 Other types of Clifford holomorphic functions • 3.4 The equation D k ƒ = 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 William E. Baylis 4.

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Of the various formalisms developed to treat relativistic phenomena, those based on Clifford's geometric algebra are especially well adapted for clear geometric interpretations and computational efficiency. Here we study relationships between formulations of special relativity in the spacetime algebra (STA) Cℓ 1,3 of the underlying Minkowski vector space, and in the algebra of physical space (APS) Cℓ 3 . STA lends itself to an absolute formulation of relativity, in which paths, fields, and other physical properties have observer-independent representations. Descriptions in APS are related by a one-to-one mapping of elements from APS to the even subalgebra STA + of STA. With this mapping, reversion in APS corresponds to hermitian conjugation in STA. The elements of STA + are all that is needed to calculate physically measurable quantities (called measurables) because only they entail the observer dependence inherent in any physical measurement. As a consequence, every relativistic physical process that can be modeled in STA also has a representation in APS, and vice versa. In the presence of two or more inertial observers, two versions of APS present themselves. In the absolute version, both the mapping to STA + and hermitian conjugation are observer dependent, and the proper basis vectors of any observer are persistent vectors that sweep out timelike planes in spacetime. To compare measurements by different inertial observers in APS, we express them in the proper algebraic basis of a single observer. This leads to the relative version of APS, which can be related to STA by assigning every inertial observer in STA to a single absolute frame in STA. The equivalence of inertial observers makes this permissible. The mapping and hermitian conjugation are then the same for all observers. Relative APS gives a covariant representation of relativistic physics with spacetime multivectors represented by multiparavectors in APS. We relate the two versions of APS as consistent models within the same algebra.

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