William Kingdon Clifford published the paper defining his "geometric algebras" in 1878, the year before his death. Clifford algebra is a generalisation to n-dimensional space of quaternions, which Hamilton used to represent scalars and vectors in real three-space: it is also a development of Grassmann's algebra, incorporating in the fundamental relations inner products defined in terms of the metric of the space. It is a strange fact that the Gibbs Heaviside vector techniques came to dominate in scientific and technical literature, while quaternions and Clifford algebras, the true associative algebras of inner-product spaces, were regarded for nearly a century simply as interesting mathematical curiosities. During this period, Pauli, Dirac and Majorana used the algebras which bear their names to describe properties of elementary particles, their spin in particular. It seems likely that none of these eminent mathematical physicists realised that they were using Clifford algebras. A few research workers such as Fueter realised the power of this algebraic scheme, but the subject only began to be appreciated more widely after the publication of Chevalley's book, 'The Algebraic Theory of Spinors' in 1954, and of Marcel Riesz' Maryland Lectures in 1959. Some of the contributors to this volume, Georges Deschamps, Erik Folke Bolinder, Albert Crumeyrolle and David Hestenes were working in this field around that time, and in their turn have persuaded others of the importance of the subject.
Table of ContentsGeneral Surveys.- A Unified Language for Mathematics and Physics.- Clifford Algebras and Spinors.- Classification of Clifford Algebras.- Pseudo-Euclidean Hurwitz Pairs and Generalized Fueter Equations.- A New Representation for Spinors in Real Clifford Algebras.- Primitive Idempotents and Indecomposable Left Ideals in Degenerate Clifford Algebras.- Spin Groups.- Groupes de Clifford et Groupes des Spineurs.- Algebres de Clifford Cr,s+ des Espaces Quadratiques Pseudo-Euclidiens standards Er,s et structures correspondantes sur les espaces de Spineurs Associes. Plongements Naturels des Quadratiques Projectives Reelles Q(E r,s) attachees aux Espaces Er,s.- Spin Groups associated with Degenerate Orthogonal Spaces.- Algebres de Clifford Separables II.- Sur une Question de Micali-Villamayor.- Clifford Analysis.- Spingroups and Spherical Monogenies.- Left Regular Polynomials in Even Dimensions, and Tensor Products of Clifford Algebras.- Spingroups and Spherical Means.- The Biregular Functions of Clifford Analysis: Some Special Topics.- Clifford Numbers and Möbius Transformations in Rn.- Mathematical Physics.- A Clifford Calculus for Physical Field Theories.- Generalized C-R Equations on Manifolds.- Integral Formulae in Complex Clifford Analysis.- Killing Vectors and Embedding of Exact Solutions in General Relativity.- From Grassmann to Clifford.- Lorentzian Applications of Pure Spinors.- The Poincaré Group.- Minimal Ideals and Clifford Algebras in the Phase Space Representation of Spin-1/2 Fields.- Some Consequences of the Clifford Algebra Approach to Physics.- Physical Models.- Algebraic Ideas in Fundamental Physics from Dirac-algebra to Superstrings.- On Two Supersymmetric Approaches to Quantum Gravity: Clifford Algebra Degeneracy v Extended Objects.- Clifford Algebra and the Interpretation of Quantum Mechanics.- Representation-free Calculations in Relativistic Quantum Mechanics.- Dirac Equation for Bispinor Densities.- Unified Spin Gauge Theory Models.- U(2,2) Spin-Gauge Theory Simplification by Use of the Dirac Algebra.- Spin(8) Gauge Field Theory.- Clifford Algebras, Projective Representations and Classification of Fundamental Particles.- Fermionic Clifford Algebras and Supersymmetry.- On Geometry and Physics of Staggered Lattice Fermions.- A System of Vectors and Spinors in Complex Spacetime and Their Application to Elementary Particle Physics.- Spinors as Components of the Metrical Tensor in 8-dimensional Relativity.- Multivector Solution to Harmonic Systems.- The Importance of Meaningful Conservation Equations in Relativistic Quantum Mechanics for the Sources of Classical Fields.- Electromagnetism.- Electromagnetic Theory and Network Theory using Clifford Algebra.- Remarks on Clifford Algebra in Classical Electromagnetism.- Quaternionic Formulation of Classical Electromagnetic Fields and Theory of Functions of a Biquaternion Variable.- Comparison of Clifford and Grassmann Algebras in Applications to Electromagnetics.- Generalisations of Clifford Algebra.- Symplectic Clifford Algebras.- Walsh Functions, Clifford Algebras and Cayley-Dickson Process.- Z(N)-Spin Systems and Generalised Clifford Algebras.- Generalized Clifford Algebras and Spin Lattice Systems.- Clifford Algebra, its Generalisations and Physical Applications.- Application of Clifford Algebras to *-products.- On Regular Functions of a Power-associative Hypercomplex Variable.- On a Geometric Torogonal Quantization Scheme.