Clifford Algebras and Their Applications in Mathematical Physics (Nato Science Series C:, 183)

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. Front Matter....Pages i-xix A Unified Language for Mathematics and Physics....Pages 1-23 Clifford Algebras and Spinors....Pages 25-37 Pseudo-Euclidean Hurwitz Pairs and Generalized Fueter Equations....Pages 39-48 A New Representation for Spinors in Real Clifford Algebras....Pages 49-60 Primitive Idempotents and Indecomposable Left Ideals in Degenerate Clifford Algebras....Pages 61-65 Groupes De Clifford Et Groupes Des Spineurs....Pages 67-78 Spin Groups Associated with Degenerate Orthogonal Spaces....Pages 79-91 Algebres De Clifford Separables II....Pages 93-102 Sur Une Question De Micali-Villamayor....Pages 103-107 Spingroups and Spherical Monogenics....Pages 109-114 Left Regular Polynomials in Even Dimensions, and Tensor Products of Clifford Algebras....Pages 115-132 Spingroups and Spherical Means....Pages 133-147 The Biregular Functions of Clifford Analysis: Some Special Topics....Pages 149-158 Clifford Numbers and Möbius Transformations in R n ....Pages 159-166 A Clifford Calculus for Physical Field Theories....Pages 167-175 Generalized C-R Equations on Manifolds....Pages 177-199 Integral Formulae in Complex Clifford Analysis....Pages 201-217 Killing Vectors and Embedding of Exact Solutions in General Relativity....Pages 219-226 From Grassmann to Clifford....Pages 227-244 Lorentzian Applications of Pure Spinors....Pages 245-255 The Poincaré Group....Pages 257-263 Minimal Ideals and Clifford Algebras in the Phase Space Representation of Spin-1/2 Fields....Pages 265-272 Some Consequences of the Clifford Algebra Approach to Physics....Pages 273-283 Algebraic Ideas in Fundamental Physics from Dirac-Algebra to Superstrings....Pages 285-292 On two Supersymmetric Approaches to Quantum Gravity: Clifford Algebra Degeneracy v Extended Objects....Pages 293-312 Clifford Algebra and the Interpretation of Quantum Mechanics....Pages 313-320 Representation-Free Calculations in Relativistic Quantum Mechanics....Pages 321-346 Dirac Equation for Bispinor Densities....Pages 347-352 Unified Spin Gauge Theory Models....Pages 353-361 U(2,2) Spin-Gauge Theory Simplification by use of the Dirac Algebra....Pages 363-370 Spin(8) Gauge Field Theory....Pages 371-376 Clifford Algebras, Projective Representations and Classification of Fundamental Particles....Pages 377-383 Fermionic Clifford Algebras and Supersymmetry....Pages 385-391 On Geometry and Physics of Staggered Lattice Fermions....Pages 393-398 A System of Vectors and Spinors in Complex Spacetime and their Application to Elementary Particle Physics....Pages 399-423 Spinors as Components of the Metrical Tensor in 8-Dimensional Relativity....Pages 425-434 Multivector Solution to Harmonic Systems....Pages 435-443 The Importance of Meaningful Conservation Equations in Relativistic Quantum Mechanics for the Sources of Classical Fields....Pages 445-454 Electromagnetic Theory and Network Theory Using Clifford Algebra....Pages 455-463 Remarks on Clifford Algebra in Classical Electromagnetism....Pages 465-483 Quaternionic Formulation of Classical Electromagnetic Fields and Theory of Functions of a Biquaternion Variable....Pages 485-493 Comparison of Clifford and Grassmann Algebras in Applications to Electromagnetics....Pages 495-500 Symplectic Clifford Algebras....Pages 501-515 Walsh Functions, Clifford Algebras and Cayley-Dickson Process....Pages 517-529 Z(N)-Spin Systems and Generalised Clifford Algebras....Pages 531-540 Generalized Clifford Algebras and Spin Lattice Systems....Pages 541-548 Clifford Algebra, Its Generalisations and Physical Applications....Pages 549-554 Application of Clifford Algebras to *-Products....Pages 555-558 On Regular Functions of a Power-Associative Hypercomplex Variable....Pages 559-564 On a Geometric Torogonal Quantization Scheme....Pages 565-572 Back Matter....Pages 573-582 ....Pages 583-592 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 GibbsƯ 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