Ebook: Clifford Algebras and Their Applications in Mathematical Physics

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.

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Content:
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
A Clifford Calculus for Physical Field Theories....Pages 159-166
Generalized C-R Equations on Manifolds....Pages 167-175
Integral Formulae in Complex Clifford Analysis....Pages 177-199
Killing Vectors and Embedding of Exact Solutions in General Relativity....Pages 201-217
From Grassmann to Clifford....Pages 219-226
Lorentzian Applications of Pure Spinors....Pages 227-244
The Poincar? Group....Pages 245-255
Minimal Ideals and Clifford Algebras in the Phase Space Representation of Spin-1/2 Fields....Pages 257-263
Some Consequences of the Clifford Algebra Approach to Physics....Pages 265-272
Algebraic Ideas in Fundamental Physics from Dirac-Algebra to Superstrings....Pages 273-283
On two Supersymmetric Approaches to Quantum Gravity: Clifford Algebra Degeneracy v Extended Objects....Pages 285-292
Clifford Algebra and the Interpretation of Quantum Mechanics....Pages 293-312
Representation-Free Calculations in Relativistic Quantum Mechanics....Pages 313-320
Dirac Equation for Bispinor Densities....Pages 321-346
Unified Spin Gauge Theory Models....Pages 347-352
U(2,2) Spin-Gauge Theory Simplification by use of the Dirac Algebra....Pages 353-361
Spin(8) Gauge Field Theory....Pages 363-370
Clifford Algebras, Projective Representations and Classification of Fundamental Particles....Pages 371-376
Fermionic Clifford Algebras and Supersymmetry....Pages 377-383
On Geometry and Physics of Staggered Lattice Fermions....Pages 385-391
A System of Vectors and Spinors in Complex Spacetime and their Application to Elementary Particle Physics....Pages 393-398
Spinors as Components of the Metrical Tensor in 8-Dimensional Relativity....Pages 399-423
Multivector Solution to Harmonic Systems....Pages 425-434
The Importance of Meaningful Conservation Equations in Relativistic Quantum Mechanics for the Sources of Classical Fields....Pages 435-443
Electromagnetic Theory and Network Theory Using Clifford Algebra....Pages 445-454
Remarks on Clifford Algebra in Classical Electromagnetism....Pages 455-463
Quaternionic Formulation of Classical Electromagnetic Fields and Theory of Functions of a Biquaternion Variable....Pages 465-483
Comparison of Clifford and Grassmann Algebras in Applications to Electromagnetics....Pages 485-493
Symplectic Clifford Algebras....Pages 495-500
Walsh Functions, Clifford Algebras and Cayley-Dickson Process....Pages 501-515
Z(N)-Spin Systems and Generalised Clifford Algebras....Pages 517-529
Generalized Clifford Algebras and Spin Lattice Systems....Pages 531-540
Clifford Algebra, Its Generalisations and Physical Applications....Pages 541-548
Application of Clifford Algebras to *-Products....Pages 549-554
On Regular Functions of a Power-Associative Hypercomplex Variable....Pages 555-558
On a Geometric Torogonal Quantization Scheme....Pages 559-564
Back Matter....Pages 565-572
....Pages 573-582