Summary Note |
The invited papers in this volume provide a detailed examination of Clifford algebras and their significance to geometry, analysis, physics, and engineering. Divided into five parts, the book's first section is devoted to Clifford analysis; here, topics encompass the Morera problem, inverse scattering associated with the Schrödinger equation, discrete Stokes equations in the plane, a symmetric functional calculus, Poincaré series, differential operators in Lipschitz domains, Paley-Wiener theorems and Shannon sampling, Bergman projections, and quaternionic calculus for a class of boundary value problems. A careful discussion of geometric applications of Clifford algebras follows, with papers on hyper-Hermitian manifolds, spin structures and Clifford bundles, differential forms on conformal manifolds, connection and torsion, Casimir elements and Bochner identities on Riemannian manifolds, Rarita-Schwinger operators, and the interface between noncommutative geometry and physics. In addition, attention is paid to the algebraic and Lie-theoretic applications of Clifford algebras---particularly their intersection with Hopf algebras, Lie algebras and representations, graded algebras, and associated mathematical structures. Symplectic Clifford algebras are also discussed. Finally, Clifford algebras play a strong role in both physics and engineering. The physics section features an investigation of geometric algebras, chiral Dirac equations, spinors and Fermions, and applications of Clifford algebras in classical mechanics and general relativity. Twistor and octonionic methods, electromagnetism and gravity, elementary particle physics, noncommutative physics, Dirac's equation, quantum spheres, and the Standard Model are among topics considered at length. The section devoted to engineering applications includes papers on twist representations for cycloidal curves, a description of an image space using Cayley-Klein geometry, pose estimation, and implementations of Clifford algebra co-processor design. While the papers collected in this volume require that the reader possess a solid knowledge of appropriate background material, they lead to the most current research topics. With its wide range of topics, well-established contributors, and excellent references and index, this book will appeal to graduate students and researchers.: |

Contents note |
I. Clifford Analysis -- 1. The Morera Problem in Clifford Algebras and the Heisenberg Group -- 2. Multidimensional Inverse Scattering Associated with the Schrödinger Equation -- 3. On Discrete Stokes and Navier—Stokes Equations in the Plane -- 4. A Symmetric Functional Calculus for Systems of Operators of Type ? -- 5. Poincaré Series in Clifford Analysis -- 6. Harmonic Analysis for General First Order Differential Operators in Lipschitz Domains -- 7. Paley—Wiener Theorems and Shannon Sampling in the Clifford Analysis Setting -- 8. Bergman Projection in Clifford Analysis -- 9. Quaternionic Calculus for a Class of Initial Boundary Value Problems -- II. Geometry -- 10. A Nahm Transform for Instantons over ALE Spaces -- 11. Hyper-Hermitian Manifolds and Connections with Skew-Symmetric Torsion -- 12. Casimir Elements and Bochner Identities on Riemannian Manifolds -- 13. Eigenvalues of Dirac and Rarita—Schwinger Operators -- 14. Differential Forms Canonically Associated to Even-Dimensional Compact Conformal Manifolds -- 15. The Interface of Noncommutative Geometry and Physics -- III. Mathematical Structures -- 16. The Method of Virtual Variables and Representations of Lie Superalgebras -- 17. Algebras Like Clifford Algebras -- 18. Grade Free Product Formulæ from Grassmann—Hopf Gebras -- 19. The Clifford Algebra in the Theory of Algebras, Quadratic Forms, and Classical Groups -- 20. Lipschitz’s Methods of 1886 Applied to Symplectic Clifford Algebras -- 21. The Group of Classes of Involutions of Graded Central Simple Algebras -- 22. A Binary Index Notation for Clifford Algebras -- 23. Transposition in Clifford Algebra: SU(3) from Reorientation Invariance -- IV. Physics -- 24. The Quantum/Classical Interface: Insights from Clifford’s (Geometric) Algebra -- 25. Standard Quantum Spheres -- 26. Clifford Algebras, Pure Spinors and the Physics of Fermions -- 27. Spinor Formulations for Gravitational Energy-Momentum -- 28. Chiral Dirac Equations -- 29. Using Octonions to Describe Fundamental Particles -- 30. Applications of Geometric Algebra in Electromagnetism, Quantum Theory and Gravity -- 31. Noncommutative Physics on Lie Algebras, (?2)n Lattices and Clifford Algebras -- 32. Dirac Operator on Quantum Homogeneous Spaces and Noncommutative Geometry -- 33. r-Fold Multivectors and Superenergy -- 34. The Cl7 Approach to the Standard Model -- V. Applications in Engineering -- 35. Implementation of a Clifford Algebra Co-Processor Design on a Field Programmable Gate Array -- 36. Image Space -- 37. Pose Estimation of Cycloidal Curves by using Twist Representations. |