Clifford Algebras: Applications to Mathematics, Physics, and Engineering

ISBN-13: 9781461273936 / Angielski / Miękka / 2011 / 626 str.

Rafal Ablamowicz

Opis

The invited papers in this volume provide a detailed examination of Clifford algebras and their significance to analysis, geometry, mathematical structures, physics, and applications in engineering. 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.

Spis treści

PREFACE * PART I. CLIFFORD ANALYSIS * 1. The Morera Problem in Clifford Algebras and the Heisenberg Group by Carlos A. Berenstein, Der-Chen Chang, and Wayne M. Eby * 2. Multidimensional Inverse Scattering Associated with the Schrödinger Equation by Swanhild Bernstein * 3. On Discrete Stokes and Navier–Stokes Equations in the Plane by Klaus Gürlebeck and Angela Hommel * 4. A Symmetric Functional Calculus for Systems of Operators of Type w by Brian Jefferies * 5. Poincaré Series in Clifford Analysis by Rolf Sören Krausshar * 6. Harmonic Analysis for General First Order Differential Operators in Lipschitz Domains by Emilio Marmolejo-Olea and Marius Mitrea * 7. Paley–Wiener Theorems and Shannon Sampling in the Clifford Analysis Setting by Tao Qian * 8. Bergman Projection in Clifford Analysis by Guangbin Ren and Helmuth R. Malonek * 9. Quaternionic Calculus for a Class of Initial Boundary Value Problems by Wolfgang Sprössig * PART II. GEOMETRY * 10. A Nahm Transform for Instantons over ALE Spaces by Claudio Bartocci and Marcos Jardim * 11. Hyper-Hermitian Manifolds and Connections with Skew-Symmetric Torsion by Gueo Grantcharov * 12. Casimir Elements and Bochner Identities on Riemannian Manifolds by Yasushi Homma * 13. Eigenvalues of Dirac and Rarita–Schwinger Operators by Doojin Hong * 14. Differential Forms Canonically Associated to Even-Dimensional Compact Conformal Manifolds by William J. Ugalde * 15. The Interface of Noncommutative Geometry and Physics by Joseph C. Várilly * PART III. MATHEMATICAL STRUCTURES * 16. The Method of Virtual Variables and Representations of Lie Superalgebras by Andrea Brini, Francesco Regonati, and Antonio Teolis * 17. Algebras Like Clifford Algebras by Michael Eastwood * 18. Grade Free Product Formulæ from Grassmann–Hopf Gebras by Bertfried Fauser * 19. The Clifford Algebra in the Theory of Algebras, Quadratic Forms, and Classical Groups by Alexander Hahn * 20. Lipschitz’s Methods of 1886 Applied to SymplecticClifford Algebras by Jacques Helmstetter * 21. The Group of Classes of Involutions of Graded Central Simple Algebras by Jacques Helmstetter * 22. A Binary Index Notation for Clifford Algebras by Dennis W. Marks * 23. Transposition in Clifford Algebra: SU(3) from Reorientation Invariance by Bernd Schmeikal * PART IV. PHYSICS * 24. The Quantum/Classical Interface: Insights from Clifford’s (Geometric) Algebra by William E. Baylis * 25. Standard Quantum Spheres by Francesco Bonechi, Nicola Ciccoli, and Marco Tarlini * 26. Clifford Algebras, Pure Spinors and the Physics of Fermions by Paolo Budinich * 27. Spinor Formulations for Gravitational Energy-Momentum by Chiang-Mei Chen, James M. Nester, and Roh-Suan Tung * 28. Chiral Dirac Equations by Claude Daviau * 29. Using Octonions to Describe Fundamental Particles by Tevian Dray and Corinne A. Manogue * 30. Applications of Geometric Algebra in Electromagnetism, Quantum Theory and Gravity by Anthony Lasenby, Chris Doran, and Elsa Arcaute * 31. Noncommutative Physics on Lie Algebras, (Z2)n Lattices and Clifford Algebras by Shahn Majid * 32. Dirac Operator on Quantum Homogeneous Spaces and Noncommutative Geometry by Robert M. Owczarek * 33. r-Fold Multivectors and Superenergy by Jose M. Pozo and Josep M. Parra * 34. The Cl7 Approach to the Standard Model by Greg Trayling and William E. Baylis * PART V. APPLICATIONS IN ENGINEERING * 35. Implementation of a Clifford Algebra Co-Processor Design on a Field Programmable Gate Array by Christian Perwass, Christian Gebken, and Gerald Sommer * 36. Image Space by Jan J. Koenderink * 37. Pose Estimation of Cycloidal Curves by using Twist Representations by Bodo Rosenhahn and Gerald Sommer * INDEX

Blurb

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.