Ebook: Clifford Algebras: Applications to Mathematics, Physics, and Engineering

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.

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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 Schr?dinger equation, discrete Stokes equations in the plane, a symmetric functional calculus, Poincar? 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.

---

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 Schr?dinger equation, discrete Stokes equations in the plane, a symmetric functional calculus, Poincar? 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.

Content:
Front Matter....Pages i-xxiv
Front Matter....Pages 1-1
The Morera Problem in Clifford Algebras and the Heisenberg Group....Pages 3-21
Multidimensional Inverse Scattering Associated with the Schr?dinger Equation....Pages 23-34
On Discrete Stokes and Navier—Stokes Equations in the Plane....Pages 35-58
A Symmetric Functional Calculus for Systems of Operators of Type ?....Pages 59-74
Poincar? Series in Clifford Analysis....Pages 75-89
Harmonic Analysis for General First Order Differential Operators in Lipschitz Domains....Pages 91-114
Paley—Wiener Theorems and Shannon Sampling in the Clifford Analysis Setting....Pages 115-124
Bergman Projection in Clifford Analysis....Pages 125-139
Quaternionic Calculus for a Class of Initial Boundary Value Problems....Pages 141-151
Front Matter....Pages 153-153
A Nahm Transform for Instantons over ALE Spaces....Pages 155-166
Hyper-Hermitian Manifolds and Connections with Skew-Symmetric Torsion....Pages 167-183
Casimir Elements and Bochner Identities on Riemannian Manifolds....Pages 185-199
Eigenvalues of Dirac and Rarita—Schwinger Operators....Pages 201-210
Differential Forms Canonically Associated to Even-Dimensional Compact Conformal Manifolds....Pages 211-225
The Interface of Noncommutative Geometry and Physics....Pages 227-242
Front Matter....Pages 243-243
The Method of Virtual Variables and Representations of Lie Superalgebras....Pages 245-263
Algebras Like Clifford Algebras....Pages 265-278
Grade Free Product Formul? from Grassmann-Hopf Gebras....Pages 279-301
The Clifford Algebra in the Theory of Algebras, Quadratic Forms, and Classical Groups....Pages 305-322
Lipschitz’s Methods of 1886 Applied to Symplectic Clifford Algebras....Pages 323-333
Front Matter....Pages 243-243
The Group of Classes of Involutions of Graded Central Simple Algebras....Pages 335-341
A Binary Index Notation for Clifford Algebras....Pages 343-350
Transposition in Clifford Algebra: SU(3) from Reorientation Invariance....Pages 351-372
Front Matter....Pages 373-373
The Quantum/Classical Interface: Insights from Clifford’s (Geometric) Algebra....Pages 375-389
Standard Quantum Spheres....Pages 393-399
Clifford Algebras, Pure Spinors and the Physics of Fermions....Pages 401-416
Spinor Formulations for Gravitational Energy-Momentum....Pages 417-430
Chiral Dirac Equations....Pages 431-450
Using Octonions to Describe Fundamental Particles....Pages 451-466
Applications of Geometric Algebra in Electromagnetism, Quantum Theory and Gravity....Pages 467-489
Dirac Operator on Quantum Homogeneous Spaces and Noncommutative Geometry....Pages 491-518
Front Matter....Pages 519-530
Implementation of a Clifford Algebra Co-Processor Design on a Field Programmable Gate Array....Pages 531-546
Image Space....Pages 547-558
Pose Estimation of Cycloidal Curves by using Twist Representations....Pages 559-559
Back Matter....Pages 561-575
....Pages 577-596