An Introduction to Clifford Algebras and Spinors

An Introduction to Clifford Algebras and Spinors
Author: Jayme Vaz Jr.
Publisher: Oxford University Press
Total Pages: 257
Release: 2016
Genre: Mathematics
ISBN: 0198782926

This work is unique compared to the existing literature. It is very didactical and accessible to both students and researchers, without neglecting the formal character and the deep algebraic completeness of the topic along with its physical applications.

Clifford Algebras: An Introduction

Clifford Algebras: An Introduction
Author: D. J. H. Garling
Publisher: Cambridge University Press
Total Pages: 209
Release: 2011-06-23
Genre: Mathematics
ISBN: 1107096383

A straightforward introduction to Clifford algebras, providing the necessary background material and many applications in mathematics and physics.

Clifford Algebra to Geometric Calculus

Clifford Algebra to Geometric Calculus
Author: David Hestenes
Publisher: Springer Science & Business Media
Total Pages: 340
Release: 1984
Genre: Mathematics
ISBN: 9789027725615

Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebra' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quaternions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.

Clifford Algebras and Lie Theory

Clifford Algebras and Lie Theory
Author: Eckhard Meinrenken
Publisher: Springer Science & Business Media
Total Pages: 331
Release: 2013-02-28
Genre: Mathematics
ISBN: 3642362168

This monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan’s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci’s proof of the Poincaré–Birkhoff–Witt theorem. This is followed by discussions of Weil algebras, Chern--Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant’s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra. Aside from these beautiful applications, the book will serve as a convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics.

Quaternions, Clifford Algebras and Relativistic Physics

Quaternions, Clifford Algebras and Relativistic Physics
Author: Patrick R. Girard
Publisher: Springer Science & Business Media
Total Pages: 177
Release: 2007-06-25
Genre: Mathematics
ISBN: 3764377917

The use of Clifford algebras in mathematical physics and engineering has grown rapidly in recent years. Whereas other developments have privileged a geometric approach, this book uses an algebraic approach that can be introduced as a tensor product of quaternion algebras and provides a unified calculus for much of physics. It proposes a pedagogical introduction to this new calculus, based on quaternions, with applications mainly in special relativity, classical electromagnetism, and general relativity.

Introduction to Clifford Analysis

Introduction to Clifford Analysis
Author: Johan Ceballos
Publisher: Nova Science Publishers
Total Pages: 182
Release: 2020-10-30
Genre:
ISBN: 9781536185331

This book pursues to exhibit how we can construct a Clifford type algebra from the classical one. The basic idea of these lecture notes is to show how to calculate fundamental solutions to either first-order differential operators of the form D=∑_(i=0)^n▒〖e_i δ_i〗or second-order elliptic differential operators ̄D D, both with constant coefficients or combinations of this kind of operators. After considering in detail how to find the fundamental solution we study the problem of integral representations in a classical Clifford algebra and in a dependent-parameter Clifford algebra which generalizes the classical one. We also propose a basic method to extend the order of the operator, for instance D^n,n∈N and how to produce integral representations for higher order operators and mixtures of them. Although the Clifford algebras have produced many applications concerning boundary value problems, initial value problems, mathematical physics, quantum chemistry, among others; in this book we do not discuss these topics as they are better discussed in other courses. Researchers and practitioners will find this book very useful as a source book.The reader is expected to have basic knowledge of partial differential equations and complex analysis. When planning and writing these lecture notes, we had in mind that they would be used as a resource by mathematics students interested in understanding how we can combine partial differential equations and Clifford analysis to find integral representations. This in turn would allow them to solve boundary value problems and initial value problems. To this end, proofs have been described in rigorous detail and we have included numerous worked examples. On the other hand, exercises have not been included.

Geometric Computing with Clifford Algebras

Geometric Computing with Clifford Algebras
Author: Gerald Sommer
Publisher: Springer Science & Business Media
Total Pages: 559
Release: 2013-06-29
Genre: Computers
ISBN: 3662046210

This monograph-like anthology introduces the concepts and framework of Clifford algebra. It provides a rich source of examples of how to work with this formalism. Clifford or geometric algebra shows strong unifying aspects and turned out in the 1960s to be a most adequate formalism for describing different geometry-related algebraic systems as specializations of one "mother algebra" in various subfields of physics and engineering. Recent work shows that Clifford algebra provides a universal and powerful algebraic framework for an elegant and coherent representation of various problems occurring in computer science, signal processing, neural computing, image processing, pattern recognition, computer vision, and robotics.

A New Approach to Differential Geometry using Clifford's Geometric Algebra

A New Approach to Differential Geometry using Clifford's Geometric Algebra
Author: John Snygg
Publisher: Springer Science & Business Media
Total Pages: 472
Release: 2011-12-09
Genre: Mathematics
ISBN: 081768283X

Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space. Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.

Clifford Algebras and Spinors

Clifford Algebras and Spinors
Author: Pertti Lounesto
Publisher: Cambridge University Press
Total Pages: 352
Release: 2001-05-03
Genre: Mathematics
ISBN: 0521005515

This is the second edition of a popular work offering a unique introduction to Clifford algebras and spinors. The beginning chapters could be read by undergraduates; vectors, complex numbers and quaternions are introduced with an eye on Clifford algebras. The next chapters will also interest physicists, and include treatments of the quantum mechanics of the electron, electromagnetism and special relativity with a flavour of Clifford algebras. This edition has three new chapters, including material on conformal invariance and a history of Clifford algebras.