Clifford Algebras and Spinors (London Mathematical Society Lecture Note Series) [Paperback]

Pertti Lounesto (Author)

Book Description

ISBN-10: 0521005515 | ISBN-13: 978-0521005517 | Publication Date: May 7, 2001 | Edition: 2

This second edition of a popular and unique introduction to Clifford algebras and spinors has three new chapters. The beginning chapters cover the basics: vectors, complex numbers and quaternions are introduced with an eye on Clifford algebras. The next chapters, which will also interest physicists, include treatments of the quantum mechanics of the electron, electromagnetism and special relativity. A new classification of spinors is introduced, based on bilinear covariants of physical observables. This reveals a new class of spinors, residing among the Weyl, Majorana and Dirac spinors. Scalar products of spinors are categorized by involutory anti-automorphisms of Clifford algebras. This leads to the chessboard of automorphism groups of scalar products of spinors. On the algebraic side, Brauer/Wall groups and Witt rings are discussed, and on the analytic, Cauchy's integral formula is generalized to higher dimensions.

Editorial Reviews

Review

"This is certainly one of the best and most useful books written about Clifford algebras and spinors." Mathematical Reviews

Book Description

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.

Product Details

• Paperback: 352 pages
• Publisher: Cambridge University Press; 2 edition (May 7, 2001)
• Language: English
• ISBN-10: 0521005515
• ISBN-13: 978-0521005517
• Product Dimensions: 8.8 x 6 x 1 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 4.6 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #339,714 in Books (See Top 100 in Books)

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23 of 23 people found the following review helpful:

4.0 out of 5 stars For the Physicist - Not the Mathematician, November 12, 2002

By A Customer

This review is from: Clifford Algebras and Spinors (London Mathematical Society Lecture Note Series) (Paperback)

Lounesto's book is replete with geometric and physical applications. The treatment is informal and non-rigorous and appears to have been designed with developing intuition in the reader. The text starts slowly working through many examples of particular Clifford algebras of interest and their relevence to physical problems. Towards the end Lounesto investigates general Clifford algebras and their associated spin groups as well as some specialized topics.

This is a good introductory text but fails to give the reader a firm mathematical basis of the material. Most striking is the almost total lack of proofs of any kind - the author is content merely to state the most important results but seldom leaves the reader with any mathematical justification. As such it is really a primer and the student of Clifford algebras must after working through the material move beyond to a rigorous algebraic text.

17 of 17 people found the following review helpful:

5.0 out of 5 stars The best introduction to Clifford algebras and spinors, December 25, 2006

By 

Prof C. R. PAIVA "Carlos Paiva" (Lisbon, Portugal) - See all my reviews
(REAL NAME)   

This review is from: Clifford Algebras and Spinors (London Mathematical Society Lecture Note Series) (Paperback)

The second edition (2001) of this book (from the late Professor Pertti Lounesto) should be considered, for those interested in Clifford algebras and their applications in physics and in engineering, as a pedagogically brilliant introduction.

Chapters 1 and 2 form a pedagogical unit for an undergraduate course on vectors (and the scalar product) and on a geometrical interpretation of complex numbers. Although only the geometric algebra of the plane is addressed in these two chapters, the student will get a firm grasp of the first (real) Clifford algebra (with dimension 4) defined on the 2D linear space R*R (with dimension 2). A clear distinction between a common associative algebra (such as any matrix representation of this first geometric algebra) and the Clifford algebra itself is stressed. In fact, by introducing the square of a vector as the square of its length, this important distinction is definitely drawn. Furthermore, with this definition, a clear interpretation of the Clifford product of vectors is made possible as well as the introduction of bivectors. Reflections and rotations in the plane can then be easily handled, although the rotor concept is not explicitly introduced. In Chapter 2 a beautiful and simple introduction to complex numbers clearly explains how the vector plane is the odd part of the first Clifford algebra, whereas the complex plane is the even part of that same Clifford algebra. Therefore, the student will be able to understand the distinction between the structure of C as a real algebra and the structure of C as the field of complex numbers. Moreover, an important distinction between the unit bivector, which anticommutes with every vector, and the number i=sqrt(-1) as the imaginary unit, which commutes with every vector, is then easily understood.

Although the author does not include Chapter 3 in the same unit as the one formed by the two previous chapters, I would certainly recommend its inclusion within the same pedagogical unit. Indeed, the second (real) Clifford algebra (with dimension 8), defined on the linear space R*R*R (with dimension 3), is a natural extension of the concept of a Clifford algebra (defined over the real field) to the ordinary 3D space. In this context it is possible to explain the important distinction between the cross product of vectors (its result being a vector) and the exterior product of vector (its result being a bivector). Through the Hodge dual, it is then possible to understand the connection between a given vector and its corresponding dual bivector as an oriented plane segment. Moreover, it is clear how the cross product does require a metric while the exterior (or outer) product does not. Indeed, the cross product satisfies the Jacobi identity which makes the linear space R*R*R a non-associative algebra called a Lie algebra. Finally, it is also transparent what distinguishes this second (real) Clifford algebra from Grassmann's exterior algebra: while the Clifford multiplication of vectors does preserve the norm, the exterior multiplication of vectors does not. In fact, this is what allows rotations to become represented as operations inside the Clifford algebra and to state that this algebra provides us with an invertible product for vectors as a direct consequence of its (graded) multivector structure. Although it is possible to define a cross product of two vectors in seven dimensions (see pages 96-98), the student will readily understand the reason why the cross product of vectors only has a unique direction in three dimensions (indeed, only in this case the dual of a vector is a bivector).

Hence, with this undergraduate pedagogical unit formed by Chapters 1-3, we have a real and consistent alternative to the elementary vector algebra solely based on the cross product - as universally promoted since Gibbs misguidedly advocated abandoning quaternions altogether.

Of course it is possible to exclaim that, to teach electromagnetism, the cross product is an invaluable tool. But then, it is also possible to defend an alternate viewpoint: electromagnetism can be easily taught with Clifford algebra as shown is Chapter 8. Indeed, if electromagnetism is to be taught in its natural framework (i.e., inside special relativity), then spacetime algebra provides the proper setting for its mathematical formulation. A very useful textbook for classical mechanics, special relativity, classical electrodynamics, quantum mechanics and gravitation using geometric algebra is «Geometric Algebra for Physicists» by Chris Doran and Anthony Lasenby (Cambridge University Press, 2003).

However, for a unique introduction to Clifford algebras and spinors, including such topics as quaternions (Chapter 5), the fourth dimension (Chapter 6), the cross product (Chapter 7), Pauli spin matrices and spinors (Chapter 4), electromagnetism (Chapter 8), Lorentz transformations (Chapter 9) and the Dirac equation (Chapter 10), this book is a real gem. Chapters 11-23 are more advanced, from a mathematical perspective, as they address technical topics such as: a rigorous definition of Clifford algebras (Chapter 14); other physical applications of spinors (Chapters 11-13); Witt rings and Brauer groups (Chapter 15); matrix representations and periodicity 8 (Chapter 16); spin groups and spinor spaces (Chapter 17); scalar products of spinors and the chessboard (Chapter 18); Möbius transformations and Vahlen matrices (Chapter 19); hypercomplex analysis (Chapter 20); binary index sets and Walsh functions (Chapter 21); Chevalley's construction and characteristic 2 (Chapter 22); octonions and triality (Chapter 23). A fine history of Clifford algebras with bibliography is presented at the end (pages 320-330) of the book.

18 of 19 people found the following review helpful:

4.0 out of 5 stars The best introduction to Clifford algebras., July 26, 1999

By A Customer

This review is from: Clifford Algebras and Spinors (London Mathematical Society Lecture Note Series) (Paperback)

A very clear and comprehensive introduction to a somewhat esoteric subject; essential for anyone in theoretical physics (especially field theory). It is possible to teach yourself the subject from this book alone--a rare feature in mathematics texts. Bravo!