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Prof C. R. PAIVA
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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.