24 internautes sur 24 ont trouvé ce commentaire utile
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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.
19 internautes sur 19 ont trouvé ce commentaire utile
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.
21 internautes sur 26 ont trouvé ce commentaire utile
Osher Doctorow, Ph.D.
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Clifford algebras make geometry and its applications to advanced physics incredibly simple, and this book is one of the best that I have read on this topic and on spinors. Readers outside physics should also study this book, if necessary with the help of a consultant or tutor to translate into more or less ordinary English, because most fields of science or industry are in need of tremendous simplification. Lounesto's approach is algebraic, as is Okubo's (see my review of his book) and Chisholm's (likewise), and Cambridge University Press as usual is at the head of the field in publishing the deepest and yet most simple topics. Spinors (the word comes from spin plus the ending -ors) describe spin in quantum theory, and Lounesto has the most detailed division of the types of spinors that I have seen in a book together with their physical applications. Weyl and Majorana spinors describe the neutrino (of the weak force, although the former only describe massless neutrinos), flagpole spinors appear to describe the strong nuclear force/interaction and appear to be related to quark confinement, Dirac spinors describe the electron (Weyl and Majorana spinors, unlike Dirac spinors, are singular with a light-like pole/current). Penrose flags (see my reviews of Roger Penrose's books) are related to Weyl and Majorana spinors. Penrose has an interesting theory of twistors which is well reviewed in some of the popular science books.
2 internautes sur 5 ont trouvé ce commentaire utile
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I would like to see more algebraic theory of spinors in this book (D. Salamon's "Spin geometry and SW invariants" possess better though somewhat one-sided insight in this field); nevertheless, the amount of multifarious information containd here some of which I have been able to get only though Internet (sometimes with controversial results and always with some frustrating notation), is outstanding.
Part of the book is dedicated to the application of spinors to physics, and this seems to be useful especially for high-school professors.