Clifford Algebras and Spinors (London Mathematical Society Lecture Note Series Book 286) 2nd Edition by Pertti Lounesto (PDF)

Description

In this book, Professor Lounesto offers a unique introduction to Clifford algebras and spinors. The initial 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 book also gives the first comprehensive survey of recent research on Clifford algebras. A new classification of spinors is introduced, based on bilinear covariants of physical observables. This reveals a new class of spinors, residing between the Weyl, Majorana and Dirac spinors. Scalar products of spinors are classified by involutory anti-automorphisms of Clifford algebras. This leads to the chessboard of automorphism groups of scalar products of spinors. On the analytic side, Brauer-Wall groups and Witt rings are discussed, and Caucy’s integral formula is generalized to higher dimensions.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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.

⭐Although it is a very readable introduction, the elaboration on contraction operation is poorly done. The author seems reluctant to give general definitions to make the material appear elementary, even when it is necessary. For example, the introduction of the three types of involution, namely the grade involution, the Clifford conjugation and the reversion should have been given an exact definition at the very beginning. It is not that the general definition is difficult to grasp, but that understanding them in a geometric way is. On the other hand, the book certainly has its merits in explaining concepts geometrically and does an amazing job on the chapter about matrix representation and periodicity of 8. It would be even nicer if the classification of even subalgebras and degenerate Clifford algebras are also touched upon. One optional suggestion would be to add a guideline to the chapters for those who wish not touch upon physical applications.

⭐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.

⭐Testo chiaro ben curato con argomenti sviluppati in modo didatticamente mirabile ed esemplare. Ottimo come riferimento e come testo principale.

⭐

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