Clifford Algebras and Spinors is an introductory book in this fascinating area of Applied Algebra. The book is gradually introducing Vector spaces, spinors, Clifford Algebras with an eye on application to electromagnetism, relativity and gravitation. It is intended for people who never heard of these subjects or for people who use them a lot without having a basic understanding of them; it is targeted mostly to engineers, physicists, mathematicians (working in geometry or analysis) although I am sure that any scientist would appreciate this extraordinary book. Applications are all over the place: inside the chapters they help to understand the new concepts introduced while the ones at the end of each section show useful methods of applying them. And the fact that each section has solutions to the exercises makes the book perfect. I will give a brief description of each chapter, outlining the most important features of it.

Chapters 1-2: could form a course in Vector Spaces and Complex Numbers for undergraduates. Chapter one contains everything about vectors (scalar and vector products, bases and coordinates, linear transformations, dimensions etc). A very elementary definition of a bivector and a Clifford algebra is given in 2 dimensions and some nice pictures define the basic operations with bivectors. Complex numbers are defined in chapter two using both the algebraic and the polar form and a nice representation in terms of bivectors and Clifford algebras is given. Although it might sound complicated the fact that everything is in 2 dimensions makes it really easy.

Chapters 3-7: Chapter 3 concentrates on bivectors, operations with them, and their Clifford algebra and exterior algebra. If you work (or plan to work) in relativity or electromagnetism, the understanding of this section is vital. After all the electromagnetic field is represented by a bivector, while the gravitational field is represented by a combination of bivectors. Chapter 4 speaks about Pauli matrices and spinors. The book makes connections with different areas of physics and applied mathematics to show the usage of the newly introduced objects. Chapter 5 and 6 describe the field of quaternions and its relation with spinors and Clifford algebras while chapter 7 generalizes the cross product to higher dimensions.

Chapters 8-10: This is the best part of the book! Real science applications for people with different backgrounds are finally given. Chapter 8 talks about electromagnetism and show how Maxwells equations can be written conveniently using a 4 dimensional bivector. Chapter 9 looks at Lorentz transformations and introduces Minkowski space, the basis for Special Relativity. Chapter 10 discusses Dirac Equation. Spinors and Clifford Algebras are all over the place and all these subjects are discussesd from this unique perspective.

Chapters 11-13: or welcome to Physics!! If you are into theoretical physics you are going to love these Chapters. They discuss Fierz identities, boomerangs, flags, poles and dipoles as well as special types of spinors (Weyl, Majorana, Dirac). This is no longer elementary and it is targeted specifically at physicists.

Chapter 14-15: define rigurously for the first time Clifford Algebras. Chapter 15 discusses Witt rings, Brauer groups.

Chapters 16-19: they give a different approach to Clifford algebras; they are viewed from the point of matrix algebras. Chapter 17 discusses spin groups and spinor spaces and chapter 19 talks about Mobius transformations and Vahlen matrices.

Chapters 20-22: discuss miscellaneous mathematical topics which find nice solutions in the Clifford algebras and spinors context.

Conclusion: The book starts with elementary description of algebras and spinors being accessible even to undergraduates. Later it becomes more advanced and knowledge of other subjects is highly recommended. I recommend this book to all people involved with science at any level.

Recommended: Yes