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Geometric Algebra for Physicists

Chris Doran and Anthony Lasenby
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Gizem Karaali
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This is the paperback edition of a 2003 hardcover book. As its title suggests it is written for physicists and students of physics. The style is typical of a good physics text: the language is simple and clear, and there are many pictures and illustrations which explain various constructions. Furthermore, the authors go over two and three dimensional examples before focusing on the more general theory valid for any dimension. A mathematician will do this in the classroom, starting with basic examples and building up the more general theory on them, but within the standard development of a mathematics text, the abstract generally takes the center stage. Not so in this book. The various examples are studied most carefully, and the development of the three dimensional case after that of two dimensions is done in just the same amount of detail. This may at first surprise the reader whose background is more in mathematics than in physics, but it certainly makes the content much more accessible.

Next question: What does the geometric algebra in the title refer to? It does not really relate much to Emil Artin's well known book of the same title, unless one reinterprets Artin's book as a study of algebraic structures that help to describe and understand geometric structures. This of course is true of Artin's book, but is a pretty vague description of its contents at best. The phrase geometric algebra is not just a shuffling of the phrase algebraic geometry, either. The algebra involved never gets that difficult, for one thing.

The geometric algebra in this book is the theory developed more than a century ago by Clifford, following in the footsteps of Hamilton and Grassman. The following is standard in an entry level linear algebra course: Students learn they can add and rescale vectors, but we tell them strictly that they cannot multiply them. Then we introduce the dot product and the cross product, and say carefully that the dot product does not yield an output of the same kind as the inputs, and the cross product is a special construction only valid for three dimensions. Some of us may go out of our way to say that a similar structure exists for seven dimensions, but that is it as far as vector multiplication.

Those students who survive to take differential geometry eventually learn of a seemingly different product, the exterior product, but to most, this fact, that you cannot multiply vectors, remains a mystery. Clifford's geometric algebra solves this problem and develops vector analysis from scratch in such a way that allows vectors to be multiplied, among all sorts of other cool stuff.

Basically Clifford's algebra consists of a graded world of scalars (grade 0), vectors (grade 1), and various levels of multivectors (higher grades). The dot product, the cross product and the exterior product can all be incorporated into this framework. And the best part is that the student does not need to wait for a course in differential geometry to see this unity! The relevant structures and constructions can be introduced at any level where we introduce vectors.

The standard interpretation of a vector in three dimensions as a directed line segment in 3-space is easily augmented by geometric interpretations of various multivectors. For instance two dimensional multivectors (bivectors) have nice geometric interpretation as directed (or oriented) areas, and three dimensional multivectors (trivectors) can be viewed as directed (or oriented) volume elements. The multiplication then can be defined easily as well, though one takes care to note that the outputs may not always be homogeneous.

Doran and Lasenby do a marvelous job of developing this whole universe from scracth. Their motivation, however, is not limited to just providing a wonderful exposition of Clifford's geometric algebra. The beauty of the mathematics is undeniable, but clearly is not their sole purpose. They are physicists, after all, and they have ulterior motives. They focus on geometric algebra because they believe (and it must be admitted that this is an ideological book) that geometric algebra provides the most successful mathematical framework for developing most of today's modern physics. The book is filled with various illustrations of this assertion. These many chapters on classical mechanics (Ch3 and Ch12), relativity (Ch5), classical electrodynamics (Ch7), basic and multi-particle quantum theory (Ch8 and Ch9), field theory (Ch14), gravitation (Ch16) are not provided as mere examples; on the contrary, they make up the main body of the book.

If its main audience is made up of physicists and physics students, then why is this book appropriate for the audience of the MAA Reviews? My answer can be found in the above paragraphs, but I will simply repeat myself here. First off, this is a very well written book about some very beautiful mathematics. It is accessible and, since it is written in the physics textbook style, it is perfect for self study, even for someone who does not have the time to do all the exercises. If one does take the time to look them over, though, it will be clear that the contents of the book can be made as rigorous and theoretical as possible. The exercises are in fact very mathematical.

The most appealing feature of this book, however, is that is can be used by a mathematician to get a relatively unified picture of various facets of modern physics, simply by building on this beautiful mathematical framework.

A small caveat here: The physics chapters do flow much better if the reader has previously been exposed to the material in some context. Without such a background, the book may be a lot more painful to read unless one is willing to let go of some of the material.

Gizem Karaali is assistant professor of Mathematics at Pomona College.

Preface; Notation; 1. Introduction; 2. Geometric algebra in two and three dimensions; 3. Classical mechanics; 4. Foundations of geometric algebra; 5. Relativity and spacetime; 6. Geometric calculus; 7. Classical electrodynamics; 8. Quantum theory and spinors; 9. Multiparticle states and quantum entanglement; 10. Geometry; 11. Further topics in calculus and group theory; 12. Lagrangian and Hamiltonian techniques; 13. Symmetry and gauge theory; 14. Gravitation; Bibliography; Index.