Geometric Algebra for Physicists [Paperback]

This is truly a great book for any one who is interested in not just physics, but physical reality. Although the ideas expressed therein have a long history and are by no means as uniquely those of its authors as were Albert Einstein's in his day, I believe that they will have comparable lasting value. Moreover the synthesis presented in this book, which builds pre-eminently on the work of Hestenes, is absolutely superb. Interested readers need not take my word for these claims, but are invited to prove it to themselves.

Although the above should be a sufficient review, my experience nevertheless indicates that it is a good idea to warn potentially enthusiastic readers against several common semantic misconceptions, lest they jump to conclusions which prevent them from ever taking that vital first step. Thus let it be clearly understood that Geometric Algebra is NOT:
(1) A replacement for linear/matrix/tensor algebra (on the contrary, it is a very nice complement to these formalisms).
(2) Identical, or even very close, to Emil Artin's earlier excellent book on bilinear forms with the title "Geometric Algebra".
(3) Another name for the enormous field "algebraic geometry" (it is indeed appropriate that the word stemming from "geometry" comes first in "geometric algebra").
(4) Just another reformulation of complex / quaternion / octonian analysis; for it connects all these purely algebraic objects, and many generalizations thereof, to Felix Klein's Erlangen Programme and Sophus Lie's theory of continuous groups.
(5) The ultimate theory of everything (although it probably will eventually be found to have something to do with it).

Geometric algebra IS a practical and natural (canonical) tool for formulating physical and mathematical problems in homogeneous spaces in a fully covariant fashion. But more importantly, you do not need to understand all those words in order to benefit from it, and this book is an excellent place for physicists of all stripes to start.

This is a well-written book on a very interesting and important subject: geometric algebra (GA) is a powerful and elegant mathematical language -- based on the works of Hamilton, Grassmann and Clifford -- that is especially well-suited for spacetime physics and several fields of engineering.

The authors adopt David Hestenes' viewpoint of a graded GA as a unified mathematical language that is coordinate-free, thereby stressing the fundamental role of geometric invariants in physics.

In fact, the elementary vector analysis -- which pervades almost all undergraduate (and even) graduate approaches to electrodynamics -- finds its roots in the misguided Gibbsian approach: Gibbs advocated abandoning Hamilton's quaternions and just work with scalar and cross products of vectors. However, the cross product has a major flaw: it only exists in three (or seven) dimensions -- if we require that (i) it should have just two factors, (ii) to be orthogonal to the factors, and (iii) to have length equal to the corresponding parallelogram.

Electrodynamics and relativistic physics, particularly, are elegantly presented through GA and otherwise cumbersome calculations may be circumvented in a simple and insightful way.

Mainstream physics and engineering cannot overlook GA anymore.

I'm reading this book somewhat in parallel with Hestenes' New Foundations for Classical Mechanics. Both are fantastic books (Hestenes' predates this one), and in some parts they are complementary, while of course they overlap in the foundations and many special topics. What is so fascinating about Geometric Algebra and Calculus? I think it's mainly the recognition that many seemingly complicated theorems of mathematical physics really become much clearer - in a sense of getting a guts feeling about the geometry. The method opens a way to look at the same thing from totally different angles: If one can't imagine something based on geometric arguments, one can take the presented formalism and translate it back into geometry, and suddenly things become clear.
Is the book (or that by Hestenes) basic and easy to understand or are they difficult? Certainly they require some work by the reader. To follow the entire book, one really can't do without learning to master the formalism of geometric algebra, which is simple, yet sometimes bizarre. I suspect though that it is only bizarre to the one who "knows it all" already: The student or scientist who has grown familiar with vector spaces, matrix notation and wiggling around with tensor notation, needs to go through the same exercises as the bloody beginner to whom even the idea of a vector may not be clear. In fact, the beginner could be at a real advantage to not being poisoned by vector calculus. For example, take the very basic notation for a geometric product of two multi-vectors: ab = a.b + a^b (the sum of inner and outer product). What's so confusing about it? Nothing, really, after one really understands what "+" here means. But it happens often enough that one only thinks about this product in terms of the right hand side of the equation, because those are totally familiar for anyone who took basic linear algebra, and then ends up making simple things complicated again. I must say that it was like loosing shadows from the eyes to see how the formulations in this book and Hestenes' work explain so well why it is that the quantum mechanical psi function needs to be complex, or better yet what really the i means in physics, and how the entire set of Maxwell equations (all 4 of them) are one simple continuity equation. That's the kind of thing that makes your head buzz. I'm not done with these books, but I have a clear feeling that in the end I will have an entry point to understand QM and parts of general relativity not just formally (especially QM) but really develop a guts feeling for it.
One thing that I'm still a bit missing in any of the books related to geometric algebra is classical continuum mechanics. This may be so because many of the authors are immersed in fields related to cosmology. In this book, one can find a tiny little bit also about elasticity (linear and nonlinear). However, I keep wondering what it would be like to reformulate the entire underlying theory of continuum mechanics (about deforming solids, elastic or viscoelastic or plastic, about fluid flow, about polarized materials, biological active materials, etc). Could something new be learned? I bet it could!