New Foundations for Classical Mechanics (Fundamental Theories of Physics) [Paperback]

David Hestenes is a forerunner of the modern development of Clifford algebra. His current research activities can be followed in the site http://modelingnts.la.asu.edu/GC_R&D.html. Probably his most important book until now (written with Garret Sobczyk) was "Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics" (Dordrecht: Kluwer Academic Publishers, 1984) also available at Amazon.com. This book on the new foundations for classical mechanics (second edition) was written as an introduction to geometric algebra. The term "geometric algebra" was coined to stress that this formulation of Clifford algebra is a unified language for physics and mathematics; it is not a matrix algebra (as used in quantum mechanics in the disguised forms of Pauli and Dirac matrices) as it uses a new property, the contraction, which makes it different from other associative algebras. A recent book on geometric algebra is "Geometric Algebra for Physicists" by Chris Doran and Anthony Lasenby (Cambridge: Cambridge University Press, 2003) - see the site http://www.mrao.cam.ac.uk/~clifford/.

Geometric algebra is a graded algebra based on the geometric product of vectors which reduces to the inner product (a scalar) when the two vectors are parallel and to the outer product (a bivector) when the two vectors are orthogonal. The geometric product is associative and can be used in spaces with any dimension (as opposed to the cross product of vectors which is not associative and can only be used in three or seven dimensions). Therefore, the geometric product is able to generate several graded algebras: (i) in two dimensions we recover the complex numbers as elements of a real algebra, not as elements of a field; (ii) in three dimensions we get a geometric algebra that is far better than the Gibbsian approach mainly due to the geometric role of rotors is reflections and rotations; (iii) in four dimensions we obtain the so-called spacetime algebra which is perfect for Minkowski spacetime within the context of special relativity - see the paper from Hestenes in American Journal of Physics (vol. 71, pp. 691-714, June 2003). Hamilton's quaternions are properly understood. Even as a new gauge theory of gravity on flat spacetime Hestenes' geometric algebra plays a very important role - see the paper from Hestenes in Foundations of Physics (vol. 25, pp. 903-970, June 2005). The clear and insightful approach that geometric algebra can bring to the Dirac equation is also remarkable.

My only problem with this book is due to Chapter 9 on relativistic mechanics. In this chapter Hestenes takes the usual approach that can be found in traditional four-vectors, by representing an event as a paravector, i.e., as a sum of a scalar and a three-dimensional vector (in Euclidean space). This kind of approach doesn't take advantage of geometric algebra (as in his article on spacetime algebra for Am. J. Phys.) because spatial vectors are not directly linked to an observer (and to its proper time) as they are in spacetime algebra where the so-called space-time split clearly leads to an invariant and proper formulation of physics. In Chapter 9, indeed, these paravectors induce a relativistic approach and not a proper approach. Nevertheless, apart from this remark, my overall comment on this book is very positive.

This is a great introduction to the Geometric (Clifford) Algebra. It's fundamentally a physics textbook, however. Those readers whose only desire is to learn the Geometric Algebra might feel some frustration at having to separate out the Geometric Algebra from the physics. Readers that prefer learning by exploring applications and examples will like this book; those that prefer explanations in the abstract will still enjoy many sections, but will have to make it through the more applied sections to get the full story.

Reading the book and working through the problems gives a firm grounding in the use of the Geometric Algebra and teaches classical mechanics besides. I could easily recommend this book as a physics textbook on its merits in that area alone.

While I found this a reasonably well-written text, I judge a book based upon how well it stands on its own - that is, can I read the book, work through the exercises, and acquire a grasp of the topic. While this is a much clearer and more accessible intro to geometric algebra than Hestene's "Clifford Algebra to Geometric Calculus", it is more the type of book which would accompany a class in GA, where the instructor fills-in the gaps, rather than a stand-alone text. The worked exercises are relatively few, and are typically of the nature: start with this, magic, more magic, resulting answer. It lacks sufficient explanation, is not self-contained, but this can be partially overcome with additional resources.