Customer Reviews

Electrodynamics: A Modern Geometric Approach (Progress in Mathematical Physics)

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Maxwell's theory of Electrodynamics is considerably more complicated than Newton's Mechanics. The latter deals with concrete objects (particles, rigid bodies, etc.) while the former deals with intangible "fields" distributed in space, a concept that took many years to evolve and gain acceptance. It is therefore not surprising that Electrodynamics has motivated a variety of important scientific developments, designed either to simplify it conceptually or to make it consistent with Mechanics. In physics, fundamental contradictions between Electrodynamics and Mechanics spurred Einstein to develop Special Relativity. In mathematics, the most well-known development is vector analysis, introduced by Gibbs to simplify Maxwell's equations. Unfortunately, the even deeper simplifications introduced by Hamilton (based on quaternions) and Clifford (based on Clifford algebra) have not gained wide acceptance because they are somewhat more technically demanding. Thus almost all physics and engineering textbooks on electrodynamics use vector analysis, and very few students and researchers are even aware of the tremendous power offered by quaternionic and Clifford analysis.

In this regard, the book by Baylis is a real blessing. By choosing the simplest geometric algebra (complex quaternions, known in quantum mechanics as the Pauli algebra), Baylis manages to use the full power of Clifford algebra without giving up the familiar notation of vector analysis (gradient, curl, divergence). The only new ingredient is the associative product of two vectors, which unifies their inner product and their cross product and leads to tremendous new possibilities. Students and workers in the field will love the resulting beauty and simplicity, and especially the continuity of notation and concepts with the mainstream literature. Although the Pauli algebra is usually associated with nonrelativistic quantum mechanics, Baylis amply demonstrates that it is equally able to handle relativistic aspects (such as the unified electromagnetic field and Lorentz transformations) when the concept of three-dimensional vectors is replaced by four-dimensional (space-time) "paravectors." Well done!

This book on electrodynamics presents an awkward geometric approach.

Indeed, electromagnetic theory is inherently relativistic -- although it is usually introduced in a nonrelativistic formalism that uses the elementary Gibbsian approach to vectors.

The author intends to avoid the tensor notation -- because it tends to disguise the corresponding geometrical significance. But he also intends to avoid spaces of nondefinite metric by using paravectors in a Clifford algebra with dimension n=3. In fact, in this process, he needs to introduce a complex structure where trivectors become imaginary scalars, bivectors are written as imaginary vectors, and every element of the algebra is a paravector -- the sum of a scalar and a vector, both of which may be complex.

One should recognize that the author has grasped the problem; his solution is, nevertheless, foggy. All in all, one cannot say -- quoting Arnold Sommerfeld -- that "the true mathematical structure of these entities will appear only now, as in a mountain landscape when the fog lifts".

Much ado about nothing: I recommend, instead, other geometric algebras that don't shy away from using indefinite metrics -- but avoid the clumsiness of complex structures.

Indeed, the real algebra of complex numbers is already an example of a Clifford algebra: "the geometric view of complex numbers is connected with the structure of C as a real algebra, and not so much as a field" (Pertti Lounesto).

David Hestenes shows how this can be done.

I quite liked the geometric algebra introduction in this book. It was one of the simplest and clearest I've read. That is, until the paravector notation is introduced. I felt overwhelmed by the variety of conjugation operations suddenly tossed out (clifford conjugation, hermitian conjugation, grade automorphism).

Having learned what I know of Geometric Algebra based E&M with the STA formalism, this approach seems more difficult. The complexity (literally) of having to deal with conjugation and scalar and vector selection seems much less intuitive.

Some of the tail end chapters almost seemed tossed in randomly. It was hard to see what value GA provided in the spherical harmonic treatment.

I'd hoped that this text would provide an easy path to learn E&M in a GA context despite a requirement for having to switch from STA notation to APS, but didn't feel the presentation provided a logical sequential flow of ideas that would facilitate this study.

In the lack of a good books on Geometric algebra and calculus (in David Hestenes' style) for physicists, the book on electrodynamics by W. Baylis looks like a Necessary evil.

The publishers did thier best to make the book at least look good. Unfortunately, the author could do better.

The annoying 'americanisms' and often inappropriate 'pseudo-scientific jargon' did not help the book.

Poor understanding of physics (not saying the geometrica algebra and analysis) is very surprising and unfortunate for the university professor.

However, with some work on the part of the reader, the book could be somewhat useful. A lot of examples and exercises is also helpful.

Highly recommend to compare Mr. Baylis' style with clear and thoughtful style of David Hestenes (see his "New foundations for physics". Unfortunately the price for that is VERY high ~ $250 )

Again, the book could be much better, but in the said vacuum, is a Necessary evil. Buy it ! ( if you have some spare money and not afraid of the strangers with professor titles)

The problem is twofold: to understand and to develop a framework to use E&M. I do not feel that this book adequately addresses either problem.

Instead, this book spends a lot of time developing a Clifford Algebra approach and the attendant notation. Very little physical feel; or, for that matter, very little calculation at all. This looks more like a pure math text than a physics book. There are few drawings or other geometric things that relate to the real world. This subject is hard enough without loading much more notation on it.