Reforming the Mathematical Language of Physics (2002) [pdf]
geocalc.clas.asu.edu
geocalc.clas.asu.edu
On page 9 he discusses what he calls the geometric product. The problem is that the geometric product of two vectors is not a vector. This means that the space he is really working is larger than the vector space he started with. This isn't explained to the reader. What is this larger space?
Equation (7) shows that the geometric product of a and b is a.b + a wedge b. From this it's clear that he is working in the exterior algebra of the vector space.
Reading further shows that what he is really doing is giving geometric meanings to the operations in the exterior algebra where V is R^2 and R^3. This is useful and I think it has merit but I also think one should start with the proper setting.
Looking on page 9 going from (11) to (12) requires quite a leap. He says to square (11) but the right hand side of (11) is not a vector and the geometric product of this object hasn't been defined. It was only defined for vectors in V and not for other elements of the exterior algebra.
For that, see his numerous other papers and books about the subject, or the several books by other people in the last 10–15 years.
It’s unfair to label this expository sketch “mathematically unsound” on the basis that it skips a bunch of steps... when there’s no space to be fully rigorous in the context.
> On page 9 he discusses what he calls the geometric product. The problem is that the geometric product of two vectors is not a vector. This means that the space he is really working is larger than the vector space he started with. This isn't explained to the reader. What is this larger space?
Anyway, you are absolutely correct that the geometric product of two vectors isn’t a vector: instead, it’s a scalar plus a bivector (scalars + bivectors form a space isomorphic to the complex numbers, conveniently).
You might notice that the conventional inner product (dot product) of two vectors is also not a vector. Likewise, while the cross product of two vectors is defined to be a vector, it’s not really the same type of vector as the two original vectors and must be treated differently, causing endless confusion, especially for students. Reducing this confusion by clearly embedding geometric products of vectors in a larger space is the whole point.
For more, see:
http://geocalc.clas.asu.edu/html/Evolution.html#References
However, I do expect well founded definitions and operations. To define an operation on two vectors and not point out that it isn't a vector while stating that the geometric algebra is different "from all other associative algebras" is misleading. Associative algebras are closed under the operations they have. So something ought to be said about the peculiarity of talking about associative algebras while not having an operation that is closed.
The only paper I've read from Hestenes is the paper in question and the perspective I have is that of a mathematician. My comment only dealt with this paper and from my perspective things were muddled a bit. He talks about confusion of physics students in graduate school on the topic of vectors. I'm assuming such students will eventually have to deal with tensor products and the exterior algebra when they study general relativity. Hence my belief that one should talk about the larger algebraic structure from the get go rather than glossing over the fact that the geometric product of a two vectors is not a vector and the mystery of squaring a wedge product.
Scalars + bivectors are isomorphic to the complex numbers provided the base field is R and the isomorphism is as vector spaces not as algebras or rings.
[0] http://en.wikipedia.org/wiki/Clifford_algebra#Introduction_a...
http://faculty.luther.edu/~macdonal/laga/ http://faculty.luther.edu/~macdonal/vagc/index.html
This paragraph I think gives context to both the HN thread author and the commenter:
Geometric algebra as practiced today originated in the 1960’s. It is currently under vigorous development. It has found important applications in computer science (in graphics, robotics, and computer vision), engineering, and physics. It is available to game developers for the Xbox and PlayStation video game consoles. This text is an attempt to keep up with these modern developments.
Putting on my must read/study list (I am proficient in linear algebra, so hoping for a gentle push into geometric algebra)
Years ago (2009) I sent an e-mail to David Hestenes asking him if he was planning on writing an introductory textbook on geometric algebra. He said it was in the planning stages.
From what I understood later, it seems he is attempting to write a 100% purely geometric algebra book unlike macdonald who reserved it for the second part of laga.
It's authors also have other articles and teaching resources on their website: http://geometry.mrao.cam.ac.uk/