

Reforming the Mathematical Language of Physics (2002) [pdf] - aethertap
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

======
yequalsx
My brief reading of this suggest it isn't mathematically sound in terms of its
presentation.

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.

~~~
jacobolus
This short lecture is not intended to be a comprehensive introduction to what
Hestenes calls “Geometric Algebra” (some other people call this the Clifford
Algebra of a real vector space, or similar).

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](http://geocalc.clas.asu.edu/html/Evolution.html#References)

[http://geometry.mrao.cam.ac.uk](http://geometry.mrao.cam.ac.uk)

[http://geometricalgebra.net](http://geometricalgebra.net)

[https://en.wikipedia.org/wiki/Geometric_algebra](https://en.wikipedia.org/wiki/Geometric_algebra)

~~~
yequalsx
I'm not a physicist and don't understand their perspective on these matters. I
say it is mathematically unsound not because it isn't rigorous. I'm not
expecting rigor.

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.

~~~
jacobolus
Scalars + bivectors (obviously talking about those generated by geometric
products of vectors in R2) are isomorphic to the complex numbers, as a ring.

~~~
yequalsx
I didn't read enough of the paper to see how bivector multiplication is
defined. I'm assuming everything is in the exterior algebra and thus made my
comment with that assumption.

------
adam930
There are two very good introductory textbooks on the subject by the same
author:

[http://faculty.luther.edu/~macdonal/laga/](http://faculty.luther.edu/~macdonal/laga/)
[http://faculty.luther.edu/~macdonal/vagc/index.html](http://faculty.luther.edu/~macdonal/vagc/index.html)

~~~
guru_meditation
The LAGA link is gold. Wish I could give more upvotes.

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)

~~~
adam930
I'm sure you will like it. Macdonald is an excellent teacher.

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.

------
jamessb
An article introducing Geometric Algebra ( _A Unified Mathematical Language
for Physics and Engineering_ ) was posted to HN
([https://news.ycombinator.com/item?id=8192054](https://news.ycombinator.com/item?id=8192054))
and may be of interest.

It's authors also have other articles and teaching resources on their website:
[http://geometry.mrao.cam.ac.uk/](http://geometry.mrao.cam.ac.uk/)

