Geometric algebra versus quaternions

Newsgroups: sci.physics.research
Subject: Re: Q: Geometric Algebra instead of Vector Algebra??
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References: <51f0sl$512@agate.berkeley.edu>
Sender: Doug Sweetser<sweetser@alum.mit.edu>
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I have an (admitted) bias toward studying 4 dimensional fields
(translation: real-valued quaternions). The properties of fields are
absolutely critical for the foundations of calculus. According to the
theorem of Frobenius, quaternions are the largest finite dimensional
field. There are infinite dimensional fields which are the focus of a
large body of work.

Physicist always acknowledge the importance of fields. When describing a
vector space or a particular algebra, the clause, "over the field of real
(or complex) numbers" is added. That way calculus can be done! It is
quite the exception for one to read "over the field of quaternions."
There is a good reason for this: quaternions don't commute. While this
may be "the work of the devil" (Lord Kelvin) from a mathematical
viewpoint, it is quite common in quantum mechanics.

One technical issue I have with Hestenes approach is that I'm not sure
his Geometric Algebra is always a field. I know for example that
quaternions with complex values are no longer a field. These
biquaternions have been used to represent the Lorentz group. To loose a
founding property of calculus is too high a price for me to pay!

If quaternions are a powerful tool, then they should be useful for
solving problems. To test that hypothesis, I am taking a undergraduate
class in special relativity at MIT and doing all the problem sets using
quaternions instead of the Lorentz transformation! I have made it through
problems dealing with time dilation, length contraction and asynchronous
moving clocks, now on to velocity addition with quaternions. This has
been an immense amount of fun, a real challenge to develop the tools in
Mathematica and then apply them (and my problem set look sooo neat typed!).

Is this approach faster, stronger, better? I solve the problem sets the
standard way using kinematic effects of special relativity and the
Lorentz transformation, and in Mathematica using tools for quaternions. The
most important thing to note is that so far, they both get from the question
to the same answer. It is quite clear though that they travel along
different roads algebraically. I have yet to exploit those difference
because I've got to do more homework now!

Doug
sweetser@alum.mit.edu