Fun playing with quaternions

Subject: Re: Quaternions
From: sweetser@alum.mit.edu (Doug B Sweetser)
Date: 1997/05/01
Message-Id: <E9HAu5.AsG@world.std.com>
Newsgroups: sci.physics.electromag,sci.physics.particle,sci.physics.relativity

Hello:

I've been having quite a bit of fun over the last year playing with
quaternions. Srdjan Budisin's simple guess about a connection between the
mathematics of quaternions and 4D spacetime was basically the one I
started out with and have been posting about for the last few months.
There is a web site (http://world.std.com/~sweetser) which shows how to
put quaternions to work to do Newton's second law, the classical simple
harmonic oscillator, tests for conservative forces, special relativity
(with over 50 problems solved explicitly), the Maxwell equations, some
beginnings for quantum mechanics and a connection to general relativity.
Sounds like I designed this place just for Srdjan!

A few points of clarification. There are infinitely many, finite
dimensional algebras out there, where an algebra is a generalization of
arithmetic. There are only three finite dimensional division algebras,
the reals, the complex numbers and the quaternions (if you drop
associativity - a BIG deal - you can add octonions to the list). A
division algebra means that an inverse will always exist. Some might say
"An inverse is nice, but I could probably live without it." Folks who
work on the foundations of quantum mechanics disagree, because propagators
for scalar fields need to have an inverse.

Adler has done work on quantum mechanics and quaternions, but it is _not_
the road I am travelling. Much work has been done with complex-valued
Hilbert spaces. He is working on quaternion-valued Hilbert spaces. In
this way, he is walking down a familiar road with a new friend. My
approach involves using the mathematics underlying quaternions to generate
the basic axioms of quantum mechanics (i.e. a Hilbert space with objects
that don't commute). Made good progress, but I need to be able to solve
actual problems, so at this point this work is preliminary but exciting.

General relativity has always been a 4D phenomenon. The difficulty of the
field is that the equations are not linear. The only progress I've made
here is to define precisely what it means to measure a distance in a
curved space of quaternions. This is known in general relativity as the
affine parameter. Again, I have not solved any problems yet, so this
work is preliminary.

I have avoided the philosophical analysis, there is too much math to
do :-) One thing I will say though. I have been really impressed with
the beauty of the formalism. I can see connections between different
areas of physics much more easily. For example, Newtonian physics
happens in the limit of c going to infinity. That makes Newtonian
physics seem totally isolated from the rest of physics. With quaternions,
the second law (in all its various forms: inertial, central force in a
plane, noninertial rotating from) is generated by a pair of quaternion
operators. These operators have d/dt's, but no spatial derivatives. The
Maxwell equations are also generated by a pair of quaternion operators.
This time though, there are spatial derivatives as well as time
derivates. The same basic math tool - a pair of quaternion operators -
chosen with care, can generate both Newton's second law and the four
Maxwell equations.

Doing physics with quaternions is not easy. In my experience, it is just
as tough as the usual road. However, the connections between fields are
much easier to spot. And I believe the number of underlying assumptions
is considerably less, because all these strange and wonderful things that
get discussed in physics are really just mathical analysis with a 4D
division algebra.


Doug Sweetser
http://world.std.com/~sweetser

Building Hamilton's dream, one equation at a time.