Real quaternions work

Subject: Re: Quaternions
From: sweetser@alum.mit.edu (Doug B Sweetser)
Date: 1997/05/01
Message-Id: <E9I890.MqJ@world.std.com>
Newsgroups: sci.physics.electromag,sci.physics.relativity
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Pertti Lounesto wrote:

>"Srdjan Budisin" <srdjan@EUnet.yu> wrote:
>-Maybe, Maxwell equations can be formulated in therms of quatrnions?
>-Maybe, Special Theory of Relativity can be formulated using Quaternions.
>-How about Quantum Mechanics?
>
>[1] No. No. Yes, if you mean real quaternions.
>[2] Yes, Yes, Yes, for complex quaternions (called also biquaternions).

I completely agree with statement [2], but have an opposite response to
[1], saying yes, yes, yes (but I kindly disagree with the approach)!

I need to change the first question however to "Can the Maxwell equations
in the Lorentz gauge be formulated in terms of quaternions?" Maxwell's
equations are a set of four first order differential equations. These
first order equations cannot be expressed as real quaternions. The Maxwell
equations in the Lorentz gauge can be expressed as four second order
differential equations in terms of the four potential phi and A. (Pertti
you should have known I would make such a claim! Do you believe my
formulation "just ain't so"? If so, please e-mail me).

My clearest success to date has been to solve problems in special
relativity using quaternions. If the question is "Can real quaternions be
used to represent the Lorentz group and then solve problems in special
relativity", Lounesto gave the correct answer, it can't be done with real
quaternions. I developed a different way, that essential makes special
relativity work locally (like it was pretending to be general relativity,
really quite funny :-) If you give me _any_ standard problem in special
relativity, then I can solve it using real quaternions, including time
dilation, length contraction, relativity of simultaneity, the pole in
the barn, additions of velocities, the Doppler effect, the twin paradox,
relativistic collisions, threshold problems and the Compton effect. Each
one of these problems is solved explicity with real quaternions at my web site in Mathematica. (Again, Pertti, you know I have made this claim. You may prefer complex quaternions, nothing wrong with that at all. Yet you should acknowledge that a new road has been opened, whether you chose to work on it or not).

Adler does not use the analytic properties of quaternions (that's what he
said at the end of his intro chapter of his book on quaternionic quantum
mechanics). To me, the great array of analytic tools that qpply to
quaternions is in fact the best reason to try and learn how to use
quaternions! Adler tosses away the power tools so he can work on the
shoulders of giants. Adler's way works, but I think there may be a
different way to go.

>A counterexample (due to Benjamin Tilly): Consider a 3- dimensional
>real algebra with basis {1,i,j} such that 1 is the unity and
>i^2=j^2=-1 but ij=ji=0. The algebra is commutative and non-associtive
>but admits inverses, although the inverses of xi+yj are not unique.

I didn't think this is a counterexample, because implicit in my statement
was that the algebra should be associative (it does belong with the
octonions which I referred to in parentheses).


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