Quaternions are not biquaternions

Subject: Re: The Maxwell equations from quaternions
From: sweetser@alum.mit.edu (Doug B Sweetser)
Date: 1997/04/13
Message-Id: <E8L9Gq.C09@world.std.com>
Newsgroups: sci.physics.research

Pertti Lounesto wrote:

>A full and comprehensive treatment on this topic [using quaternions
to derive the Maxwell equations] is the monograph:
>
>K. Imaeda: Quaternionic Formulation of Classical Electrodynamics and
>the Theory of Functions of a Biquaternion Variable. Department of
>Electronic Science, Okayama University of Science, 1983.

It seems like every time I post about my research on quaternions, I
need to make this point: quaternions are not biquaternions (which
also go by the name "complex-valued quaternions"). There is a
substantial body of work under the latter, the above cited work and
Mark Hopkins' post (dejanews 1997/02/13) included, but not the
former which is area of study.

Let me approach this distinction from an aesthetic point of view. I
treat events as quaternions where

t + x I + y J + z K t, x, y, z are real

These can be added together, subtracted, multiplied, and inverted.
Calculus can be done with this field. But my friend the
mathematician thinks there is something horrible about quaternions:
they don't commute. This flaw can muddy simple proofs and fast
calculations. The abominations are natural for a student of physics
because there are many pairs of variables that don't commute in
quantum mechanics. Even classical rotations in 3D don't commute.
Math should reflect nature (or is it the other way around?)

The I, J and K in a quaternion are 3 orthogonal imaginary vectors.
Each one, when paired with a real number, behaves like a complex
number should. Quaternions structured the dance of these three pairs.
The dance is quite intricate - particularly because they don't
commute - but that is the source of the elegance, at least in my
eyes :-)

Biquaternions not only have the imaginary basis vectors I, J, and K,
but they allow t, x, y, and z to take the complex values of the form
a + b i. But what is the relationship between I, J, K and i? Paired
with a real number, I, J, and K used to be able to do all the work of
complex numbers. Now what happens when i cuts in? And why i
and not j? j might stand for "jealous" ;-) It all has been worked out,
set out in the algebra. The dance card overflows with scalars,
vectors, bivectors, pseudovectors and pseudoscalars who must be
carefully matched up. Biquaternions are bound to be less hostile
about commuting, but that doesn't justify redundancy. I like a close
shave with Occam's razor. There is no need in my work for complex
values within quaternions.

One can accept the Maxwell equations as an article of faith whose
implications have been tested and confirmed in the smallest of detail.
Or one can look to biquaternions for mathematical confirmation of
what is been proven beyond a shadow of a doubt. Or one can
look to quaternions for a similar but distinct support. There is no
overwhelming reason to chose between the second and third choices.
Only if one or the other becomes a to guide through the foundations
of general relativity might there be a way to make a rational choice.
That will have to wait for another post...


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


"Quaternions came from Hamilton after his really good work had
been done; and though beautifully ingenious, have been an unmixed
evil to those who have touched them in any way, including Maxwell."
- Lord Kelvin

It's time to tame the evil, oh Lord K., because the Millennium is fast
approaching.